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;;;; This file contains all the irrational functions. (Actually, most
;;;; of the work is done by calling out to C.)
;;;; 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")
;;;; miscellaneous constants, utility functions, and macros
(defconstant pi
#!+long-float 3.14159265358979323846264338327950288419716939937511l0
#!-long-float 3.14159265358979323846264338327950288419716939937511d0)
;;; Make these INLINE, since the call to C is at least as compact as a
;;; Lisp call, and saves number consing to boot.
(eval-when (:compile-toplevel :execute)
(sb!xc:defmacro def-math-rtn (name num-args &optional wrapper)
(let ((function (symbolicate "%" (string-upcase name)))
(args (loop for i below num-args
collect (intern (format nil "ARG~D" i)))))
(declaim (inline ,function))
(defun ,function ,args
(extern-alien ,(format nil "~:[~;sb_~]~a" wrapper name)
(function double-float
,@(loop repeat num-args
collect 'double-float)))
(defun handle-reals (function var)
`((((foreach fixnum single-float bignum ratio))
(coerce (,function (coerce ,var 'double-float)) 'single-float))
(,function ,var))))
(defun handle-complex (form)
`((((foreach (complex double-float) (complex single-float) (complex rational)))
#!+x86 ;; for constant folding
(macrolet ((def (name ll)
`(defun ,name ,ll (,name ,@ll))))
(def %atan2 (x y))
(def %atan (x))
(def %tan (x))
(def %tan-quick (x))
(def %cos (x))
(def %cos-quick (x))
(def %sin (x))
(def %sin-quick (x))
(def %sqrt (x))
(def %log (x))
(def %exp (x)))
#!+(or x86-64 arm-vfp arm64) ;; for constant folding
(macrolet ((def (name ll)
`(defun ,name ,ll (,name ,@ll))))
(def %sqrt (x)))
;;;; stubs for the Unix math library
;;;; Many of these are unnecessary on the X86 because they're built
;;;; into the FPU.
;;; trigonometric
#!-x86 (def-math-rtn "sin" 1)
#!-x86 (def-math-rtn "cos" 1)
#!-x86 (def-math-rtn "tan" 1)
#!-x86 (def-math-rtn "atan" 1)
#!-x86 (def-math-rtn "atan2" 2)
(def-math-rtn "acos" 1 #!+win32 t)
(def-math-rtn "asin" 1 #!+win32 t)
(def-math-rtn "cosh" 1 #!+win32 t)
(def-math-rtn "sinh" 1 #!+win32 t)
(def-math-rtn "tanh" 1 #!+win32 t)
(def-math-rtn "asinh" 1 #!+win32 t)
(def-math-rtn "acosh" 1 #!+win32 t)
(def-math-rtn "atanh" 1 #!+win32 t)
;;; exponential and logarithmic
(def-math-rtn "hypot" 2 #!+win32 t)
#!-x86 (def-math-rtn "exp" 1)
#!-x86 (def-math-rtn "log" 1)
#!-x86 (def-math-rtn "log10" 1)
(def-math-rtn "pow" 2)
#!-(or x86 x86-64 arm-vfp arm64) (def-math-rtn "sqrt" 1)
#!-x86 (def-math-rtn "log1p" 1)
;;;; power functions
(defun exp (number)
"Return e raised to the power NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(handle-reals %exp number)
(* (exp (realpart number))
(cis (imagpart number))))))
;;; INTEXP -- Handle the rational base, integer power case.
(declaim (type (or integer null) *intexp-maximum-exponent*))
(defparameter *intexp-maximum-exponent* nil)
;;; This function precisely calculates base raised to an integral
;;; power. It separates the cases by the sign of power, for efficiency
;;; reasons, as powers can be calculated more efficiently if power is
;;; a positive integer. Values of power are calculated as positive
;;; integers, and inverted if negative.
(defun intexp (base power)
(when (and *intexp-maximum-exponent*
(> (abs power) *intexp-maximum-exponent*))
(error "The absolute value of ~S exceeds ~S."
power '*intexp-maximum-exponent*))
(cond ((minusp power)
(/ (intexp base (- power))))
((eql base 2)
(ash 1 power))
(do ((nextn (ash power -1) (ash power -1))
(total (if (oddp power) base 1)
(if (oddp power) (* base total) total)))
((zerop nextn) total)
(setq base (* base base))
(setq power nextn)))))
;;; If an integer power of a rational, use INTEXP above. Otherwise, do
;;; floating point stuff. If both args are real, we try %POW right
;;; off, assuming it will return 0 if the result may be complex. If
;;; so, we call COMPLEX-POW which directly computes the complex
;;; result. We also separate the complex-real and real-complex cases
;;; from the general complex case.
(defun expt (base power)
"Return BASE raised to the POWER."
(declare (explicit-check))
(if (zerop power)
(if (and (zerop base) (floatp power))
(error 'arguments-out-of-domain-error
:operands (list base power)
:operation 'expt
:references (list '(:ansi-cl :function expt)))
(let ((result (1+ (* base power))))
(if (and (floatp result) (float-nan-p result))
(float 1 result)
(labels (;; determine if the double float is an integer.
;; 0 - not an integer
;; 1 - an odd int
;; 2 - an even int
(isint (ihi lo)
(declare (type (unsigned-byte 31) ihi)
(type (unsigned-byte 32) lo)
(optimize (speed 3) (safety 0)))
(let ((isint 0))
(declare (type fixnum isint))
(cond ((>= ihi #x43400000) ; exponent >= 53
(setq isint 2))
((>= ihi #x3ff00000)
(let ((k (- (ash ihi -20) #x3ff))) ; exponent
(declare (type (mod 53) k))
(cond ((> k 20)
(let* ((shift (- 52 k))
(j (logand (ash lo (- shift))))
(j2 (ash j shift)))
(declare (type (mod 32) shift)
(type (unsigned-byte 32) j j2))
(when (= j2 lo)
(setq isint (- 2 (logand j 1))))))
((= lo 0)
(let* ((shift (- 20 k))
(j (ash ihi (- shift)))
(j2 (ash j shift)))
(declare (type (mod 32) shift)
(type (unsigned-byte 31) j j2))
(when (= j2 ihi)
(setq isint (- 2 (logand j 1))))))))))
(real-expt (x y rtype)
(let ((x (coerce x 'double-float))
(y (coerce y 'double-float)))
(declare (double-float x y))
(let* ((x-hi (double-float-high-bits x))
(x-lo (double-float-low-bits x))
(x-ihi (logand x-hi #x7fffffff))
(y-hi (double-float-high-bits y))
(y-lo (double-float-low-bits y))
(y-ihi (logand y-hi #x7fffffff)))
(declare (type (signed-byte 32) x-hi y-hi)
(type (unsigned-byte 31) x-ihi y-ihi)
(type (unsigned-byte 32) x-lo y-lo))
;; y==zero: x**0 = 1
(when (zerop (logior y-ihi y-lo))
(return-from real-expt (coerce 1d0 rtype)))
;; +-NaN return x+y
;; FIXME: Hardcoded qNaN/sNaN values are not portable.
(when (or (> x-ihi #x7ff00000)
(and (= x-ihi #x7ff00000) (/= x-lo 0))
(> y-ihi #x7ff00000)
(and (= y-ihi #x7ff00000) (/= y-lo 0)))
(return-from real-expt (coerce (+ x y) rtype)))
(let ((yisint (if (< x-hi 0) (isint y-ihi y-lo) 0)))
(declare (type fixnum yisint))
;; special value of y
(when (and (zerop y-lo) (= y-ihi #x7ff00000))
;; y is +-inf
(return-from real-expt
(cond ((and (= x-ihi #x3ff00000) (zerop x-lo))
;; +-1**inf is NaN
(coerce (- y y) rtype))
((>= x-ihi #x3ff00000)
;; (|x|>1)**+-inf = inf,0
(if (>= y-hi 0)
(coerce y rtype)
(coerce 0 rtype)))
;; (|x|<1)**-,+inf = inf,0
(if (< y-hi 0)
(coerce (- y) rtype)
(coerce 0 rtype))))))
(let ((abs-x (abs x)))
(declare (double-float abs-x))
;; special value of x
(when (and (zerop x-lo)
(or (= x-ihi #x7ff00000) (zerop x-ihi)
(= x-ihi #x3ff00000)))
;; x is +-0,+-inf,+-1
(let ((z (if (< y-hi 0)
(/ 1 abs-x) ; z = (1/|x|)
(declare (double-float z))
(when (< x-hi 0)
(cond ((and (= x-ihi #x3ff00000) (zerop yisint))
;; (-1)**non-int
(let ((y*pi (* y pi)))
(declare (double-float y*pi))
(return-from real-expt
(coerce (%cos y*pi) rtype)
(coerce (%sin y*pi) rtype)))))
((= yisint 1)
;; (x<0)**odd = -(|x|**odd)
(setq z (- z)))))
(return-from real-expt (coerce z rtype))))
(if (>= x-hi 0)
;; x>0
(coerce (%pow x y) rtype)
;; x<0
(let ((pow (%pow abs-x y)))
(declare (double-float pow))
(case yisint
(1 ; odd
(coerce (* -1d0 pow) rtype))
(2 ; even
(coerce pow rtype))
(t ; non-integer
(let ((y*pi (* y pi)))
(declare (double-float y*pi))
(coerce (* pow (%cos y*pi))
(coerce (* pow (%sin y*pi))
(complex-expt (base power)
(if (and (zerop base) (plusp (realpart power)))
(* base power)
(exp (* power (log base))))))
(declare (inline real-expt complex-expt))
(number-dispatch ((base number) (power number))
(((foreach fixnum (or bignum ratio) (complex rational)) integer)
(intexp base power))
(((foreach single-float double-float) rational)
(real-expt base power '(dispatch-type base)))
(((foreach fixnum (or bignum ratio) single-float)
(foreach ratio single-float))
(real-expt base power 'single-float))
(((foreach fixnum (or bignum ratio) single-float double-float)
(real-expt base power 'double-float))
((double-float single-float)
(real-expt base power 'double-float))
;; Handle (expt <complex> <rational>), except the case dealt with
;; in the first clause above, (expt <(complex rational)> <integer>).
(((foreach (complex rational))
(* (expt (abs base) power)
(cis (* power (phase base)))))
(((foreach (complex single-float) (complex double-float))
(foreach fixnum (or bignum ratio)))
(* (expt (abs base) power)
(cis (* power (phase base)))))
;; The next three clauses handle (expt <real> <complex>).
(((foreach fixnum (or bignum ratio) single-float)
(foreach (complex single-float) (complex rational)))
(complex-expt base power))
(((foreach fixnum (or bignum ratio) single-float)
(complex double-float))
(complex-expt (coerce base 'double-float) power))
((double-float complex)
(complex-expt base power))
;; The next three clauses handle (expt <complex> <float>) and
;; (expt <complex> <complex>).
(((foreach (complex single-float) (complex rational))
(foreach (complex single-float) (complex rational) single-float))
(complex-expt base power))
(((foreach (complex single-float) (complex rational))
(foreach (complex double-float) double-float))
(complex-expt (coerce base '(complex double-float)) power))
(((complex double-float)
(foreach complex double-float single-float))
(complex-expt base power))))))
;;; FIXME: Maybe rename this so that it's clearer that it only works
;;; on integers?
(defun log2 (x)
(declare (type integer x))
;; CMUCL comment:
;; Write x = 2^n*f where 1/2 < f <= 1. Then log2(x) = n +
;; log2(f). So we grab the top few bits of x and scale that
;; appropriately, take the log of it and add it to n.
;; Motivated by an attempt to get LOG to work better on bignums.
(let ((n (integer-length x)))
(if (< n sb!vm:double-float-digits)
(log (coerce x 'double-float) 2.0d0)
(let ((f (ldb (byte sb!vm:double-float-digits
(- n sb!vm:double-float-digits))
(+ n (log (scale-float (coerce f 'double-float)
(- sb!vm:double-float-digits))
(defun log (number &optional (base nil base-p))
"Return the logarithm of NUMBER in the base BASE, which defaults to e."
(declare (explicit-check))
(if base-p
((zerop base)
(if (or (typep number 'double-float) (typep base 'double-float))
((and (typep number '(integer (0) *))
(typep base '(integer (0) *)))
(coerce (/ (log2 number) (log2 base)) 'single-float))
((and (typep number 'integer) (typep base 'double-float))
;; No single float intermediate result
(/ (log2 number) (log base 2.0d0)))
((and (typep number 'double-float) (typep base 'integer))
(/ (log number 2.0d0) (log2 base)))
(/ (log number) (log base))))
(number-dispatch ((number number))
(((foreach fixnum bignum))
(if (minusp number)
(complex (log (- number)) (coerce pi 'single-float))
(coerce (/ (log2 number) (log (exp 1.0d0) 2.0d0)) 'single-float)))
(if (minusp number)
(complex (log (- number)) (coerce pi 'single-float))
(let ((numerator (numerator number))
(denominator (denominator number)))
(if (= (integer-length numerator)
(integer-length denominator))
(coerce (%log1p (coerce (- number 1) 'double-float))
(coerce (/ (- (log2 numerator) (log2 denominator))
(log (exp 1.0d0) 2.0d0))
(((foreach single-float double-float))
;; Is (log -0) -infinity (libm.a) or -infinity + i*pi (Kahan)?
;; Since this doesn't seem to be an implementation issue
;; I (pw) take the Kahan result.
(if (< (float-sign number)
(coerce 0 '(dispatch-type number)))
(complex (log (- number)) (coerce pi '(dispatch-type number)))
(coerce (%log (coerce number 'double-float))
'(dispatch-type number))))
(complex-log number)))))
(defun sqrt (number)
"Return the square root of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(((foreach fixnum bignum ratio))
(if (minusp number)
(complex-sqrt number)
(coerce (%sqrt (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (minusp number)
(complex-sqrt (complex number))
(coerce (%sqrt (coerce number 'double-float))
'(dispatch-type number))))
(complex-sqrt number))))
;;;; trigonometic and related functions
(defun abs (number)
"Return the absolute value of the number."
(declare (explicit-check))
(number-dispatch ((number number))
(((foreach single-float double-float fixnum rational))
(abs number))
(let ((rx (realpart number))
(ix (imagpart number)))
(etypecase rx
(sqrt (+ (* rx rx) (* ix ix))))
(coerce (%hypot (coerce rx 'double-float)
(coerce (truly-the single-float ix) 'double-float))
(%hypot rx (truly-the double-float ix))))))))
(defun phase (number)
"Return the angle part of the polar representation of a complex number.
For complex numbers, this is (atan (imagpart number) (realpart number)).
For non-complex positive numbers, this is 0. For non-complex negative
numbers this is PI."
(declare (explicit-check))
(number-dispatch ((number number))
(if (minusp number)
(coerce pi 'single-float)
(if (minusp (float-sign number))
(coerce pi 'single-float)
(if (minusp (float-sign number))
(coerce pi 'double-float)
(atan (imagpart number) (realpart number)))))
(defun sin (number)
"Return the sine of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(handle-reals %sin number)
(let ((x (realpart number))
(y (imagpart number)))
(complex (* (sin x) (cosh y))
(* (cos x) (sinh y)))))))
(defun cos (number)
"Return the cosine of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(handle-reals %cos number)
(let ((x (realpart number))
(y (imagpart number)))
(complex (* (cos x) (cosh y))
(- (* (sin x) (sinh y))))))))
(defun tan (number)
"Return the tangent of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(handle-reals %tan number)
;; tan z = -i * tanh(i*z)
(let* ((result (complex-tanh (complex (- (imagpart number))
(realpart number)))))
(complex (imagpart result)
(- (realpart result)))))))
(defun cis (theta)
"Return cos(Theta) + i sin(Theta), i.e. exp(i Theta)."
(declare (explicit-check ))
(number-dispatch ((theta real))
(((foreach single-float double-float rational))
(complex (cos theta) (sin theta)))))
(defun asin (number)
"Return the arc sine of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(if (or (> number 1) (< number -1))
(complex-asin number)
(coerce (%asin (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (or (> number (coerce 1 '(dispatch-type number)))
(< number (coerce -1 '(dispatch-type number))))
(complex-asin (complex number))
(coerce (%asin (coerce number 'double-float))
'(dispatch-type number))))
(complex-asin number))))
(defun acos (number)
"Return the arc cosine of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(if (or (> number 1) (< number -1))
(complex-acos number)
(coerce (%acos (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (or (> number (coerce 1 '(dispatch-type number)))
(< number (coerce -1 '(dispatch-type number))))
(complex-acos (complex number))
(coerce (%acos (coerce number 'double-float))
'(dispatch-type number))))
(complex-acos number))))
(defun atan (y &optional (x nil xp))
"Return the arc tangent of Y if X is omitted or Y/X if X is supplied."
(declare (explicit-check))
(if xp
(flet ((atan2 (y x)
(declare (type double-float y x)
(values double-float))
(if (zerop x)
(if (zerop y)
(if (plusp (float-sign x))
(float-sign y pi))
(float-sign y (/ pi 2)))
(%atan2 y x))))
(number-dispatch ((y real) (x real))
(foreach double-float single-float fixnum bignum ratio))
(atan2 y (coerce x 'double-float)))
(((foreach single-float fixnum bignum ratio)
(atan2 (coerce y 'double-float) x))
(((foreach single-float fixnum bignum ratio)
(foreach single-float fixnum bignum ratio))
(coerce (atan2 (coerce y 'double-float) (coerce x 'double-float))
(number-dispatch ((y number))
(handle-reals %atan y)
(complex-atan y)))))
;;; It seems that every target system has a C version of sinh, cosh,
;;; and tanh. Let's use these for reals because the original
;;; implementations based on the definitions lose big in round-off
;;; error. These bad definitions also mean that sin and cos for
;;; complex numbers can also lose big.
(defun sinh (number)
"Return the hyperbolic sine of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(handle-reals %sinh number)
(let ((x (realpart number))
(y (imagpart number)))
(complex (* (sinh x) (cos y))
(* (cosh x) (sin y)))))))
(defun cosh (number)
"Return the hyperbolic cosine of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(handle-reals %cosh number)
(let ((x (realpart number))
(y (imagpart number)))
(complex (* (cosh x) (cos y))
(* (sinh x) (sin y)))))))
(defun tanh (number)
"Return the hyperbolic tangent of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(handle-reals %tanh number)
(complex-tanh number))))
(defun asinh (number)
"Return the hyperbolic arc sine of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
(handle-reals %asinh number)
(complex-asinh number))))
(defun acosh (number)
"Return the hyperbolic arc cosine of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
;; acosh is complex if number < 1
(if (< number 1)
(complex-acosh number)
(coerce (%acosh (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (< number (coerce 1 '(dispatch-type number)))
(complex-acosh (complex number))
(coerce (%acosh (coerce number 'double-float))
'(dispatch-type number))))
(complex-acosh number))))
(defun atanh (number)
"Return the hyperbolic arc tangent of NUMBER."
(declare (explicit-check))
(number-dispatch ((number number))
;; atanh is complex if |number| > 1
(if (or (> number 1) (< number -1))
(complex-atanh number)
(coerce (%atanh (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (or (> number (coerce 1 '(dispatch-type number)))
(< number (coerce -1 '(dispatch-type number))))
(complex-atanh (complex number))
(coerce (%atanh (coerce number 'double-float))
'(dispatch-type number))))
(complex-atanh number))))
;;;; not-OLD-SPECFUN stuff
;;;; (This was conditional on #-OLD-SPECFUN in the CMU CL sources,
;;;; but OLD-SPECFUN was mentioned nowhere else, so it seems to be
;;;; the standard special function system.)
;;;; This is a set of routines that implement many elementary
;;;; transcendental functions as specified by ANSI Common Lisp. The
;;;; implementation is based on Kahan's paper.
;;;; I believe I have accurately implemented the routines and are
;;;; correct, but you may want to check for your self.
;;;; These functions are written for CMU Lisp and take advantage of
;;;; some of the features available there. It may be possible,
;;;; however, to port this to other Lisps.
;;;; Some functions are significantly more accurate than the original
;;;; definitions in CMU Lisp. In fact, some functions in CMU Lisp
;;;; give the wrong answer like (acos #c(-2.0 0.0)), where the true
;;;; answer is pi + i*log(2-sqrt(3)).
;;;; All of the implemented functions will take any number for an
;;;; input, but the result will always be a either a complex
;;;; single-float or a complex double-float.
;;;; general functions:
;;;; complex-sqrt
;;;; complex-log
;;;; complex-atanh
;;;; complex-tanh
;;;; complex-acos
;;;; complex-acosh
;;;; complex-asin
;;;; complex-asinh
;;;; complex-atan
;;;; utility functions:
;;;; logb
;;;; internal functions:
;;;; square coerce-to-complex-type cssqs complex-log-scaled
;;;; references:
;;;; Kahan, W. "Branch Cuts for Complex Elementary Functions, or Much
;;;; Ado About Nothing's Sign Bit" in Iserles and Powell (eds.) "The
;;;; State of the Art in Numerical Analysis", pp. 165-211, Clarendon
;;;; Press, 1987
;;;; The original CMU CL code requested:
;;;; Please send any bug reports, comments, or improvements to
;;;; Raymond Toy at <email address deleted during 2002 spam avalanche>.
;;; FIXME: In SBCL, the floating point infinity constants like
;;; constants at cross-compile time, because the cross-compilation
;;; host might not have support for floating point infinities. Thus,
;;; they're effectively implemented as special variable references,
;;; and the code below which uses them might be unnecessarily
;;; inefficient. Perhaps some sort of MAKE-LOAD-TIME-VALUE hackery
;;; should be used instead? (KLUDGED 2004-03-08 CSR, by replacing the
;;; special variable references with (probably equally slow)
;;; constructors)
;;; FIXME: As of 2004-05, when PFD noted that IMAGPART and COMPLEX
;;; differ in their interpretations of the real line, IMAGPART was
;;; patch, which without a certain amount of effort would have altered
;;; all the branch cut treatment. Clients of these COMPLEX- routines
;;; were patched to use explicit COMPLEX, rather than implicitly
;;; passing in real numbers for treatment with IMAGPART, and these
;;; COMPLEX- functions altered to require arguments of type COMPLEX;
;;; however, someone needs to go back to Kahan for the definitive
;;; answer for treatment of negative real floating point numbers and
;;; branch cuts. If adjustment is needed, it is probably the removal
;;; of explicit calls to COMPLEX in the clients of irrational
;;; functions. -- a slightly bitter CSR, 2004-05-16
(declaim (inline square))
(defun square (x)
(declare (double-float x))
(* x x))
;;; original CMU CL comment, apparently re. LOGB and
;;; perhaps CSSQS:
;;; If you have these functions in libm, perhaps they should be used
;;; instead of these Lisp versions. These versions are probably good
;;; enough, especially since they are portable.
;;; This is like LOGB, but X is not infinity and non-zero and not a
;;; NaN, so we can always return an integer.
(declaim (inline logb-finite))
(defun logb-finite (x)
(declare (type double-float x))
(multiple-value-bind (signif exponent sign)
(decode-float x)
(declare (ignore signif sign))
;; DECODE-FLOAT is almost right, except that the exponent is off
;; by one.
(1- exponent)))
;;; Compute an integer N such that 1 <= |2^N * x| < 2.
;;; For the special cases, the following values are used:
;;; x logb
;;; NaN NaN
;;; +/- infinity +infinity
;;; 0 -infinity
(defun logb (x)
(declare (type double-float x))
(cond ((float-nan-p x)
((float-infinity-p x)
(double-from-bits 0 (1+ sb!vm:double-float-normal-exponent-max) 0))
((zerop x)
;; The answer is negative infinity, but we are supposed to
;; signal divide-by-zero, so do the actual division
(/ -1.0d0 x)
(logb-finite x))))
;;; This function is used to create a complex number of the
;;; appropriate type:
;;; Create complex number with real part X and imaginary part Y
;;; such that has the same type as Z. If Z has type (complex
;;; rational), the X and Y are coerced to single-float.
#!+long-float (eval-when (:compile-toplevel :load-toplevel :execute)
(error "needs work for long float support"))
(declaim (inline coerce-to-complex-type))
(defun coerce-to-complex-type (x y z)
(declare (double-float x y)
(number z))
(if (typep (realpart z) 'double-float)
(complex x y)
;; Convert anything that's not already a DOUBLE-FLOAT (because
;; the initial argument was a (COMPLEX DOUBLE-FLOAT) and we
;; haven't done anything to lose precision) to a SINGLE-FLOAT.
(complex (float x 1f0)
(float y 1f0))))
;;; Compute |(x+i*y)/2^k|^2 scaled to avoid over/underflow. The
;;; result is r + i*k, where k is an integer.
#!+long-float (eval-when (:compile-toplevel :load-toplevel :execute)
(error "needs work for long float support"))
(defun cssqs (z)
(declare (muffle-conditions t))
(let ((x (float (realpart z) 1d0))
(y (float (imagpart z) 1d0)))
;; Would this be better handled using an exception handler to
;; catch the overflow or underflow signal? For now, we turn all
;; traps off and look at the accrued exceptions to see if any
;; signal would have been raised.
(with-float-traps-masked (:underflow :overflow)
(let ((rho (+ (square x) (square y))))
(declare (optimize (speed 3) (space 0)))
(cond ((and (or (float-nan-p rho)
(float-infinity-p rho))
(or (float-infinity-p (abs x))
(float-infinity-p (abs y))))
(double-from-bits 0 (1+ sb!vm:double-float-normal-exponent-max) 0)
((let ((threshold
;; (/ least-positive-double-float double-float-epsilon)
(make-double-float #x1fffff #xfffffffe)
(error "(/ least-positive-long-float long-float-epsilon)")))
(traps (ldb sb!vm::float-sticky-bits
;; Overflow raised or (underflow raised and rho <
;; lambda/eps)
(or (not (zerop (logand sb!vm:float-overflow-trap-bit traps)))
(and (not (zerop (logand sb!vm:float-underflow-trap-bit
(< rho threshold))))
;; If we're here, neither x nor y are infinity and at
;; least one is non-zero.. Thus logb returns a nice
;; integer.
(let ((k (- (logb-finite (max (abs x) (abs y))))))
(values (+ (square (scale-float x k))
(square (scale-float y k)))
(- k))))
(values rho 0)))))))
;;; principal square root of Z
;;; Z may be RATIONAL or COMPLEX; the result is always a COMPLEX.
(defun complex-sqrt (z)
;; KLUDGE: Here and below, we can't just declare Z to be of type
;; COMPLEX, because one-arg COMPLEX on rationals returns a rational.
;; Since there isn't a rational negative zero, this is OK from the
;; point of view of getting the right answer in the face of branch
;; cuts, but declarations of the form (OR RATIONAL COMPLEX) are
;; still ugly. -- CSR, 2004-05-16
(declare (type (or complex rational) z))
(multiple-value-bind (rho k)
(cssqs z)
(declare (type (or (member 0d0) (double-float 0d0)) rho)
(type fixnum k))
(let ((x (float (realpart z) 1.0d0))
(y (float (imagpart z) 1.0d0))
(eta 0d0)
(nu 0d0))
(declare (double-float x y eta nu)
;; get maybe-inline functions inlined.
(optimize (space 0)))
(if (not (float-nan-p x))
(setf rho (+ (scale-float (abs x) (- k)) (sqrt rho))))
(cond ((oddp k)
(setf k (ash k -1)))
(setf k (1- (ash k -1)))
(setf rho (+ rho rho))))
(setf rho (scale-float (sqrt rho) k))
(setf eta rho)
(setf nu y)
(when (/= rho 0d0)
(when (not (float-infinity-p (abs nu)))
(setf nu (/ (/ nu rho) 2d0)))
(when (< x 0d0)
(setf eta (abs nu))
(setf nu (float-sign y rho))))
(coerce-to-complex-type eta nu z))))
;;; Compute log(2^j*z).
;;; This is for use with J /= 0 only when |z| is huge.
(defun complex-log-scaled (z j)
(declare (muffle-conditions t))
(declare (type (or rational complex) z)
(fixnum j))
;; The constants t0, t1, t2 should be evaluated to machine
;; precision. In addition, Kahan says the accuracy of log1p
;; influences the choices of these constants but doesn't say how to
;; choose them. We'll just assume his choices matches our
;; implementation of log1p.
(let ((t0 (load-time-value
(make-double-float #x3fe6a09e #x667f3bcd)
(error "(/ (sqrt 2l0))")))
;; KLUDGE: if repeatable fasls start failing under some weird
;; xc host, this 1.2d0 might be a good place to examine: while
;; it _should_ be the same in all vaguely-IEEE754 hosts, 1.2
;; is not exactly representable, so something could go wrong.
(t1 1.2d0)
(t2 3d0)
(ln2 (load-time-value
(make-double-float #x3fe62e42 #xfefa39ef)
(error "(log 2l0)")))
(x (float (realpart z) 1.0d0))
(y (float (imagpart z) 1.0d0)))
(multiple-value-bind (rho k)
(cssqs z)
(declare (optimize (speed 3)))
(let ((beta (max (abs x) (abs y)))
(theta (min (abs x) (abs y))))
(coerce-to-complex-type (if (and (zerop k)
(< t0 beta)
(or (<= beta t1)
(< rho t2)))
(/ (%log1p (+ (* (- beta 1.0d0)
(+ beta 1.0d0))
(* theta theta)))
(+ (/ (log rho) 2d0)
(* (+ k j) ln2)))
(atan y x)
;;; log of Z = log |Z| + i * arg Z
;;; Z may be any number, but the result is always a complex.
(defun complex-log (z)
(declare (type (or rational complex) z))
(complex-log-scaled z 0))
;;; KLUDGE: Let us note the following "strange" behavior. atanh 1.0d0
;;; is +infinity, but the following code returns approx 176 + i*pi/4.
;;; The reason for the imaginary part is caused by the fact that arg
;;; i*y is never 0 since we have positive and negative zeroes. -- rtoy
;;; Compute atanh z = (log(1+z) - log(1-z))/2.
(defun complex-atanh (z)
(declare (muffle-conditions t))
(declare (type (or rational complex) z))
(let* (;; constants
(theta (/ (sqrt most-positive-double-float) 4.0d0))
(rho (/ 4.0d0 (sqrt most-positive-double-float)))
(half-pi (/ pi 2.0d0))
(rp (float (realpart z) 1.0d0))
(beta (float-sign rp 1.0d0))
(x (* beta rp))
(y (* beta (- (float (imagpart z) 1.0d0))))
(eta 0.0d0)
(nu 0.0d0))
;; Shouldn't need this declare.
(declare (double-float x y))
(declare (optimize (speed 3)))
(cond ((or (> x theta)
(> (abs y) theta))
;; To avoid overflow...
(setf nu (float-sign y half-pi))
;; ETA is real part of 1/(x + iy). This is x/(x^2+y^2),
;; which can cause overflow. Arrange this computation so
;; that it won't overflow.
(setf eta (let* ((x-bigger (> x (abs y)))
(r (if x-bigger (/ y x) (/ x y)))
(d (+ 1.0d0 (* r r))))
(if x-bigger
(/ (/ x) d)
(/ (/ r y) d)))))
((= x 1.0d0)
;; Should this be changed so that if y is zero, eta is set
;; to +infinity instead of approx 176? In any case
;; tanh(176) is 1.0d0 within working precision.
(let ((t1 (+ 4d0 (square y)))
(t2 (+ (abs y) rho)))
(setf eta (log (/ (sqrt (sqrt t1))
(sqrt t2))))
(setf nu (* 0.5d0
(float-sign y
(+ half-pi (atan (* 0.5d0 t2))))))))
(let ((t1 (+ (abs y) rho)))
;; Normal case using log1p(x) = log(1 + x)
(setf eta (* 0.25d0
(%log1p (/ (* 4.0d0 x)
(+ (square (- 1.0d0 x))
(square t1))))))
(setf nu (* 0.5d0
(atan (* 2.0d0 y)
(- (* (- 1.0d0 x)
(+ 1.0d0 x))
(square t1))))))))
(coerce-to-complex-type (* beta eta)
(- (* beta nu))
;;; Compute tanh z = sinh z / cosh z.
(defun complex-tanh (z)
(declare (muffle-conditions t))
(declare (type (or rational complex) z))
(let ((x (float (realpart z) 1.0d0))
(y (float (imagpart z) 1.0d0)))
;; space 0 to get maybe-inline functions inlined
(declare (optimize (speed 3) (space 0)))
(cond ((> (abs x)
(make-double-float #x406633ce #x8fb9f87e)
(error "(/ (+ (log 2l0) (log most-positive-long-float)) 4l0)")))
(coerce-to-complex-type (float-sign x)
(float-sign y) z))
(let* ((tv (%tan y))
(beta (+ 1.0d0 (* tv tv)))
(s (sinh x))
(rho (sqrt (+ 1.0d0 (* s s)))))
(if (float-infinity-p (abs tv))
(coerce-to-complex-type (/ rho s)
(/ tv)
(let ((den (+ 1.0d0 (* beta s s))))
(coerce-to-complex-type (/ (* beta rho s)
(/ tv den)
;;; Compute acos z = pi/2 - asin z.
;;; Z may be any NUMBER, but the result is always a COMPLEX.
(defun complex-acos (z)
;; Kahan says we should only compute the parts needed. Thus, the
;; REALPART's below should only compute the real part, not the whole
;; complex expression. Doing this can be important because we may get
;; spurious signals that occur in the part that we are not using.
;; However, we take a pragmatic approach and just use the whole
;; expression.
;; NOTE: The formula given by Kahan is somewhat ambiguous in whether
;; it's the conjugate of the square root or the square root of the
;; conjugate. This needs to be checked.
;; I checked. It doesn't matter because (conjugate (sqrt z)) is the
;; same as (sqrt (conjugate z)) for all z. This follows because
;; (conjugate (sqrt z)) = exp(0.5*log |z|)*exp(-0.5*j*arg z).
;; (sqrt (conjugate z)) = exp(0.5*log|z|)*exp(0.5*j*arg conj z)
;; and these two expressions are equal if and only if arg conj z =
;; -arg z, which is clearly true for all z.
(declare (type (or rational complex) z))
(let ((sqrt-1+z (complex-sqrt (+ 1 z)))
(sqrt-1-z (complex-sqrt (- 1 z))))
(with-float-traps-masked (:divide-by-zero)
(complex (* 2 (atan (/ (realpart sqrt-1-z)
(realpart sqrt-1+z))))
(asinh (imagpart (* (conjugate sqrt-1+z)
;;; Compute acosh z = 2 * log(sqrt((z+1)/2) + sqrt((z-1)/2))
;;; Z may be any NUMBER, but the result is always a COMPLEX.
(defun complex-acosh (z)
(declare (type (or rational complex) z))
(let ((sqrt-z-1 (complex-sqrt (- z 1)))
(sqrt-z+1 (complex-sqrt (+ z 1))))
(with-float-traps-masked (:divide-by-zero)
(complex (asinh (realpart (* (conjugate sqrt-z-1)
(* 2 (atan (/ (imagpart sqrt-z-1)
(realpart sqrt-z+1))))))))
;;; Compute asin z = asinh(i*z)/i.
;;; Z may be any NUMBER, but the result is always a COMPLEX.
(defun complex-asin (z)
(declare (type (or rational complex) z))
(let ((sqrt-1-z (complex-sqrt (- 1 z)))
(sqrt-1+z (complex-sqrt (+ 1 z))))
(with-float-traps-masked (:divide-by-zero)
(complex (atan (/ (realpart z)
(realpart (* sqrt-1-z sqrt-1+z))))
(asinh (imagpart (* (conjugate sqrt-1-z)
;;; Compute asinh z = log(z + sqrt(1 + z*z)).
;;; Z may be any number, but the result is always a complex.
(defun complex-asinh (z)
(declare (type (or rational complex) z))
;; asinh z = -i * asin (i*z)
(let* ((iz (complex (- (imagpart z)) (realpart z)))
(result (complex-asin iz)))
(complex (imagpart result)
(- (realpart result)))))
;;; Compute atan z = atanh (i*z) / i.
;;; Z may be any number, but the result is always a complex.
(defun complex-atan (z)
(declare (type (or rational complex) z))
;; atan z = -i * atanh (i*z)
(let* ((iz (complex (- (imagpart z)) (realpart z)))
(result (complex-atanh iz)))
(complex (imagpart result)
(- (realpart result)))))