;;;; This file contains the definitions of most number functions.

;;;; This software is part of the SBCL system. See the README file for
;;;; more information.
;;;;
;;;; This software is derived from the CMU CL system, which was
;;;; written at Carnegie Mellon University and released into the
;;;; public domain. The software is in the public domain and is
;;;; provided with absolutely no warranty. See the COPYING and CREDITS
;;;; files for more information.

(in-package "SB-KERNEL")

;;;; canonicalization utilities

;;; If IMAGPART is 0, return REALPART, otherwise make a complex. This is
;;; used when we know that REALPART and IMAGPART are the same type, but
;;; rational canonicalization might still need to be done.
(declaim (inline canonical-complex))
(defun canonical-complex (realpart imagpart)
  (if (eql imagpart 0)
      realpart
      (cond #+long-float
            ((and (typep realpart 'long-float)
                  (typep imagpart 'long-float))
             (truly-the (complex long-float) (complex realpart imagpart)))
            ((and (typep realpart 'double-float)
                  (typep imagpart 'double-float))
             (truly-the (complex double-float) (complex realpart imagpart)))
            ((and (typep realpart 'single-float)
                  (typep imagpart 'single-float))
             (truly-the (complex single-float) (complex realpart imagpart)))
            (t
             (%make-complex realpart imagpart)))))

;;; Given a numerator and denominator with the GCD already divided
;;; out, make a canonical rational. We make the denominator positive,
;;; and check whether it is 1.
(declaim (inline build-ratio))
(defun build-ratio (num den)
  (multiple-value-bind (num den)
      (if (minusp den)
          (values (- num) (- den))
          (values num den))
    (cond
      ((eql den 0)
       (error 'division-by-zero
              :operands (list num den)
              :operation '/))
      ((eql den 1) num)
      (t (%make-ratio num den)))))

;;; Truncate X and Y, but bum the case where Y is 1.
(declaim (inline maybe-truncate))
(defun maybe-truncate (x y)
  (if (eql y 1)
      x
      (truncate x y)))

;;;; COMPLEXes

(defun complex (realpart &optional (imagpart 0))
  "Return a complex number with the specified real and imaginary components."
  (declare (explicit-check))
  (flet ((%%make-complex (realpart imagpart)
           (cond #+long-float
                 ((and (typep realpart 'long-float)
                       (typep imagpart 'long-float))
                  (truly-the (complex long-float)
                             (complex realpart imagpart)))
                 ((and (typep realpart 'double-float)
                       (typep imagpart 'double-float))
                  (truly-the (complex double-float)
                             (complex realpart imagpart)))
                 ((and (typep realpart 'single-float)
                       (typep imagpart 'single-float))
                  (truly-the (complex single-float)
                             (complex realpart imagpart)))
                 (t
                  (%make-complex realpart imagpart)))))
  (number-dispatch ((realpart real) (imagpart real))
    ((rational rational)
     (canonical-complex realpart imagpart))
    (float-contagion %%make-complex realpart imagpart (rational)))))

(defun realpart (number)
  "Extract the real part of a number."
  (etypecase number
    #+long-float
    ((complex long-float)
     (truly-the long-float (realpart number)))
    ((complex double-float)
     (truly-the double-float (realpart number)))
    ((complex single-float)
     (truly-the single-float (realpart number)))
    ((complex rational)
     (%realpart number))
    (number
     number)))

(defun imagpart (number)
  "Extract the imaginary part of a number."
  (etypecase number
    #+long-float
    ((complex long-float)
     (truly-the long-float (imagpart number)))
    ((complex double-float)
     (truly-the double-float (imagpart number)))
    ((complex single-float)
     (truly-the single-float (imagpart number)))
    ((complex rational)
     (%imagpart number))
    (single-float 0f0)
    (double-float 0d0)
    (t 0)))

(defun conjugate (number)
  "Return the complex conjugate of NUMBER. For non-complex numbers, this is
  an identity."
  (declare (type number number) (explicit-check))
  (if (complexp number)
      (complex (realpart number) (- (imagpart number)))
      number))

(defun signum (number)
  "If NUMBER is zero, return NUMBER, else return (/ NUMBER (ABS NUMBER))."
  (declare (explicit-check))
  (if (zerop number)
      number
      (number-dispatch ((number number))
        (((foreach fixnum rational single-float double-float))
         (if (plusp number)
             (coerce 1 '(dispatch-type number))
             (coerce -1 '(dispatch-type number))))
        ((complex)
         (/ number (abs number))))))

;;;; ratios

(defun numerator (number)
  "Return the numerator of NUMBER, which must be rational."
  (numerator number))

(defun denominator (number)
  "Return the denominator of NUMBER, which must be rational."
  (denominator number))

;;;; arithmetic operations
;;;;
;;;; IMPORTANT NOTE: Accessing &REST arguments with NTH is actually extremely
;;;; efficient in SBCL, as is taking their LENGTH -- so this code is very
;;;; clever instead of being charmingly naive. Please check that "obvious"
;;;; improvements don't actually ruin performance.
;;;;
;;;; (Granted that the difference between very clever and charmingly naivve
;;;; can sometimes be sliced exceedingly thing...)

(macrolet ((define-arith (op init doc)
             `(defun ,op (&rest numbers)
                (declare (explicit-check))
                ,doc
                (if numbers
                    (let ((result (the number (fast-&rest-nth 0 numbers))))
                      (do-rest-arg ((n) numbers 1 result)
                        (setq result (,op result n))))
                    ,init))))
  (define-arith + 0
    "Return the sum of its arguments. With no args, returns 0.")
  (define-arith * 1
    "Return the product of its arguments. With no args, returns 1."))

(defun - (number &rest more-numbers)
  "Subtract the second and all subsequent arguments from the first;
  or with one argument, negate the first argument."
  (declare (explicit-check))
  (if more-numbers
      (let ((result number))
        (do-rest-arg ((n) more-numbers 0 result)
          (setf result (- result n))))
      (- number)))

(defun / (number &rest more-numbers)
  "Divide the first argument by each of the following arguments, in turn.
  With one argument, return reciprocal."
  (declare (explicit-check))
  (if more-numbers
      (let ((result number))
        (do-rest-arg ((n) more-numbers 0 result)
          (setf result (/ result n))))
      (/ number)))

(defun 1+ (number)
  "Return NUMBER + 1."
  (declare (explicit-check))
  (1+ number))

(defun 1- (number)
  "Return NUMBER - 1."
  (declare (explicit-check))
  (1- number))

(defmacro two-arg-+/- (name op big-op)
  `(defun ,name (x y)
     (declare (explicit-check))
     (number-dispatch ((x number) (y number))
       (bignum-cross-fixnum ,op ,big-op)
       (float-contagion ,op x y)

       ((complex complex)
        (canonical-complex (,op (realpart x) (realpart y))
                           (,op (imagpart x) (imagpart y))))
       (((foreach bignum fixnum ratio single-float double-float
                  #+long-float long-float) complex)
        (complex (,op x (realpart y)) (,op 0 (imagpart y))))
       ((complex (or rational float))
        (complex (,op (realpart x) y) (,op (imagpart x) 0)))

       (((foreach fixnum bignum) ratio)
        (let* ((dy (denominator y))
               (n (,op (* x dy) (numerator y))))
          (%make-ratio n dy)))
       ((ratio integer)
        (let* ((dx (denominator x))
               (n (,op (numerator x) (* y dx))))
          (%make-ratio n dx)))
       ((ratio ratio)
        (let* ((nx (numerator x))
               (dx (denominator x))
               (ny (numerator y))
               (dy (denominator y))
               (g1 (gcd dx dy)))
          (if (eql g1 1)
              (%make-ratio (,op (* nx dy) (* dx ny)) (* dx dy))
              (let* ((t2 (truncate dx g1))
                     (t1 (,op (* nx (truncate dy g1)) (* t2 ny)))
                     (g2 (gcd t1 g1)))
                (cond ((eql t1 0) 0)
                      ((eql g2 1)
                       (%make-ratio t1 (* t2 dy)))
                      (t (let* ((nn (truncate t1 g2))
                                (t3 (truncate dy g2))
                                (nd (if (eql t2 1) t3 (* t2 t3))))
                           (if (eql nd 1) nn (%make-ratio nn nd))))))))))))

(two-arg-+/- two-arg-+ + add-bignums)
(two-arg-+/- two-arg-- - subtract-bignum)

(defun two-arg-* (x y)
  (declare (explicit-check))
  (flet ((integer*ratio (x y)
           (if (eql x 0) 0
               (let* ((ny (numerator y))
                      (dy (denominator y))
                      (gcd (gcd x dy)))
                 (if (eql gcd 1)
                     (%make-ratio (* x ny) dy)
                     (let ((nn (* (truncate x gcd) ny))
                           (nd (truncate dy gcd)))
                       (if (eql nd 1)
                           nn
                           (%make-ratio nn nd)))))))
         (complex*real (x y)
           (canonical-complex (* (realpart x) y) (* (imagpart x) y))))
    (number-dispatch ((x number) (y number))
      (float-contagion * x y)
      ((fixnum fixnum) (* x y))
      ((bignum fixnum) (multiply-bignum-and-fixnum x y))
      ((fixnum bignum) (multiply-bignum-and-fixnum y x))
      ((bignum bignum) (multiply-bignums x y))

      ((complex complex)
       (let* ((rx (realpart x))
              (ix (imagpart x))
              (ry (realpart y))
              (iy (imagpart y)))
         (canonical-complex (- (* rx ry) (* ix iy)) (+ (* rx iy) (* ix ry)))))
      (((foreach bignum fixnum ratio single-float double-float
                 #+long-float long-float)
        complex)
       (complex*real y x))
      ((complex (or rational float))
       (complex*real x y))

      (((foreach bignum fixnum) ratio) (integer*ratio x y))
      ((ratio integer) (integer*ratio y x))
      ((ratio ratio)
       (let* ((nx (numerator x))
              (dx (denominator x))
              (ny (numerator y))
              (dy (denominator y))
              (g1 (gcd nx dy))
              (g2 (gcd dx ny)))
         (build-ratio (* (maybe-truncate nx g1)
                         (maybe-truncate ny g2))
                      (* (maybe-truncate dx g2)
                         (maybe-truncate dy g1))))))))

(defun *-by-fixnum-to-fixnum (n f)
  (declare (explicit-check n))
  (flet ((err ()
           (the fixnum (two-arg-* n f))
           (sb-impl::unreachable)))
    (typecase n
      (ratio
       (if (= f 0)
           0
           (let ((num (numerator n))
                 (den (denominator n)))
             (if (and (typep num 'fixnum)
                      (typep den 'fixnum))
                 (multiple-value-bind (q r) (truncate f den)
                   (if (zerop r)
                       (the fixnum (* q num))
                       (err)))
                 (err)))))
      (bignum
       (if (= f 0)
           0
           (if (and (typep n 'sb-vm:signed-word)
                    (= f -1)
                    (= n (- most-negative-fixnum)))
               most-negative-fixnum
               (err))))
      ((complex rational)
       (if (= f 0)
           0
           (err)))
      (t
       (err)))))

(defun %negate (n)
  (declare (explicit-check))
  (number-dispatch ((n number))
    (((foreach fixnum single-float double-float #+long-float long-float))
     (%negate n))
    ((bignum)
     (negate-bignum n))
    ((ratio)
     (%make-ratio (- (numerator n)) (denominator n)))
    ((complex)
     (complex (- (realpart n)) (- (imagpart n))))))

(defun %multiply-high (x y)
  (declare (type word x y))
  (%multiply-high x y))

(defmacro compare-two-ratios (op numerator-x denominator-x numerator-y denominator-y)
  `(dispatch-two-ratios
       (numerator-x ,numerator-x)
       (denominator-x ,denominator-x)
       (numerator-y ,numerator-y)
       (denominator-y ,denominator-y)
       (or
        ,@(case op
            ((<= >=)
             `((and (eql numerator-x numerator-y)
                    (eql denominator-x denominator-y)))))
        (,(case op
            (<= '<)
            (>= '>)
            (t op))
         (* numerator-x denominator-y)
         (* numerator-y denominator-x)))
       ,@(or
          (sb-c::when-vop-existsp (:named sb-vm::signed-multiply-low-high)
            `((multiple-value-bind (low1 high1) (%primitive sb-vm::signed-multiply-low-high numerator-x denominator-y)
                (multiple-value-bind (low2 high2) (%primitive sb-vm::signed-multiply-low-high numerator-y denominator-x)
                  (cond ((= high1 high2)
                         (,op low1
                              low2))
                        ((,(case op
                             (<= '<)
                             (>= '>)
                             (t op))
                          high1 high2)
                         t))))))
          (sb-c::when-vop-existsp (:translate %signed-multiply-high)
            `((let ((high1 (%signed-multiply-high numerator-x denominator-y))
                    (high2 (%signed-multiply-high numerator-y denominator-x)))
                (cond ((= high1 high2)
                       (,op (logand most-positive-word (* numerator-x denominator-y))
                            (logand most-positive-word (* numerator-y denominator-x))))
                      ((,(case op
                           (<= '<)
                           (>= '>)
                           (t op))
                        high1 high2)
                       t))))))))


;;;; TRUNCATE and friends

(declaim (maybe-inline truncate floor ceiling round fround))

(defun truncate (number &optional (divisor 1))
  "Return number/divisor as an integer, rounded toward 0.
The second returned value is the remainder."
  (declare (explicit-check)
           (maybe-inline truncate))
  (macrolet ((truncate-float (rtype)
               `(truncate (coerce number ',rtype)
                          (coerce divisor ',rtype)))
             (single-digit-bignum-p (x)
               #+(or x86-64 x86 ppc64 arm64)
               `(or (typep ,x 'word)
                    (typep ,x 'sb-vm:signed-word))
               ;; Other backends don't have native double-word/word division,
               ;; and their bigfloor implementation doesn't handle
               ;; full-width divisors.
               #-(or x86-64 x86 ppc64 arm64)
               `(or (typep ,x '(unsigned-byte ,(1- sb-vm:n-word-bits)))
                    (typep ,x '(integer ,(- 1 (expt 2 (- sb-vm:n-word-bits 1)))
                                ,(1- (expt 2 (- sb-vm:n-word-bits 1))))))))
    (number-dispatch ((number real) (divisor real))
      ((fixnum fixnum) (truncate number divisor))
      ((fixnum bignum)
       (if (single-digit-bignum-p divisor)
           (bignum-truncate-single-digit (make-small-bignum number) divisor)
           (bignum-truncate (make-small-bignum number) divisor)))
      ((ratio (foreach (eql 1) fixnum bignum))
       (dispatch-ratio (number numerator denominator)
         (multiple-value-bind (q rem) (truncate numerator
                                                (* denominator divisor))
           (values q (%make-ratio rem denominator)))))
      (((foreach fixnum bignum ratio) ratio)
       (dispatch-two-ratios
           (n-num (numerator number))
           (n-den (denominator number))
           (d-num (numerator divisor))
           (d-den (denominator divisor))
           (let ((q-num (* n-num d-den))
                 (q-den (* n-den d-num)))
             (multiple-value-bind (q rem) (truncate q-num q-den)
               (values q (/ rem (* n-den d-den)))))))
      ((bignum fixnum)
       (bignum-truncate-single-digit number divisor))
      ((bignum bignum)
       (if (single-digit-bignum-p divisor)
           (bignum-truncate-single-digit number divisor)
           (bignum-truncate number divisor)))
      (((foreach single-float double-float #+long-float long-float)
        (foreach fixnum rational single-float))
       (if (eql divisor 1)
           (let ((res (unary-truncate number)))
             (values res (- number (coerce res '(dispatch-type number)))))
           (truncate-float (dispatch-type number))))
      #+long-float
      ((long-float (or single-float double-float long-float))
       (truncate-float long-float))
      #+long-float
      (((foreach double-float single-float) long-float)
       (truncate-float long-float))
      ((double-float double-float)
       (truncate-float double-float))
      ((single-float double-float)
       (truncate-float double-float))
      (((foreach fixnum bignum ratio)
        (foreach single-float double-float #+long-float long-float))
       (truncate-float (dispatch-type divisor))))))

(defun floor (number &optional (divisor 1))
  "Return number/divisor as an integer, rounded toward negative infinity.
The second returned value is the remainder."
  (declare (explicit-check)
           (maybe-inline floor))
  (macrolet ((truncate-float (rtype)
               `(floor (coerce number ',rtype) (coerce divisor ',rtype)))
             (single-digit-bignum-p (x)
               #+(or x86-64 x86 ppc64)
               `(or (typep ,x 'word)
                    (typep ,x 'sb-vm:signed-word))
               ;; Other backends don't have native double-word/word division,
               ;; and their bigfloor implementation doesn't handle
               ;; full-width divisors.
               #-(or x86-64 x86 ppc64)
               `(or (typep ,x '(unsigned-byte ,(1- sb-vm:n-word-bits)))
                    (typep ,x '(integer ,(- 1 (expt 2 (- sb-vm:n-word-bits 1)))
                                ,(1- (expt 2 (- sb-vm:n-word-bits 1)))))))
             (fixup (form)
               `(multiple-value-bind (tru rem) ,form
                  (if (if (minusp divisor)
                          (> rem 0)
                          (< rem 0))
                      (values (1- tru) (+ rem divisor))
                      (values tru rem)))))
    (number-dispatch ((number real) (divisor real))
      ((fixnum fixnum) (floor number divisor))
      ((fixnum bignum)
       (fixup
        (if (single-digit-bignum-p divisor)
            (bignum-truncate-single-digit (make-small-bignum number) divisor)
            (bignum-truncate (make-small-bignum number) divisor))))
      ((ratio (foreach (eql 1) fixnum bignum))
       (dispatch-ratio (number numerator denominator)
         (let* ((q-num numerator)
                (q-den (* denominator divisor))
                (rem-den denominator))
           (multiple-value-bind (q rem) (truncate q-num q-den)
             (if (if (minusp divisor)
                     (> rem 0)
                     (< rem 0))
                 (values (1- q) (%make-ratio (+ rem q-den) rem-den))
                 (values q (%make-ratio rem rem-den)))))))
      (((foreach fixnum bignum ratio) ratio)
       (dispatch-two-ratios
           (n-num (numerator number))
           (n-den (denominator number))
           (d-num (numerator divisor))
           (d-den (denominator divisor))
           (let ((q-num (* n-num d-den))
                 (q-den (* n-den d-num))
                 (rem-den (* n-den d-den)))
             (multiple-value-bind (q rem) (truncate q-num q-den)
               (if (if (minusp d-num)
                       (> rem 0)
                       (< rem 0))
                   (values (1- q) (/ (+ rem q-den) rem-den))
                   (values q (/ rem rem-den)))))))
      ((bignum fixnum)
       (fixup (bignum-truncate-single-digit number divisor)))
      ((bignum bignum)
       (fixup (if (single-digit-bignum-p divisor)
                  (bignum-truncate-single-digit number divisor)
                  (bignum-truncate number divisor))))
      (((foreach single-float double-float #+long-float long-float)
        (foreach fixnum rational single-float))
       (truncate-float (dispatch-type number)))
      #+long-float
      ((long-float (or single-float double-float long-float))
       (truncate-float long-float))
      #+long-float
      (((foreach double-float single-float) long-float)
       (truncate-float long-float))
      ((double-float double-float)
       (truncate-float double-float))
      ((single-float double-float)
       (truncate-float double-float))
      (((foreach fixnum bignum ratio)
        (foreach single-float double-float #+long-float long-float))
       (truncate-float (dispatch-type divisor))))))

(defun ceiling (number &optional (divisor 1))
  "Return number/divisor as an integer, rounded toward positive infinity.
The second returned value is the remainder."
  (declare (explicit-check)
           (maybe-inline ceiling))
  (macrolet ((truncate-float (rtype)
               `(ceiling (coerce number ',rtype) (coerce divisor ',rtype)))
             (single-digit-bignum-p (x)
               #+(or x86-64 x86 ppc64)
               `(or (typep ,x 'word)
                    (typep ,x 'sb-vm:signed-word))
               ;; Other backends don't have native double-word/word division,
               ;; and their bigfloor implementation doesn't handle
               ;; full-width divisors.
               #-(or x86-64 x86 ppc64)
               `(or (typep ,x '(unsigned-byte ,(1- sb-vm:n-word-bits)))
                    (typep ,x '(integer ,(- 1 (expt 2 (- sb-vm:n-word-bits 1)))
                                ,(1- (expt 2 (- sb-vm:n-word-bits 1)))))))
             (fixup (form)
               `(multiple-value-bind (tru rem) ,form
                  (if (if (minusp divisor)
                          (< rem 0)
                          (> rem 0))
                      (values (+ tru 1) (- rem divisor))
                      (values tru rem)))))
    (number-dispatch ((number real) (divisor real))
      ((fixnum fixnum) (ceiling number divisor))
      ((fixnum bignum)
       (fixup
        (if (single-digit-bignum-p divisor)
            (bignum-truncate-single-digit (make-small-bignum number) divisor)
            (bignum-truncate (make-small-bignum number) divisor))))
      ((ratio (foreach (eql 1) fixnum bignum))
       (dispatch-ratio (number numerator denominator)
         (let* ((q-num numerator)
                (q-den (* denominator divisor))
                (rem-den denominator))
           (multiple-value-bind (q rem) (truncate q-num q-den)
             (if (if (minusp divisor)
                     (< rem 0)
                     (> rem 0))
                 (values (1+ q) (%make-ratio (- rem q-den) rem-den))
                 (values q (%make-ratio rem rem-den)))))))
      (((foreach fixnum bignum ratio) ratio)
       (dispatch-two-ratios
           (n-num (numerator number))
           (n-den (denominator number))
           (d-num (numerator divisor))
           (d-den (denominator divisor))
           (let ((q-num (* n-num d-den))
                 (q-den (* n-den d-num))
                 (rem-den (* n-den d-den)))
             (multiple-value-bind (q rem) (truncate q-num q-den)
               (if (if (minusp d-num)
                       (< rem 0)
                       (> rem 0))
                   (values (1+ q) (/ (- rem q-den) rem-den))
                   (values q (/ rem rem-den)))))))
      ((bignum fixnum)
       (fixup (bignum-truncate-single-digit number divisor)))
      ((bignum bignum)
       (fixup
        (if (single-digit-bignum-p divisor)
            (bignum-truncate-single-digit number divisor)
            (bignum-truncate number divisor))))
      (((foreach single-float double-float #+long-float long-float)
        (foreach fixnum rational single-float))
       (truncate-float (dispatch-type number)))
      #+long-float
      ((long-float (or single-float double-float long-float))
       (truncate-float long-float))
      #+long-float
      (((foreach double-float single-float) long-float)
       (truncate-float long-float))
      ((double-float double-float)
       (truncate-float double-float))
      ((single-float double-float)
       (truncate-float double-float))
      (((foreach fixnum bignum ratio)
        (foreach single-float double-float #+long-float long-float))
       (truncate-float (dispatch-type divisor))))))

(defun truncate1 (number divisor)
  (declare (explicit-check)
           (inline truncate))
  (values (truncate number divisor)))

(defun floor1 (number divisor)
  (declare (explicit-check)
           (inline floor))
  (values (floor number divisor)))

(defun ceiling1 (number divisor)
  (declare (explicit-check)
           (inline ceiling))
  (values (ceiling number divisor)))

(defun rem (number divisor)
  "Return second result of TRUNCATE."
  (declare (explicit-check))
  (rem number divisor))

(defun mod (number divisor)
  "Return second result of FLOOR."
  (declare (explicit-check))
  (mod number divisor))

(defun round (number &optional (divisor 1))
  "Rounds number (or number/divisor) to nearest integer.
  The second returned value is the remainder."
  (declare (explicit-check)
           (maybe-inline round))
  (macrolet ((round-float (rtype)
               `(round (coerce number ',rtype) (coerce divisor ',rtype))))
    (number-dispatch ((number real) (divisor real))
      (((foreach single-float double-float)
        (foreach single-float fixnum rational))
       (round-float (dispatch-type number)))
      ((double-float double-float)
       (round-float double-float))
      ((single-float double-float)
       (round-float double-float))
      (((foreach fixnum bignum ratio)
        (foreach single-float double-float))
       (round-float (dispatch-type divisor)))
      ((fixnum (foreach bignum fixnum))
       #1=
       (multiple-value-bind (tru rem) (truncate number divisor)
         (if (zerop rem)
             (values tru rem)
             (let ((thresh (* (abs rem) 2))
                   (abs-divisor (abs divisor)))
               (cond ((or (> thresh abs-divisor)
                          (and (= thresh abs-divisor)
                               (oddp tru)))
                      (if (or (minusp tru)
                              (and (zerop tru)
                                   (neq (minusp number) (minusp divisor))))
                          (values (- tru 1) (+ rem divisor))
                          (values (+ tru 1) (- rem divisor))))
                     (t
                      (values tru rem)))))))
      ((bignum integer)
       #1#)
      ((ratio
        (foreach (eql 1) fixnum bignum))
       (dispatch-two-ratios
           (n-num (numerator number))
           (n-den (denominator number))
           (d-num (numerator divisor))
           (d-den (denominator divisor))
           (let ((q-num (* n-num d-den))
                 (q-den (* n-den d-num)))
             (multiple-value-bind (q rem) (round q-num q-den)
               (values q (%make-ratio rem (* n-den d-den)))))))
      (((foreach ratio fixnum bignum) ratio)
       (dispatch-two-ratios
           (n-num (numerator number))
           (n-den (denominator number))
           (d-num (numerator divisor))
           (d-den (denominator divisor))
           (let ((q-num (* n-num d-den))
                 (q-den (* n-den d-num)))
             (multiple-value-bind (q rem) (round q-num q-den)
               (values q (/ rem (* n-den d-den))))))))))

(defun round1 (number divisor)
  (declare (explicit-check)
           (inline round))
  (values (round number divisor)))

(defun %unary-round (number)
  (declare (explicit-check)
           (inline round))
  (values (round number 1)))

(defun %unary-truncate (number)
  (declare (explicit-check)
           (inline truncate))
  (values (truncate number 1)))

(defun unary-truncate (number)
  (declare (explicit-check)
           (inline truncate))
  (truncate number 1))

(defmacro !define-float-rounding-function (name op doc)
  `(defun ,name (number &optional (divisor 1))
     ,doc
     (multiple-value-bind (res rem) (,op number divisor)
       (values (float res (if (floatp rem) rem 1.0)) rem))))

#-round-float
(defun fround (number &optional (divisor 1))
  "Same as ROUND, but returns first value as a float."
  (declare (explicit-check)
           (maybe-inline fround))
  (macrolet ((fround-float (rtype)
               `(let* ((float-div (coerce divisor ',rtype))
                       (res (%unary-fround (/ number float-div))))
                  (values res
                          (- number
                             (* (coerce res ',rtype) float-div))))))
    (number-dispatch ((number real) (divisor real))
      (((foreach fixnum bignum ratio) (or fixnum bignum ratio))
       (multiple-value-bind (q r)
           (round number divisor)
         (if (and (zerop q) (or (and (minusp number) (not (minusp divisor)))
                                (and (not (minusp number)) (minusp divisor))))
             (values -0f0 r)
             (values (float q) r))))
      (((foreach single-float double-float #+long-float long-float)
        (or rational single-float))
       (if (eql divisor 1)
           (let ((res (%unary-fround number)))
             (values res (- number (coerce res '(dispatch-type number)))))
           (fround-float (dispatch-type number))))
      #+long-float
      ((long-float (or single-float double-float long-float))
       (fround-float long-float))
      #+long-float
      (((foreach double-float single-float) long-float)
       (fround-float long-float))
      ((double-float (or single-float double-float))
       (fround-float double-float))
      ((single-float double-float)
       (fround-float double-float))
      (((foreach fixnum bignum ratio)
        (foreach single-float double-float #+long-float long-float))
       (fround-float (dispatch-type divisor))))))

#-round-float
(defun fround1 (number divisor)
  (declare (explicit-check)
           (inline fround))
  (values (fround number divisor)))

(eval-when (:compile-toplevel :load-toplevel :execute)
  (dolist (s '(truncate floor ceiling round fround))
    (clear-info :function :inlining-data s)
    (clear-info :function :inlinep s)
    (clear-info :source-location :declaration s)))

(macrolet ((def (name mode docstring &optional values)
             `(defun ,name (number ,@(if values
                                         `(divisor)
                                         `(&optional (divisor 1))))
                (declare (explicit-check))
                ,docstring
                ,(wrap-if
                  values '(values)
                  `(macrolet ((ftruncate-float (rtype)
                                `(let* ((float-div (coerce divisor ',rtype))
                                        (res (,(case rtype
                                                 (double-float 'sb-kernel:round-double)
                                                 (single-float 'sb-kernel:round-single))
                                              (/ number float-div)
                                              ,,mode)))
                                   (values res
                                           (- number
                                              (* (coerce res ',rtype) float-div)))))
                              (unary-ftruncate-float (rtype)
                                `(let* ((res (,(case rtype
                                                 (double-float 'sb-kernel:round-double)
                                                 (single-float 'sb-kernel:round-single))
                                              number
                                              ,,mode)))
                                   (values res (- number res)))))
                     (number-dispatch ((number real) (divisor real))
                       (((foreach fixnum bignum ratio) (or fixnum bignum ratio))
                        (multiple-value-bind (q r)
                            (,(find-symbol (string mode) :cl) number divisor)
                          (if (and (zerop q) (or (and (minusp number) (not (minusp divisor)))
                                                 (and (not (minusp number)) (minusp divisor))))
                              (values -0f0 r)
                              (values (float q) r))))
                       (((foreach single-float double-float)
                         (foreach fixnum rational single-float))
                        (if (eql divisor 1)
                            (unary-ftruncate-float (dispatch-type number))
                            (ftruncate-float (dispatch-type number))))
                       ((double-float double-float)
                        (ftruncate-float double-float))
                       ((single-float double-float)
                        (ftruncate-float double-float))
                       (((foreach fixnum bignum ratio)
                         (foreach single-float double-float))
                        (ftruncate-float (dispatch-type divisor)))))))))
  (def ftruncate :truncate
    "Same as TRUNCATE, but returns first value as a float.")

  (def ffloor :floor
    "Same as FLOOR, but returns first value as a float.")

  (def fceiling :ceiling
    "Same as CEILING, but returns first value as a float.")

  (def ftruncate1 :truncate nil t)

  (def ffloor1 :floor nil t)

  (def fceiling1 :ceiling nil t)

  #+round-float
  (def fround :round
    "Same as ROUND, but returns first value as a float.")

  #+round-float
  (def fround1 :round nil t))

(macrolet ((def (name)
             `(defun ,name (x mode)
                (ecase mode
                  ,@(loop for m in '(#+round-float :round :floor :ceiling :truncate)
                          collect `(,m (,name x ,m)))))))


  (def round-single)
  (def round-double))

;;;; comparisons

(defun = (number &rest more-numbers)
  "Return T if all of its arguments are numerically equal, NIL otherwise."
  (declare (number number) (explicit-check))
  (do-rest-arg ((n i) more-numbers 0 t)
    (unless (= number n)
      (return (do-rest-arg ((n) more-numbers (1+ i))
                (the number n)))))) ; for effect

(defun /= (number &rest more-numbers)
  "Return T if no two of its arguments are numerically equal, NIL otherwise."
  (declare (number number) (explicit-check))
  (if more-numbers
      (do ((n number (nth i more-numbers))
            (i 0 (1+ i)))
          ((>= i (length more-numbers))
           t)
        (do-rest-arg ((n2) more-numbers i)
          (when (= n n2)
            (return-from /= nil))))
      t))

(macrolet ((def (op doc)
             `(defun ,op (number &rest more-numbers)
                ,doc
                (declare (explicit-check))
                (let ((n1 number))
                  (declare (real n1))
                  (do-rest-arg ((n2 i) more-numbers 0 t)
                    (if (,op n1 n2)
                        (setf n1 n2)
                        (return (do-rest-arg ((n) more-numbers (1+ i))
                                  (the real n))))))))) ; for effect
  (def <  "Return T if its arguments are in strictly increasing order, NIL otherwise.")
  (def >  "Return T if its arguments are in strictly decreasing order, NIL otherwise.")
  (def <= "Return T if arguments are in strictly non-decreasing order, NIL otherwise.")
  (def >= "Return T if arguments are in strictly non-increasing order, NIL otherwise."))

(defun max (number &rest more-numbers)
  "Return the greatest of its arguments; among EQUALP greatest, return
the first."
  (declare (explicit-check))
  (let ((n number))
    (declare (real n))
    (do-rest-arg ((arg) more-numbers 0 n)
      (when (> arg n)
        (setf n arg)))))

(defun min (number &rest more-numbers)
  "Return the least of its arguments; among EQUALP least, return
the first."
  (declare (explicit-check))
  (let ((n number))
    (declare (real n))
    (do-rest-arg ((arg) more-numbers 0 n)
      (when (< arg n)
        (setf n arg)))))

#+64-bit
(defmacro float-bignum-= (float bignum type)
  (macrolet ((sym (name)
               `(package-symbolicate #.(find-package "SB-VM") type '- ',name)))
    `(let* ((float ,float)
            (bignum ,bignum)
            (bignum-length (%bignum-length bignum))
            (bits (,(sym bits) float))
            (exp (ldb ,(sym exponent-byte) bits)))
       (cond ((< exp (+ sb-vm:n-fixnum-bits ,(sym bias)))
              nil)
             (t
              (let* ((exp (- exp ,(sym bias) ,(sym digits)))
                     (sig (logior ,(sym hidden-bit)
                                  (ldb ,(sym significand-byte) bits)))
                     (int (if (minusp bits)
                              (- sig)
                              sig))
                     (int-length (integer-length int)))
                (and (= (truly-the bignum-length (sb-bignum::bignum-buffer-integer-length bignum bignum-length))
                        (+ int-length exp))
                     (sb-bignum::bignum-lower-bits-zero-p bignum exp bignum-length)
                     (= int
                        (truly-the fixnum
                                   (sb-bignum::last-bignum-part=>fixnum (- (1- sb-bignum::digit-size) int-length) exp bignum))))))))))

#+64-bit
(defmacro float-bignum-< (float bignum type)
  (macrolet ((sym (name)
               `(package-symbolicate #.(find-package "SB-VM") type '- ',name)))
    `(let* ((float ,float)
            (bignum ,bignum)
            (bignum-length (%bignum-length bignum))
            (bits (,(sym bits) float))
            (exp (ldb ,(sym exponent-byte) bits)))
       (cond ((< exp (+ sb-vm:n-fixnum-bits ,(sym bias)))
              (sb-bignum::%bignum-0-or-plusp bignum bignum-length))
             (t
              (let ((sig (logior ,(sym hidden-bit)
                                  (ldb ,(sym significand-byte) bits)))
                    (exp (- exp ,(sym bias) ,(sym digits))))
                (let* ((int (if (minusp bits)
                                (- sig)
                                sig))
                       (int-length (integer-length int))
                       (length-diff (- (truly-the bignum-length (sb-bignum::bignum-buffer-integer-length bignum bignum-length))
                                       (+ int-length exp))))
                  (cond
                    ((plusp length-diff) (sb-bignum::%bignum-0-or-plusp bignum bignum-length))
                    ((minusp length-diff) (minusp float))
                    (t
                     (let ((diff (- (truly-the fixnum
                                               (sb-bignum::last-bignum-part=>fixnum (- (1- sb-bignum::digit-size) int-length) exp bignum))
                                    int)))
                       (cond ((plusp diff) t)
                             ((minusp diff) nil)
                             (t
                              (not (sb-bignum::bignum-lower-bits-zero-p bignum exp bignum-length))))))))))))))

#+64-bit
(defmacro float-bignum-> (float bignum type)
  (macrolet ((sym (name)
               `(package-symbolicate #.(find-package "SB-VM") type '- ',name)))
    `(let* ((float ,float)
            (bignum ,bignum)
            (bignum-length (%bignum-length bignum))
            (bits (,(sym bits) float))
            (exp (ldb ,(sym exponent-byte) bits)))
       (cond ((< exp (+ sb-vm:n-fixnum-bits ,(sym bias)))
              (not (sb-bignum::%bignum-0-or-plusp bignum bignum-length)))
             (t
              (let ((sig (logior ,(sym hidden-bit)
                                 (ldb ,(sym significand-byte) bits)))
                    (exp (- exp ,(sym bias) ,(sym digits))))
                (let* ((int
                         (if (minusp bits)
                             (- sig)
                             sig))
                       (int-length (integer-length int))
                       (length-diff (- (truly-the bignum-length (sb-bignum::bignum-buffer-integer-length bignum bignum-length))
                                       (+ int-length exp))))
                  (cond ((plusp length-diff)
                         (not (sb-bignum::%bignum-0-or-plusp bignum bignum-length)))
                        ((minusp length-diff)
                         (not (minusp float)))
                        (t
                         (< (truly-the fixnum (sb-bignum::last-bignum-part=>fixnum (- (1- sb-bignum::digit-size) int-length) exp bignum))
                            int))))))))))

(defmacro make-fixnum-float-comparer (operation integer float float-type)
  (multiple-value-bind (min max)
      (ecase float-type
        (single-float
         (values most-negative-fixnum-single-float most-positive-fixnum-single-float))
        (double-float
         (values most-negative-fixnum-double-float most-positive-fixnum-double-float)))
    ` (cond ((> ,float ,max)
             ,(ecase operation
                ((= > >=) nil)
                ((< <=) t)))
            ((< ,float ,min)
             ,(ecase operation
                ((= < <=) nil)
                ((> >=) t)))
            (t
             (sb-c::if-vop-existsp (:translate %unary-floor)
               ,(ecase operation
                  (=
                   `(let ((quot (%unary-truncate ,float)))
                      (and (= quot ,integer)
                           (= (float quot ,float) ,float))))
                  (>
                   `(,operation ,integer (truly-the fixnum (%unary-floor ,float))))
                  (<
                   `(,operation ,integer (truly-the fixnum (%unary-ceiling ,float))))
                  (>=
                   `(let ((quot (truly-the fixnum (%unary-floor ,float))))
                      (or (> ,integer quot)
                          (and (= ,integer quot)
                               (= (float quot ,float) ,float)))))
                  (<=
                   `(let ((quot (truly-the fixnum (%unary-ceiling ,float))))
                      (or (< ,integer quot)
                          (and (= ,integer quot)
                               (= (float quot ,float) ,float))))))
               (let ((quot (%unary-truncate ,float)))
                 ,(ecase operation
                    (=
                     `(and (= quot ,integer)
                           (= (float quot ,float) ,float)))
                    (>
                     `(cond ((> ,integer quot))
                            ((< ,integer quot)
                             nil)
                            ((<= ,integer 0)
                             (> (float quot ,float) ,float))))
                    (<
                     `(cond ((< ,integer quot))
                            ((> ,integer quot)
                             nil)
                            ((>= ,integer 0)
                             (< (float quot ,float) ,float))))
                    (>=
                     `(cond ((> ,integer quot))
                            ((< ,integer quot)
                             nil)
                            (t
                             (or (< ,float 0)
                                 (= (float quot ,float) ,float)))))
                    (<=
                     `(cond ((< ,integer quot))
                            ((> ,integer quot)
                             nil)
                            (t
                             (or (>= ,float 0)
                                 (= (float quot ,float) ,float))))))))))))

(eval-when (:compile-toplevel :load-toplevel :execute)
;;; The INFINITE-X-FINITE-Y and INFINITE-Y-FINITE-X args tell us how
;;; to handle the case when X or Y is a floating-point infinity and
;;; the other arg is a rational. (Section 12.1.4.1 of the ANSI spec
;;; says that comparisons are done by converting the float to a
;;; rational when comparing with a rational, but infinities can't be
;;; converted to a rational, so we show some initiative and do it this
;;; way instead.)
  (defun basic-compare (op &key infinite-x-finite-y infinite-y-finite-x)
    `(((fixnum fixnum) (,op x y))
      ((single-float single-float) (,op x y))
      (((foreach single-float double-float) double-float)
       (,op (coerce x 'double-float) y))
      #+long-float
      (((foreach single-float double-float long-float) long-float)
       (,op (coerce x 'long-float) y))
      #+long-float
      ((long-float (foreach single-float double-float))
       (,op x (coerce y 'long-float)))
      ((fixnum (foreach single-float double-float))
       (with-float-inf-or-nan-test y
         ,infinite-y-finite-x
         nil
         (make-fixnum-float-comparer ,op x y (dispatch-type y))))
      (((foreach single-float double-float) fixnum)
       (if (eql y 0)
           (,op x (coerce 0 '(dispatch-type x)))
           (with-float-inf-or-nan-test x
             ,infinite-x-finite-y
             nil
             (make-fixnum-float-comparer ,(case op
                                            (> '<)
                                            (< '>)
                                            (>= '<=)
                                            (<= '>=)
                                            (= '=))
                                         y x (dispatch-type x)))))
      ((double-float single-float)
       (,op x (coerce y 'double-float)))
      (((foreach single-float double-float #+long-float long-float) ratio)
       (with-float-inf-or-nan-test x
         ,infinite-x-finite-y
         nil
         ;; Avoid converting the float into a rational, since it
         ;; will be taken apart later anyway.
         (multiple-value-bind (bits exp) (integer-decode-float x)
           (if (eql bits 0)
               (,op 0 y)
               (let ((int (if (minusp x) (- bits) bits)))
                 (if (minusp exp)
                     (,op (* int (denominator y))
                          (ash (numerator y) (- exp)))
                     (,op (ash int exp) y)))))))
      ((ratio (foreach single-float double-float))
       (with-float-inf-or-nan-test y
         ,infinite-y-finite-x
         nil
         (multiple-value-bind (bits exp) (integer-decode-float y)
           (if (eql bits 0)
               (,op x 0)
               (let ((int (if (minusp y) (- bits) bits)))
                 (if (minusp exp)
                     (,op (ash (numerator x) (- exp))
                          (* int (denominator x)))
                     (,op x (ash int exp))))))))
      (((foreach single-float double-float) bignum)
       (with-float-inf-or-nan-test x
         ,infinite-x-finite-y
         nil
         #+64-bit
         ,(case op
            (> '(float-bignum-> x y (dispatch-type x)))
            (< '(float-bignum-< x y (dispatch-type x)))
            (>= '(not (float-bignum-< x y (dispatch-type x))))
            (<= '(not (float-bignum-> x y (dispatch-type x))))
            (= '(float-bignum-= x y (dispatch-type x))))
         #-64-bit
         (,op (rational x) y)))
      ((bignum (foreach single-float double-float))
       (with-float-inf-or-nan-test y
         ,infinite-y-finite-x
         nil
         #+64-bit
         ,(case op
            (> '(float-bignum-< y x (dispatch-type y)))
            (< '(float-bignum-> y x (dispatch-type y)))
            (>= '(not (float-bignum-> y x (dispatch-type y))))
            (<= '(not (float-bignum-< y x (dispatch-type y))))
            (= '(float-bignum-= y x (dispatch-type y))))
         #-64-bit
         (,op x (rational y))))))
  )                                     ; EVAL-WHEN

(macrolet ((def-two-arg-</> (name op ratio-arg1 ratio-arg2 &rest cases)
             `(defun ,name (x y)
                (declare (explicit-check))
                (number-dispatch ((x real) (y real))
                  (basic-compare
                   ,op
                   :infinite-x-finite-y
                   (,op x (coerce 0 '(dispatch-type x)))
                   :infinite-y-finite-x
                   (,op (coerce 0 '(dispatch-type y)) y))
                  (((foreach fixnum bignum) ratio)
                   (dispatch-ratio (y numerator denominator)
                       (,(case op
                           (<= '<)
                           (>= '>)
                           (t op))
                        x (,ratio-arg2 numerator
                                       denominator))))
                  ((ratio (foreach fixnum bignum))
                   (dispatch-ratio (x numerator denominator)
                     (,(case op
                         (<= '<)
                         (>= '>)
                         (t op))
                      (,ratio-arg1 numerator
                                   denominator)
                      y)))
                  ((ratio ratio)
                   (compare-two-ratios ,op (numerator x) (denominator x)
                                       (numerator y) (denominator y)))
                  ,@cases))))
  (def-two-arg-</> two-arg-< < floor ceiling
    ((fixnum bignum)
     (bignum-plus-p y))
    ((bignum fixnum)
     (not (bignum-plus-p x)))
    ((bignum bignum)
     (minusp (bignum-compare x y))))
  (def-two-arg-</> two-arg-> > ceiling floor
    ((fixnum bignum)
     (not (bignum-plus-p y)))
    ((bignum fixnum)
     (bignum-plus-p x))
    ((bignum bignum)
     (plusp (bignum-compare x y))))
  (def-two-arg-</> two-arg-<= <= floor ceiling
    ((fixnum bignum)
     (bignum-plus-p y))
    ((bignum fixnum)
     (not (bignum-plus-p x)))
    ((bignum bignum)
     (<= (bignum-compare x y) 0)))
  (def-two-arg-</> two-arg->= >= ceiling floor
    ((fixnum bignum)
     (not (bignum-plus-p y)))
    ((bignum fixnum)
     (bignum-plus-p x))
    ((bignum bignum)
     (>= (bignum-compare x y) 0))))

(defun two-arg-= (x y)
  (declare (explicit-check))
  (number-dispatch ((x number) (y number))
    (basic-compare =
                   ;; An infinite value is never equal to a finite value.
                   :infinite-x-finite-y nil
                   :infinite-y-finite-x nil)
    ((fixnum (or bignum ratio)) nil)

    ((bignum (or fixnum ratio)) nil)
    ((bignum bignum)
     (zerop (bignum-compare x y)))

    ((ratio integer) nil)
    ((ratio ratio)
     (and (eql (numerator x) (numerator y))
          (eql (denominator x) (denominator y))))

    ((complex complex)
     (and (= (realpart x) (realpart y))
          (= (imagpart x) (imagpart y))))
    (((foreach fixnum bignum ratio single-float double-float
               #+long-float long-float) complex)
     (and (= x (realpart y))
          (zerop (imagpart y))))
    ((complex (or float rational))
     (and (= (realpart x) y)
          (zerop (imagpart x))))))

(macrolet ((make-float-fixnum-range-comparer (operation low high float float-type)
             (multiple-value-bind (min max)
                 (ecase float-type
                   (single-float
                    (values most-negative-fixnum-single-float most-positive-fixnum-single-float))
                   (double-float
                    (values most-negative-fixnum-double-float most-positive-fixnum-double-float)))
               `(cond
                  ((or (> ,float ,max)
                       (< ,float ,min)
                       (float-inf-or-nan-test ,float nil nil))
                   nil)
                  (t
                   (let ((quot (%unary-truncate ,float)))
                     ,(ecase operation
                        (<
                         `(cond ((or (> quot ,high)
                                     (< quot ,low)
                                     (= low high))
                                 nil)
                                ((= quot high)
                                 (and (minusp ,float)
                                      (/= ,float (float quot))))
                                ((= quot low)
                                 (and (plusp ,float)
                                      (/= ,float (float quot))))
                                (t)))
                        (<=
                         `(cond ((or (> quot ,high)
                                     (< quot ,low))
                                 nil)
                                ((= quot high)
                                 (or (and (minusp ,float)
                                          (/= low high))
                                     (= ,float (float quot))))
                                ((= quot low)
                                 (or (plusp ,float)
                                     (= ,float (float quot))))
                                (t)))
                        (<<=
                         `(cond ((or (> quot ,high)
                                     (< quot ,low)
                                     (= low high))
                                 nil)
                                ((= quot high)
                                 (or (minusp ,float)
                                     (= ,float (float quot))))
                                ((= quot low)
                                 (and (plusp ,float)
                                      (/= ,float (float quot))))
                                (t)))
                        (<=<
                         `(cond ((or (> quot ,high)
                                     (< quot ,low)
                                     (= low high))
                                 nil)
                                ((= quot high)
                                 (and (minusp ,float)
                                      (/= ,float (float quot))))
                                ((= quot low)
                                 (or (plusp ,float)
                                     (= ,float (float quot))))
                                (t)))))))))
           (def-range (name op &optional (low-op op) (high-op op))
             `(progn
                (defun ,name (low x high)
                  (declare (explicit-check)
                           (fixnum low high))
                  (number-dispatch ((x real))
                    ((fixnum) (and (,low-op low x)
                                   (,high-op x high)))
                    (((foreach single-float double-float))
                     (make-float-fixnum-range-comparer ,op low high x (dispatch-type x)))
                    ((bignum) nil)
                    ((ratio)
                     (let* ((numerator (numerator x))
                            (minusp (minusp numerator))
                            (x (truncate numerator (denominator x))))
                       (cond ((or (not (fixnump x))
                                  (< x low)
                                  (> x high))
                              nil)
                             ((= low high)
                              nil)
                             ((= x high)
                              minusp)
                             ((= x low)
                              (not minusp))
                             (t)))))))))
  (def-range range< <)
  (def-range range<= <=)
  (def-range range<<= <<= < <=)
  (def-range range<=< <=< <= <))

(sb-c::when-vop-existsp (:translate check-range<=)
  (defun check-range<= (low x high)
    (when (typep x 'fixnum)
      (range<= low x high))))


;;;; logicals

(macrolet ((def (op init doc)
             `(defun ,op (&rest integers)
                ,doc
                (declare (explicit-check))
                (if integers
                    (do ((result (fast-&rest-nth 0 integers)
                                 (,op result (fast-&rest-nth i integers)))
                         (i 1 (1+ i)))
                        ((>= i (length integers))
                         result)
                      (declare (integer result)))
                    ,init))))
  (def logior 0 "Return the bit-wise or of its arguments. Args must be integers.")
  (def logxor 0 "Return the bit-wise exclusive or of its arguments. Args must be integers.")
  (def logand -1 "Return the bit-wise and of its arguments. Args must be integers.")
  (def logeqv -1 "Return the bit-wise equivalence of its arguments. Args must be integers."))

(defun lognot (number)
  "Return the bit-wise logical not of integer."
  (declare (explicit-check))
  (etypecase number
    (fixnum (lognot (truly-the fixnum number)))
    (bignum (bignum-logical-not number))))

(macrolet ((def (name op big-op &optional doc)
             `(defun ,name (integer1 integer2)
                ,@(when doc (list doc))
                (declare (explicit-check))
                (let ((x integer1)
                      (y integer2))
                  (number-dispatch ((x integer) (y integer))
                    (bignum-cross-fixnum ,op ,big-op))))))
  (def two-arg-and logand bignum-logical-and)
  (def two-arg-ior logior bignum-logical-ior)
  (def two-arg-xor logxor bignum-logical-xor)
  ;; BIGNUM-LOGICAL-{AND,IOR,XOR} need not return a bignum, so must
  ;; call the generic LOGNOT...
  (def two-arg-eqv logeqv (lambda (x y) (lognot (bignum-logical-xor x y))))
  (def lognand lognand
       (lambda (x y) (lognot (bignum-logical-and x y)))
       "Complement the logical AND of INTEGER1 and INTEGER2.")
  (def lognor lognor
       (lambda (x y) (lognot (bignum-logical-ior x y)))
       "Complement the logical OR of INTEGER1 and INTEGER2.")
  ;; ... but BIGNUM-LOGICAL-NOT on a bignum will always return a bignum
  (def logandc1 logandc1
       (lambda (x y) (bignum-logical-and (bignum-logical-not x) y))
       "Bitwise AND (LOGNOT INTEGER1) with INTEGER2.")
  (def logandc2 logandc2
       (lambda (x y) (bignum-logical-and x (bignum-logical-not y)))
       "Bitwise AND INTEGER1 with (LOGNOT INTEGER2).")
  (def logorc1 logorc1
       (lambda (x y) (bignum-logical-ior (bignum-logical-not x) y))
       "Bitwise OR (LOGNOT INTEGER1) with INTEGER2.")
  (def logorc2 logorc2
       (lambda (x y) (bignum-logical-ior x (bignum-logical-not y)))
       "Bitwise OR INTEGER1 with (LOGNOT INTEGER2)."))

(defun logcount (integer)
  "Count the number of 1 bits if INTEGER is non-negative,
and the number of 0 bits if INTEGER is negative."
  (declare (explicit-check))
  (etypecase integer
    (fixnum
     (logcount #-(or x86-64 arm64)
               (truly-the (integer 0
                                   #.(max most-positive-fixnum
                                          (lognot most-negative-fixnum)))
                          (if (minusp (truly-the fixnum integer))
                              (lognot (truly-the fixnum integer))
                              integer))
               ;; The VOP handles that case better
               #+(or x86-64 arm64) integer))
    (bignum
     (bignum-logcount integer))))

(defun logtest (integer1 integer2)
  "Predicate which returns T if logand of integer1 and integer2 is not zero."
  (logtest integer1 integer2))

(defun logbitp (index integer)
  "Predicate returns T if bit index of integer is a 1."
  (declare (explicit-check))
  (number-dispatch ((index unsigned-byte) (integer integer))
    ((fixnum fixnum) (if (< index sb-vm:n-positive-fixnum-bits)
                         (not (zerop (logand integer (ash 1 index))))
                         (minusp integer)))
    ((fixnum bignum) (bignum-logbitp index integer))
    ((bignum (foreach fixnum bignum)) (minusp integer))))

(defun ash (integer count)
  "Shifts integer left by count places preserving sign. - count shifts right."
  (declare (explicit-check))
  (etypecase integer
    (fixnum
     (cond ((fixnump count)
            (if (zerop integer)
                0
                (let ((length (integer-length (truly-the fixnum integer)))
                      (count (truly-the fixnum count)))
                  (declare (fixnum length count))
                  (cond ((and (plusp count)
                              (>= (+ length count)
                                  sb-vm:n-word-bits))
                         (bignum-ashift-left-fixnum integer count))
                        (t
                         (truly-the sb-vm:signed-word
                                    (ash (truly-the fixnum integer) count)))))))
           ((minusp (the integer count))
            (if (minusp integer) -1 0))
           (t
            (bignum-ashift-left (make-small-bignum integer) count))))
    (bignum
     (if (plusp (the integer count))
         (bignum-ashift-left integer count)
         (bignum-ashift-right integer (- count))))))

(defun ash-right (integer count)
  (declare (explicit-check))
  (number-dispatch ((integer integer))
    ((fixnum)
     (ash integer (- (truly-the (mod #.sb-vm:n-word-bits) count))))
    ((bignum)
     (bignum-ashift-right integer count))))

(sb-c::when-vop-existsp (:translate ash-left-add)
  (defun ash-left-add (integer count add)
    (etypecase integer
      (fixnum
       (ash-left-add (truly-the fixnum integer) count add))
      (bignum
       (sb-bignum::bignum-ashift-left-add integer count add))))

  (defun ash-left-word-add (integer add)
    (etypecase integer
      (fixnum
       (ash-left-word-add integer add))
      (bignum
       (sb-bignum::bignum-ashift-left-add integer sb-vm:n-word-bits add)))))

(defun ash-right-two-words (word2 word1 count)
  (logand most-positive-word
          (ash (dpb word2 (byte sb-vm:n-word-bits sb-vm:n-word-bits) word1) (- count))))

(defun integer-length (integer)
  "Return the number of non-sign bits in the twos-complement representation
  of INTEGER."
  (declare (explicit-check))
  (etypecase integer
    (fixnum
     (integer-length (truly-the fixnum integer)))
    (bignum
     (bignum-integer-length integer))))

;;;; BYTE, bytespecs, and related operations

(defun byte (size position)
  "Return a byte specifier which may be used by other byte functions
  (e.g. LDB)."
  (byte size position))

(defun byte-size (bytespec)
  "Return the size part of the byte specifier bytespec."
  (byte-size bytespec))

(defun byte-position (bytespec)
  "Return the position part of the byte specifier bytespec."
  (byte-position bytespec))

(defun ldb (bytespec integer)
  "Extract the specified byte from integer, and right justify result."
  (ldb bytespec integer))

(defun ldb-test (bytespec integer)
  "Return T if any of the specified bits in integer are 1's."
  (ldb-test bytespec integer))

(defun mask-field (bytespec integer)
  "Extract the specified byte from integer,  but do not right justify result."
  (mask-field bytespec integer))

(defun dpb (newbyte bytespec integer)
  "Return new integer with newbyte in specified position, newbyte is right justified."
  (dpb newbyte bytespec integer))

(defun deposit-field (newbyte bytespec integer)
  "Return new integer with newbyte in specified position, newbyte is not right justified."
  (deposit-field newbyte bytespec integer))

(defun %ldb (size posn integer)
  ;; The naive algorithm is horrible in the general case.
  ;; Consider (LDB (BYTE 1 2) (SOME-GIANT-BIGNUM)) which has to shift the
  ;; input rightward 2 bits, consing a new bignum just to read 1 bit.
  (cond ((<= 0 size sb-vm:n-positive-fixnum-bits)
         (if (fixnump integer)
             (logand (ash integer (- posn))
                     (1- (ash 1 size)))
             (sb-bignum::ldb-bignum=>fixnum size posn integer)))
        (t
         (logand (ash integer (- posn))
                 (1- (ash 1 size))))))

(defun %mask-field (size posn integer)
  (declare (type bit-index size posn) (explicit-check))
  (logand integer (ash (1- (ash 1 size)) posn)))

(defun %dpb (newbyte size posn integer)
  (declare (type bit-index size posn) (explicit-check))
  (let ((mask (1- (ash 1 size))))
    (logior (logand integer (lognot (ash mask posn)))
            (ash (logand newbyte mask) posn))))

(defun %deposit-field (newbyte size posn integer)
  (declare (type bit-index size posn) (explicit-check))
  (let ((mask (ash (ldb (byte size 0) -1) posn)))
    (logior (logand newbyte mask)
            (logand integer (lognot mask)))))

(defun sb-c::mask-signed-field (size integer)
  "Extract SIZE lower bits from INTEGER, considering them as a
2-complement SIZE-bits representation of a signed integer."
  (macrolet ((msf (size integer)
               `(if (logbitp (1- ,size) ,integer)
                    (dpb ,integer (byte (1- ,size) 0) -1)
                    (ldb (byte (1- ,size) 0) ,integer))))
    (typecase size
      ((eql 0) 0)
      ((integer 1 #.sb-vm:n-fixnum-bits)
       (number-dispatch ((integer integer))
         ((fixnum) (msf size integer))
         ((bignum) (let ((fix (sb-c::mask-signed-field #.sb-vm:n-fixnum-bits (%bignum-ref integer 0))))
                     (if (= size #.sb-vm:n-fixnum-bits)
                         fix
                         (msf size fix))))))
      ((integer (#.sb-vm:n-fixnum-bits) #.sb-vm:n-word-bits)
       (number-dispatch ((integer integer))
         ((fixnum) integer)
         ((bignum) (let ((word (sb-c::mask-signed-field #.sb-vm:n-word-bits (%bignum-ref integer 0))))
                     (if (= size #.sb-vm:n-word-bits)
                         word
                         (msf size word))))))
      ((unsigned-byte) (msf size integer)))))

;;;; BOOLE

(defun boole (op integer1 integer2)
  "Bit-wise boolean function on two integers. Function chosen by OP:
        0       BOOLE-CLR
        1       BOOLE-SET
        2       BOOLE-1
        3       BOOLE-2
        4       BOOLE-C1
        5       BOOLE-C2
        6       BOOLE-AND
        7       BOOLE-IOR
        8       BOOLE-XOR
        9       BOOLE-EQV
        10      BOOLE-NAND
        11      BOOLE-NOR
        12      BOOLE-ANDC1
        13      BOOLE-ANDC2
        14      BOOLE-ORC1
        15      BOOLE-ORC2"
  (case op
    (0 (boole 0 integer1 integer2))
    (1 (boole 1 integer1 integer2))
    (2 (boole 2 integer1 integer2))
    (3 (boole 3 integer1 integer2))
    (4 (boole 4 integer1 integer2))
    (5 (boole 5 integer1 integer2))
    (6 (boole 6 integer1 integer2))
    (7 (boole 7 integer1 integer2))
    (8 (boole 8 integer1 integer2))
    (9 (boole 9 integer1 integer2))
    (10 (boole 10 integer1 integer2))
    (11 (boole 11 integer1 integer2))
    (12 (boole 12 integer1 integer2))
    (13 (boole 13 integer1 integer2))
    (14 (boole 14 integer1 integer2))
    (15 (boole 15 integer1 integer2))
    (t (error 'type-error :datum op :expected-type '(mod 16)))))

;;;; GCD and LCM

(defun gcd (&rest integers)
  "Return the greatest common divisor of the arguments, which must be
  integers. GCD with no arguments is defined to be 0."
  (declare (explicit-check))
  (case (length integers)
    (0 0)
    (1 (abs (the integer (fast-&rest-nth 0 integers))))
    (otherwise
     (do ((result (fast-&rest-nth 0 integers)
                  ;; Call TWO-ARG-GCD directly, because self calls are
                  ;; recognized before REWRITE-FULL-CALL can act,
                  ;; because GCD has no templates to prevent self
                  ;; call optimization.
                  (two-arg-gcd result (the integer (fast-&rest-nth i integers))))
          (i 1 (1+ i)))
         ((>= i (length integers))
          result)
       (declare (integer result))))))

(defun lcm (&rest integers)
  "Return the least common multiple of one or more integers. LCM of no
  arguments is defined to be 1."
  (declare (explicit-check))
  (case (length integers)
    (0 1)
    (1 (abs (the integer (fast-&rest-nth 0 integers))))
    (otherwise
     (do ((result (fast-&rest-nth 0 integers)
                  (two-arg-lcm result (the integer (fast-&rest-nth i integers))))
          (i 1 (1+ i)))
         ((>= i (length integers))
          result)
       (declare (integer result))))))

(defun two-arg-lcm (n m)
  (declare (integer n m))
  (if (or (zerop n) (zerop m))
      0
      ;; KLUDGE: I'm going to assume that it was written this way
      ;; originally for a reason.  However, this is a somewhat
      ;; complicated way of writing the algorithm in the CLHS page for
      ;; LCM, and I don't know why.  To be investigated.  -- CSR,
      ;; 2003-09-11
      ;;
      ;;    It seems to me that this is written this way to avoid
      ;;    unnecessary bignumification of intermediate results.
      ;;        -- TCR, 2008-03-05
      (let ((m (abs m))
            (n (abs n)))
        (multiple-value-bind (max min)
            (if (> m n)
                (values m n)
                (values n m))
          (* (truncate max (gcd n m)) min)))))

;;; Actually do a simple Euclidean algorithm instead of a binary GCD,
;;; binary GCD is slow on large numbers, using count-trailing-zeros
;;; helps, but only when there's a lot of trailing zeros.
;;; Using division before doing binary GCD, like some sources suggest,
;;; doesn't speed things up uniformly.
(declaim (maybe-inline fixnum-gcd))
#-arm
(defun fixnum-gcd (u v)
  (cond ((eql u 0) (abs v))
        ((eql v 0) (abs u))
        (t
         (let ((u (abs u))
               (v (abs v)))
           (declare (sb-vm:signed-word u v))
           (loop (setf (values u v)
                       (values v (rem u (the (not (eql 0)) v))))
                 (if (zerop v)
                     (return (truly-the (integer 0 #.(1+ most-positive-fixnum)) u))))))))

;;; But 32-bit arm has no division instruction
#+arm
(defun fixnum-gcd (u v)
  (locally
      (declare (optimize (speed 3) (safety 0)))
    (cond ((eql u 0) (abs v))
          ((eql v 0) (abs u))
          (t
           (do ((k 0 (1+ k))
                (u (abs u) (ash u -1))
                (v (abs v) (ash v -1)))
               ((oddp (logior u v))
                (do ((temp (if (oddp u) (- v) (ash u -1))
                           (ash temp -1)))
                    (nil)
                  (declare (fixnum temp))
                  (when (oddp temp)
                    (if (plusp temp)
                        (setq u temp)
                        (setq v (- temp)))
                    (setq temp (- u v))
                    (when (zerop temp)
                      (return (the (integer 0 #.(1+ most-positive-fixnum)) (ash u k)))))))
             (declare (type (mod #.sb-vm:n-word-bits) k)
                      (type sb-vm:signed-word u v)))))))

;;; Do the GCD of two integer arguments.
;;; We pick off the special case
;;; of 0 before the dispatch so that the bignum code doesn't have to worry
;;; about "small bignum" zeros.
(defun two-arg-gcd (u v)
  (declare (muffle-conditions compiler-note)
           (inline fixnum-gcd))
  (cond ((eql u 0) (abs v))
        ((eql v 0) (abs u))
        (t
         (number-dispatch ((u integer) (v integer))
           ((fixnum fixnum)
            (fixnum-gcd u v))
           ((bignum bignum)
            (bignum-gcd u v))
           ((bignum fixnum)
            (bignum-gcd u (make-small-bignum v)))
           ((fixnum bignum)
            (bignum-gcd (make-small-bignum u) v))))))

;;; Divide two integers, producing a canonical rational. If a fixnum,
;;; we see whether they divide evenly before trying the GCD. In the
;;; bignum case, we don't bother, since bignum division is expensive,
;;; and the test is not very likely to succeed.
(defun integer-/-integer (x y)
  (declare (explicit-check)
           (inline fixnum-gcd))
  (if (and (typep x 'fixnum) (typep y 'fixnum))
      (cond ((= y 0)
             (error 'division-by-zero
                    :operands (list x y)
                    :operation '/))
            ((= x 0)
             0)
            (t
             (let ((abs-x (abs x))
                   (abs-y (abs y)))
               (if (> abs-y abs-x)
                   ;; It is a ratio, truncate anyway, leaving less work for GCD.
                   (multiple-value-bind (quo rem) (truncate y x)
                     (if (zerop rem)
                         (build-ratio 1 (truly-the (not (integer 0 1)) quo))
                         (let* ((gcd (gcd x rem)))
                           (multiple-value-bind (num den)
                               (if (eql gcd 1)
                                   (values x y)
                                   (values (truncate x gcd) (truncate y gcd)))
                             (build-ratio num (truly-the (not (integer 0 1)) den))))))
                   (multiple-value-bind (quo rem) (truncate x y)
                     (if (zerop rem)
                         quo
                         (let* ((gcd (gcd y rem)))
                           (multiple-value-bind (num den)
                               (if (eql gcd 1)
                                   (values x y)
                                   (values (truncate x gcd) (truncate y gcd)))
                             (build-ratio num (truly-the (not (integer 0 1)) den))))))))))
      (let ((gcd (gcd x y)))
        (multiple-value-bind (num den)
            (if (eql gcd 1)
                (values x y)
                (values (truncate x gcd) (truncate y gcd)))
          (build-ratio num den)))))

(defun two-arg-/ (x y)
  (declare (explicit-check))
  (number-dispatch ((x number) (y number))
    (float-contagion / x y (ratio integer))
    (((foreach integer ratio single-float double-float complex) complex)
     (let* ((rx (realpart x))
            (ix (imagpart x))
            (ry (realpart y))
            (iy (imagpart y)))
       (if (> (abs ry) (abs iy))
           (let* ((r (/ iy ry))
                  (dn (+ ry (* r iy))))
             (complex (/ (+ rx (* ix r)) dn)
                      (/ (- ix (* rx r)) dn)))
           (let* ((r (/ ry iy))
                  (dn (+ iy (* r ry))))
             (complex (/ (+ (* rx r) ix) dn)
                      (/ (- (* ix r) rx) dn))))))
    ((complex (or rational float))
     (canonical-complex (/ (realpart x) y)
                        (/ (imagpart x) y)))

    ((ratio ratio)
     (let* ((nx (numerator x))
            (dx (denominator x))
            (ny (numerator y))
            (dy (denominator y))
            (g1 (gcd nx ny))
            (g2 (gcd dx dy)))
       (build-ratio (* (maybe-truncate nx g1) (maybe-truncate dy g2))
                    (* (maybe-truncate dx g2) (maybe-truncate ny g1)))))

    ((integer integer)
     (integer-/-integer x y))

    ((integer ratio)
     (if (zerop x)
         0
         (let* ((ny (numerator y))
                (dy (denominator y))
                (gcd (gcd x ny)))
           (build-ratio (* (maybe-truncate x gcd) dy)
                        (maybe-truncate ny gcd)))))

    ((ratio integer)
     (let* ((nx (numerator x))
            (gcd (gcd nx y)))
       (build-ratio (maybe-truncate nx gcd)
                    (* (maybe-truncate y gcd) (denominator x)))))))

;;; This uses the algorithm employed by python in
;;; https://github.com/python/cpython/blob/3.13/Modules/mathmodule.c#L1494
(defun isqrt (n)
  "Return the greatest integer less than or equal to the square root of N."
  (declare (explicit-check))
  (labels ((approximate-isqrt (n)
             (declare ((unsigned-byte 64) n))
             (let ((table #.(coerce (loop for i from 64 below 256
                                          collect (min (round (sqrt (+ (* i 256) 128))) 255))
                                    '(vector (unsigned-byte 8)))))

               (let* ((u (aref table (truly-the index (- (ash n -56) 64))))
                      (u (+ (ash u 7)
                            (truncate (ash n -41) u))))
                 (+ (ash u 15)
                    (truncate (ash n -17) u))))))
    (declare (inline approximate-isqrt))
    (macrolet ((word-sized-sqrt (type)
                 `(let* ((c (truncate (1- (integer-length n)) 2))
                         (shift (- 31 c))
                         (approximate (truly-the ,type
                                                 (ash (approximate-isqrt
                                                       (truly-the (unsigned-byte 64) (ash n (* shift 2))))
                                                      (- shift)))))
                    (if (> (truly-the ,type (* approximate approximate)) n)
                        (1- approximate)
                        approximate))))

      (number-dispatch ((n unsigned-byte))
        ((fixnum)
         (cond ((zerop n)
                0)
               (t
                (word-sized-sqrt fixnum))))
        ((word)
         (word-sized-sqrt (unsigned-byte 64)))
        ((unsigned-byte)
         (let* ((bit-length (integer-length n))
                (c (floor (1- bit-length) 2))
                (c-bit-length (integer-length c))
                (d (ash c (- 5 c-bit-length)))
                (m #-64-bit (ash n (- 62 (* 2 c)))
                   #+64-bit (sb-bignum::last-bignum-part=>word (logand bit-length 1) (- (* 2 c) 62) n))
                (a (ash (approximate-isqrt m) (truly-the (integer * 0) (- d 31)))))
           (loop for s from (- c-bit-length 6) downto 0
                 for e = d
                 do
                 (setf d (ash c (- s)))
                 (let* ((shift-amount (- (+ (* 2 c) 1) d e))
                        (n-shifted (ash n (- shift-amount)))
                        (q (truncate n-shifted a)))

                   (setf a (+ (ash a (- d 1 e)) q))))
           (when (< n (* a a))
             (decf a))
           a))))))

;;;; miscellaneous number predicates

(macrolet ((def (name var doc)
             `(defun ,name (,var) ,doc
                (declare (explicit-check))
                (,name ,var))))
  (def zerop number "Is this number zero?")
  (def plusp number "Is this real number strictly positive?")
  (def minusp number "Is this real number strictly negative?")
  (def oddp integer "Is this integer odd?")
  (def evenp integer "Is this integer even?"))

;;;; modular functions
#.
(collect ((forms))
  (flet ((unsigned-definition (name lambda-list width)
           (let ((pattern (1- (ash 1 width))))
             `(defun ,name ,(copy-list lambda-list)
               (flet ((prepare-argument (x)
                        (declare (integer x))
                        (etypecase x
                          ((unsigned-byte ,width) x)
                          (fixnum (logand x ,pattern))
                          (bignum (logand x ,pattern)))))
                 (,name ,@(loop for arg in lambda-list
                                collect `(prepare-argument ,arg)))))))
         (signed-definition (name lambda-list width)
           `(defun ,name ,(copy-list lambda-list)
              (flet ((prepare-argument (x)
                       (declare (integer x))
                       (etypecase x
                         ((signed-byte ,width) x)
                         (bignum (sb-c::mask-signed-field ,width x)))))
                (,name ,@(loop for arg in lambda-list
                               collect `(prepare-argument ,arg)))))))
    (flet ((do-mfuns (class)
             (loop for infos being each hash-value of (sb-c::modular-class-funs class)
                   ;; FIXME: We need to process only "toplevel" functions
                   when (listp infos)
                   do (loop for info in infos
                            for name = (sb-c::modular-fun-info-name info)
                            and width = (sb-c::modular-fun-info-width info)
                            and signedp = (sb-c::modular-fun-info-signedp info)
                            and lambda-list = (sb-c::modular-fun-info-lambda-list info)
                            if signedp
                            do (forms (signed-definition name lambda-list width))
                            else
                            do (forms (unsigned-definition name lambda-list width))))))
      (do-mfuns sb-c::*untagged-unsigned-modular-class*)
      (do-mfuns sb-c::*untagged-signed-modular-class*)
      (do-mfuns sb-c::*tagged-modular-class*)))
  `(progn ,@(sort (forms) #'string< :key #'cadr)))

#-(or 64-bit 64-bit-registers)
(defun sb-vm::ash-left-mod32 (integer amount)
  (etypecase integer
    ((unsigned-byte 32) (ldb (byte 32 0) (ash integer amount)))
    (fixnum (ldb (byte 32 0) (ash (logand integer #xffffffff) amount)))
    (bignum (ldb (byte 32 0) (ash (logand integer #xffffffff) amount)))))
#+(or 64-bit 64-bit-registers)
(defun sb-vm::ash-left-mod64 (integer amount)
  (etypecase integer
    ((unsigned-byte 64) (ldb (byte 64 0) (ash integer amount)))
    (fixnum (ldb (byte 64 0) (ash (logand integer #xffffffffffffffff) amount)))
    (bignum (ldb (byte 64 0)
                 (ash (logand integer #xffffffffffffffff) amount)))))

#+(or x86 x86-64 arm arm64)
(defun sb-vm::ash-left-modfx (integer amount)
  (let ((fixnum-width (- sb-vm:n-word-bits sb-vm:n-fixnum-tag-bits)))
    (etypecase integer
      (fixnum (sb-c::mask-signed-field fixnum-width (ash integer amount)))
      (integer (sb-c::mask-signed-field fixnum-width (ash (sb-c::mask-signed-field fixnum-width integer) amount))))))

(sb-c::when-vop-existsp (:translate sb-vm::ash-modfx)
  (defun sb-vm::ash-mod64 (integer amount)
    (ldb (byte 64 0) (ash integer amount)))

  (defun sb-vm::ash-modfx (integer amount)
    (if (minusp integer)
        (sb-c::mask-signed-field sb-vm:n-fixnum-bits (ash integer amount))
        (logand most-positive-fixnum (ash integer amount)))))

(defun sb-vm::truncate-mod64 (a b)
  (multiple-value-bind (q r) (truncate a b)
    (values (ldb (byte 64 0) q) r)))
