1 ;;;; This file contains all the irrational functions. (Actually, most
2 ;;;; of the work is done by calling out to C.)
4 ;;;; This software is part of the SBCL system. See the README file for
7 ;;;; This software is derived from the CMU CL system, which was
8 ;;;; written at Carnegie Mellon University and released into the
9 ;;;; public domain. The software is in the public domain and is
10 ;;;; provided with absolutely no warranty. See the COPYING and CREDITS
11 ;;;; files for more information.
13 (in-package "SB!KERNEL")
15 ;;;; miscellaneous constants, utility functions, and macros
17 (defconstant pi 3.14159265358979323846264338327950288419716939937511L0)
18 ;(defconstant e 2.71828182845904523536028747135266249775724709369996L0)
20 ;;; Make these INLINE, since the call to C is at least as compact as a
21 ;;; Lisp call, and saves number consing to boot.
22 (eval-when (:compile-toplevel :execute)
24 (sb!xc:defmacro def-math-rtn (name num-args)
25 (let ((function (symbolicate "%" (string-upcase name))))
27 (proclaim '(inline ,function))
28 (sb!alien:def-alien-routine (,name ,function) double-float
29 ,@(let ((results nil))
30 (dotimes (i num-args (nreverse results))
31 (push (list (intern (format nil "ARG-~D" i))
35 (defun handle-reals (function var)
36 `((((foreach fixnum single-float bignum ratio))
37 (coerce (,function (coerce ,var 'double-float)) 'single-float))
43 ;;;; stubs for the Unix math library
45 ;;; Please refer to the Unix man pages for details about these routines.
48 #!-x86 (def-math-rtn "sin" 1)
49 #!-x86 (def-math-rtn "cos" 1)
50 #!-x86 (def-math-rtn "tan" 1)
51 (def-math-rtn "asin" 1)
52 (def-math-rtn "acos" 1)
53 #!-x86 (def-math-rtn "atan" 1)
54 #!-x86 (def-math-rtn "atan2" 2)
55 (def-math-rtn "sinh" 1)
56 (def-math-rtn "cosh" 1)
57 (def-math-rtn "tanh" 1)
58 (def-math-rtn "asinh" 1)
59 (def-math-rtn "acosh" 1)
60 (def-math-rtn "atanh" 1)
62 ;;; Exponential and Logarithmic.
63 #!-x86 (def-math-rtn "exp" 1)
64 #!-x86 (def-math-rtn "log" 1)
65 #!-x86 (def-math-rtn "log10" 1)
66 (def-math-rtn "pow" 2)
67 #!-x86 (def-math-rtn "sqrt" 1)
68 (def-math-rtn "hypot" 2)
69 #!-(or hpux x86) (def-math-rtn "log1p" 1)
71 #!+x86 ;; These are needed for use by byte-compiled files.
74 (declare (double-float x)
75 (values double-float))
78 (declare (double-float x)
79 (values double-float))
82 (declare (double-float x)
83 (values double-float))
86 (declare (double-float x)
87 (values double-float))
90 (declare (double-float x)
91 (values double-float))
94 (declare (double-float x)
95 (values double-float))
98 (declare (double-float x)
99 (values double-float))
102 (declare (double-float x y)
103 (values double-float))
106 (declare (double-float x)
107 (values double-float))
110 (declare (double-float x)
111 (values double-float))
114 (declare (double-float x)
115 (values double-float))
119 (declare (type (double-float 0d0) x)
121 (values (double-float 0d0)))
124 (declare (double-float x)
125 (values double-float))
127 (defun %scalbn (f ex)
128 (declare (double-float f)
129 (type (signed-byte 32) ex)
130 (values double-float))
133 (declare (double-float f ex)
134 (values double-float))
137 (declare (double-float x)
138 (values double-float))
141 (declare (double-float x)
142 (values double-float))
150 "Return e raised to the power NUMBER."
151 (number-dispatch ((number number))
152 (handle-reals %exp number)
154 (* (exp (realpart number))
155 (cis (imagpart number))))))
157 ;;; INTEXP -- Handle the rational base, integer power case.
159 ;;; FIXME: As long as the
160 ;;; system dies on stack overflow or memory exhaustion, it seems reasonable
161 ;;; to have this, but its default should be NIL, and when it's NIL,
162 ;;; anything should be accepted.
163 (defparameter *intexp-maximum-exponent* 10000)
165 ;;; This function precisely calculates base raised to an integral power. It
166 ;;; separates the cases by the sign of power, for efficiency reasons, as powers
167 ;;; can be calculated more efficiently if power is a positive integer. Values
168 ;;; of power are calculated as positive integers, and inverted if negative.
169 (defun intexp (base power)
170 (when (> (abs power) *intexp-maximum-exponent*)
171 ;; FIXME: should be ordinary error, not CERROR. (Once we set the
172 ;; default for the variable to NIL, the un-continuable error will
173 ;; be less obnoxious.)
174 (cerror "Continue with calculation."
175 "The absolute value of ~S exceeds ~S."
176 power '*intexp-maximum-exponent* base power))
177 (cond ((minusp power)
178 (/ (intexp base (- power))))
182 (do ((nextn (ash power -1) (ash power -1))
183 (total (if (oddp power) base 1)
184 (if (oddp power) (* base total) total)))
185 ((zerop nextn) total)
186 (setq base (* base base))
187 (setq power nextn)))))
189 ;;; If an integer power of a rational, use INTEXP above. Otherwise, do
190 ;;; floating point stuff. If both args are real, we try %POW right off,
191 ;;; assuming it will return 0 if the result may be complex. If so, we call
192 ;;; COMPLEX-POW which directly computes the complex result. We also separate
193 ;;; the complex-real and real-complex cases from the general complex case.
194 (defun expt (base power)
196 "Returns BASE raised to the POWER."
199 (labels (;; determine if the double float is an integer.
200 ;; 0 - not an integer
204 (declare (type (unsigned-byte 31) ihi)
205 (type (unsigned-byte 32) lo)
206 (optimize (speed 3) (safety 0)))
208 (declare (type fixnum isint))
209 (cond ((>= ihi #x43400000) ; exponent >= 53
212 (let ((k (- (ash ihi -20) #x3ff))) ; exponent
213 (declare (type (mod 53) k))
215 (let* ((shift (- 52 k))
216 (j (logand (ash lo (- shift))))
218 (declare (type (mod 32) shift)
219 (type (unsigned-byte 32) j j2))
221 (setq isint (- 2 (logand j 1))))))
223 (let* ((shift (- 20 k))
224 (j (ash ihi (- shift)))
226 (declare (type (mod 32) shift)
227 (type (unsigned-byte 31) j j2))
229 (setq isint (- 2 (logand j 1))))))))))
231 (real-expt (x y rtype)
232 (let ((x (coerce x 'double-float))
233 (y (coerce y 'double-float)))
234 (declare (double-float x y))
235 (let* ((x-hi (sb!kernel:double-float-high-bits x))
236 (x-lo (sb!kernel:double-float-low-bits x))
237 (x-ihi (logand x-hi #x7fffffff))
238 (y-hi (sb!kernel:double-float-high-bits y))
239 (y-lo (sb!kernel:double-float-low-bits y))
240 (y-ihi (logand y-hi #x7fffffff)))
241 (declare (type (signed-byte 32) x-hi y-hi)
242 (type (unsigned-byte 31) x-ihi y-ihi)
243 (type (unsigned-byte 32) x-lo y-lo))
245 (when (zerop (logior y-ihi y-lo))
246 (return-from real-expt (coerce 1d0 rtype)))
248 (when (or (> x-ihi #x7ff00000)
249 (and (= x-ihi #x7ff00000) (/= x-lo 0))
251 (and (= y-ihi #x7ff00000) (/= y-lo 0)))
252 (return-from real-expt (coerce (+ x y) rtype)))
253 (let ((yisint (if (< x-hi 0) (isint y-ihi y-lo) 0)))
254 (declare (type fixnum yisint))
255 ;; special value of y
256 (when (and (zerop y-lo) (= y-ihi #x7ff00000))
258 (return-from real-expt
259 (cond ((and (= x-ihi #x3ff00000) (zerop x-lo))
261 (coerce (- y y) rtype))
262 ((>= x-ihi #x3ff00000)
263 ;; (|x|>1)**+-inf = inf,0
268 ;; (|x|<1)**-,+inf = inf,0
271 (coerce 0 rtype))))))
273 (let ((abs-x (abs x)))
274 (declare (double-float abs-x))
275 ;; special value of x
276 (when (and (zerop x-lo)
277 (or (= x-ihi #x7ff00000) (zerop x-ihi)
278 (= x-ihi #x3ff00000)))
279 ;; x is +-0,+-inf,+-1
280 (let ((z (if (< y-hi 0)
281 (/ 1 abs-x) ; z = (1/|x|)
283 (declare (double-float z))
285 (cond ((and (= x-ihi #x3ff00000) (zerop yisint))
287 (let ((y*pi (* y pi)))
288 (declare (double-float y*pi))
289 (return-from real-expt
291 (coerce (%cos y*pi) rtype)
292 (coerce (%sin y*pi) rtype)))))
294 ;; (x<0)**odd = -(|x|**odd)
296 (return-from real-expt (coerce z rtype))))
300 (coerce (sb!kernel::%pow x y) rtype)
302 (let ((pow (sb!kernel::%pow abs-x y)))
303 (declare (double-float pow))
306 (coerce (* -1d0 pow) rtype))
310 (let ((y*pi (* y pi)))
311 (declare (double-float y*pi))
313 (coerce (* pow (%cos y*pi)) rtype)
314 (coerce (* pow (%sin y*pi)) rtype)))))))))))))
315 (declare (inline real-expt))
316 (number-dispatch ((base number) (power number))
317 (((foreach fixnum (or bignum ratio) (complex rational)) integer)
319 (((foreach single-float double-float) rational)
320 (real-expt base power '(dispatch-type base)))
321 (((foreach fixnum (or bignum ratio) single-float)
322 (foreach ratio single-float))
323 (real-expt base power 'single-float))
324 (((foreach fixnum (or bignum ratio) single-float double-float)
326 (real-expt base power 'double-float))
327 ((double-float single-float)
328 (real-expt base power 'double-float))
329 (((foreach (complex rational) (complex float)) rational)
330 (* (expt (abs base) power)
331 (cis (* power (phase base)))))
332 (((foreach fixnum (or bignum ratio) single-float double-float)
334 (if (and (zerop base) (plusp (realpart power)))
336 (exp (* power (log base)))))
337 (((foreach (complex float) (complex rational))
338 (foreach complex double-float single-float))
339 (if (and (zerop base) (plusp (realpart power)))
341 (exp (* power (log base)))))))))
343 (defun log (number &optional (base nil base-p))
345 "Return the logarithm of NUMBER in the base BASE, which defaults to e."
349 (/ (log number) (log base)))
350 (number-dispatch ((number number))
351 (((foreach fixnum bignum ratio))
353 (complex (log (- number)) (coerce pi 'single-float))
354 (coerce (%log (coerce number 'double-float)) 'single-float)))
355 (((foreach single-float double-float))
356 ;; Is (log -0) -infinity (libm.a) or -infinity + i*pi (Kahan)?
357 ;; Since this doesn't seem to be an implementation issue
358 ;; I (pw) take the Kahan result.
359 (if (< (float-sign number)
360 (coerce 0 '(dispatch-type number)))
361 (complex (log (- number)) (coerce pi '(dispatch-type number)))
362 (coerce (%log (coerce number 'double-float))
363 '(dispatch-type number))))
365 (complex-log number)))))
369 "Return the square root of NUMBER."
370 (number-dispatch ((number number))
371 (((foreach fixnum bignum ratio))
373 (complex-sqrt number)
374 (coerce (%sqrt (coerce number 'double-float)) 'single-float)))
375 (((foreach single-float double-float))
377 (complex-sqrt number)
378 (coerce (%sqrt (coerce number 'double-float))
379 '(dispatch-type number))))
381 (complex-sqrt number))))
383 ;;;; trigonometic and related functions
387 "Returns the absolute value of the number."
388 (number-dispatch ((number number))
389 (((foreach single-float double-float fixnum rational))
392 (let ((rx (realpart number))
393 (ix (imagpart number)))
396 (sqrt (+ (* rx rx) (* ix ix))))
398 (coerce (%hypot (coerce rx 'double-float)
399 (coerce ix 'double-float))
404 (defun phase (number)
406 "Returns the angle part of the polar representation of a complex number.
407 For complex numbers, this is (atan (imagpart number) (realpart number)).
408 For non-complex positive numbers, this is 0. For non-complex negative
413 (coerce pi 'single-float)
416 (if (minusp (float-sign number))
417 (coerce pi 'single-float)
420 (if (minusp (float-sign number))
421 (coerce pi 'double-float)
424 (atan (imagpart number) (realpart number)))))
428 "Return the sine of NUMBER."
429 (number-dispatch ((number number))
430 (handle-reals %sin number)
432 (let ((x (realpart number))
433 (y (imagpart number)))
434 (complex (* (sin x) (cosh y))
435 (* (cos x) (sinh y)))))))
439 "Return the cosine of NUMBER."
440 (number-dispatch ((number number))
441 (handle-reals %cos number)
443 (let ((x (realpart number))
444 (y (imagpart number)))
445 (complex (* (cos x) (cosh y))
446 (- (* (sin x) (sinh y))))))))
450 "Return the tangent of NUMBER."
451 (number-dispatch ((number number))
452 (handle-reals %tan number)
454 (complex-tan number))))
458 "Return cos(Theta) + i sin(Theta), AKA exp(i Theta)."
459 (declare (type real theta))
460 (complex (cos theta) (sin theta)))
464 "Return the arc sine of NUMBER."
465 (number-dispatch ((number number))
467 (if (or (> number 1) (< number -1))
468 (complex-asin number)
469 (coerce (%asin (coerce number 'double-float)) 'single-float)))
470 (((foreach single-float double-float))
471 (if (or (> number (coerce 1 '(dispatch-type number)))
472 (< number (coerce -1 '(dispatch-type number))))
473 (complex-asin number)
474 (coerce (%asin (coerce number 'double-float))
475 '(dispatch-type number))))
477 (complex-asin number))))
481 "Return the arc cosine of NUMBER."
482 (number-dispatch ((number number))
484 (if (or (> number 1) (< number -1))
485 (complex-acos number)
486 (coerce (%acos (coerce number 'double-float)) 'single-float)))
487 (((foreach single-float double-float))
488 (if (or (> number (coerce 1 '(dispatch-type number)))
489 (< number (coerce -1 '(dispatch-type number))))
490 (complex-acos number)
491 (coerce (%acos (coerce number 'double-float))
492 '(dispatch-type number))))
494 (complex-acos number))))
496 (defun atan (y &optional (x nil xp))
498 "Return the arc tangent of Y if X is omitted or Y/X if X is supplied."
501 (declare (type double-float y x)
502 (values double-float))
505 (if (plusp (float-sign x))
508 (float-sign y (/ pi 2)))
510 (number-dispatch ((y number) (x number))
512 (foreach double-float single-float fixnum bignum ratio))
513 (atan2 y (coerce x 'double-float)))
514 (((foreach single-float fixnum bignum ratio)
516 (atan2 (coerce y 'double-float) x))
517 (((foreach single-float fixnum bignum ratio)
518 (foreach single-float fixnum bignum ratio))
519 (coerce (atan2 (coerce y 'double-float) (coerce x 'double-float))
521 (number-dispatch ((y number))
522 (handle-reals %atan y)
526 ;; It seems that everyone has a C version of sinh, cosh, and
527 ;; tanh. Let's use these for reals because the original
528 ;; implementations based on the definitions lose big in round-off
529 ;; error. These bad definitions also mean that sin and cos for
530 ;; complex numbers can also lose big.
535 "Return the hyperbolic sine of NUMBER."
536 (/ (- (exp number) (exp (- number))) 2))
540 "Return the hyperbolic sine of NUMBER."
541 (number-dispatch ((number number))
542 (handle-reals %sinh number)
544 (let ((x (realpart number))
545 (y (imagpart number)))
546 (complex (* (sinh x) (cos y))
547 (* (cosh x) (sin y)))))))
552 "Return the hyperbolic cosine of NUMBER."
553 (/ (+ (exp number) (exp (- number))) 2))
557 "Return the hyperbolic cosine of NUMBER."
558 (number-dispatch ((number number))
559 (handle-reals %cosh number)
561 (let ((x (realpart number))
562 (y (imagpart number)))
563 (complex (* (cosh x) (cos y))
564 (* (sinh x) (sin y)))))))
568 "Return the hyperbolic tangent of NUMBER."
569 (number-dispatch ((number number))
570 (handle-reals %tanh number)
572 (complex-tanh number))))
574 (defun asinh (number)
576 "Return the hyperbolic arc sine of NUMBER."
577 (number-dispatch ((number number))
578 (handle-reals %asinh number)
580 (complex-asinh number))))
582 (defun acosh (number)
584 "Return the hyperbolic arc cosine of NUMBER."
585 (number-dispatch ((number number))
587 ;; acosh is complex if number < 1
589 (complex-acosh number)
590 (coerce (%acosh (coerce number 'double-float)) 'single-float)))
591 (((foreach single-float double-float))
592 (if (< number (coerce 1 '(dispatch-type number)))
593 (complex-acosh number)
594 (coerce (%acosh (coerce number 'double-float))
595 '(dispatch-type number))))
597 (complex-acosh number))))
599 (defun atanh (number)
601 "Return the hyperbolic arc tangent of NUMBER."
602 (number-dispatch ((number number))
604 ;; atanh is complex if |number| > 1
605 (if (or (> number 1) (< number -1))
606 (complex-atanh number)
607 (coerce (%atanh (coerce number 'double-float)) 'single-float)))
608 (((foreach single-float double-float))
609 (if (or (> number (coerce 1 '(dispatch-type number)))
610 (< number (coerce -1 '(dispatch-type number))))
611 (complex-atanh number)
612 (coerce (%atanh (coerce number 'double-float))
613 '(dispatch-type number))))
615 (complex-atanh number))))
617 ;;; HP-UX does not supply a C version of log1p, so
618 ;;; use the definition.
621 #!-sb-fluid (declaim (inline %log1p))
623 (defun %log1p (number)
624 (declare (double-float number)
625 (optimize (speed 3) (safety 0)))
626 (the double-float (log (the (double-float 0d0) (+ number 1d0)))))