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Revision 1.26 - (hide annotations)
Thu Feb 19 03:49:49 1998 UTC (16 years, 2 months ago) by dtc
Branch: MAIN
Changes since 1.25: +4 -4 lines
Fix float-trapping-nan-p which was returning T for quiet NaN and Nil
of trapping NaN.
1 wlott 1.1 ;;; -*- Mode: Lisp; Package: KERNEL; Log: code.log -*-
2     ;;;
3     ;;; **********************************************************************
4 ram 1.7 ;;; This code was written as part of the CMU Common Lisp project at
5     ;;; Carnegie Mellon University, and has been placed in the public domain.
6     ;;;
7     (ext:file-comment
8 dtc 1.26 "$Header: /tiger/var/lib/cvsroots/cmucl/src/code/irrat.lisp,v 1.26 1998/02/19 03:49:49 dtc Exp $")
9 ram 1.7 ;;;
10 wlott 1.1 ;;; **********************************************************************
11     ;;;
12     ;;; This file contains all the irrational functions. Actually, most of the
13     ;;; work is done by calling out to C...
14     ;;;
15     ;;; Author: William Lott.
16     ;;;
17    
18     (in-package "KERNEL")
19    
20    
21     ;;;; Random constants, utility functions, and macros.
22    
23     (defconstant pi 3.14159265358979323846264338327950288419716939937511L0)
24 wlott 1.2 ;(defconstant e 2.71828182845904523536028747135266249775724709369996L0)
25 wlott 1.1
26 ram 1.5 ;;; Make these INLINE, since the call to C is at least as compact as a Lisp
27     ;;; call, and saves number consing to boot.
28     ;;;
29 wlott 1.1 (defmacro def-math-rtn (name num-args)
30     (let ((function (intern (concatenate 'simple-string
31     "%"
32     (string-upcase name)))))
33 ram 1.4 `(progn
34 ram 1.5 (proclaim '(inline ,function))
35 ram 1.4 (export ',function)
36 wlott 1.10 (alien:def-alien-routine (,name ,function) double-float
37 ram 1.4 ,@(let ((results nil))
38     (dotimes (i num-args (nreverse results))
39     (push (list (intern (format nil "ARG-~D" i))
40     'double-float)
41     results)))))))
42 wlott 1.1
43     (eval-when (compile load eval)
44    
45     (defun handle-reals (function var)
46     `((((foreach fixnum single-float bignum ratio))
47     (coerce (,function (coerce ,var 'double-float)) 'single-float))
48     ((double-float)
49     (,function ,var))))
50    
51     ); eval-when (compile load eval)
52    
53    
54     ;;;; Stubs for the Unix math library.
55    
56     ;;; Please refer to the Unix man pages for details about these routines.
57    
58     ;;; Trigonometric.
59 ram 1.17 #-x86 (def-math-rtn "sin" 1)
60     #-x86 (def-math-rtn "cos" 1)
61     #-x86 (def-math-rtn "tan" 1)
62 wlott 1.1 (def-math-rtn "asin" 1)
63     (def-math-rtn "acos" 1)
64 ram 1.17 #-x86 (def-math-rtn "atan" 1)
65     #-x86 (def-math-rtn "atan2" 2)
66 wlott 1.1 (def-math-rtn "sinh" 1)
67     (def-math-rtn "cosh" 1)
68     (def-math-rtn "tanh" 1)
69 pw 1.18 (def-math-rtn "asinh" 1)
70     (def-math-rtn "acosh" 1)
71     (def-math-rtn "atanh" 1)
72 wlott 1.1
73     ;;; Exponential and Logarithmic.
74 dtc 1.19 #-x86 (def-math-rtn "exp" 1)
75     #-x86 (def-math-rtn "log" 1)
76     #-x86 (def-math-rtn "log10" 1)
77 wlott 1.1 (def-math-rtn "pow" 2)
78 ram 1.17 #-x86 (def-math-rtn "sqrt" 1)
79 wlott 1.1 (def-math-rtn "hypot" 2)
80 pw 1.18 #-hpux
81 ram 1.17 (def-math-rtn "log1p" 1)
82    
83     #+x86 ;; These are needed for use by byte-compiled files.
84     (progn
85     (defun %sin (x)
86 dtc 1.19 (declare (double-float x)
87 ram 1.17 (values double-float))
88     (%sin x))
89     (defun %sin-quick (x)
90     (declare (double-float x)
91     (values double-float))
92     (%sin-quick x))
93     (defun %cos (x)
94 dtc 1.19 (declare (double-float x)
95 ram 1.17 (values double-float))
96     (%cos x))
97     (defun %cos-quick (x)
98     (declare (double-float x)
99     (values double-float))
100     (%cos-quick x))
101     (defun %tan (x)
102     (declare (double-float x)
103     (values double-float))
104     (%tan x))
105     (defun %tan-quick (x)
106     (declare (double-float x)
107     (values double-float))
108     (%tan-quick x))
109     (defun %atan (x)
110     (declare (double-float x)
111     (values double-float))
112     (%atan x))
113     (defun %atan2 (x y)
114     (declare (double-float x y)
115     (values double-float))
116     (%atan2 x y))
117     (defun %exp (x)
118 dtc 1.19 (declare (double-float x)
119 ram 1.17 (values double-float))
120     (%exp x))
121     (defun %log (x)
122 dtc 1.19 (declare (double-float x)
123 ram 1.17 (values double-float))
124     (%log x))
125     (defun %log10 (x)
126 dtc 1.19 (declare (double-float x)
127 ram 1.17 (values double-float))
128     (%log10 x))
129     #+nil ;; notyet
130     (defun %pow (x y)
131     (declare (type (double-float 0d0) x)
132 dtc 1.19 (double-float y)
133 ram 1.17 (values (double-float 0d0)))
134     (%pow x y))
135     (defun %sqrt (x)
136 dtc 1.19 (declare (double-float x)
137 ram 1.17 (values double-float))
138     (%sqrt x))
139     (defun %scalbn (f ex)
140 dtc 1.19 (declare (double-float f)
141 ram 1.17 (type (signed-byte 32) ex)
142     (values double-float))
143     (%scalbn f ex))
144     (defun %scalb (f ex)
145     (declare (double-float f ex)
146     (values double-float))
147     (%scalb f ex))
148     (defun %logb (x)
149 dtc 1.19 (declare (double-float x)
150     (values double-float))
151     (%logb x))
152 ram 1.17 ) ; progn
153 wlott 1.1
154    
155     ;;;; Power functions.
156    
157     (defun exp (number)
158     "Return e raised to the power NUMBER."
159     (number-dispatch ((number number))
160     (handle-reals %exp number)
161     ((complex)
162     (* (exp (realpart number))
163     (cis (imagpart number))))))
164    
165     ;;; INTEXP -- Handle the rational base, integer power case.
166    
167     (defparameter *intexp-maximum-exponent* 10000)
168    
169 ram 1.6 ;;; This function precisely calculates base raised to an integral power. It
170     ;;; separates the cases by the sign of power, for efficiency reasons, as powers
171     ;;; can be calculated more efficiently if power is a positive integer. Values
172     ;;; of power are calculated as positive integers, and inverted if negative.
173     ;;;
174 wlott 1.1 (defun intexp (base power)
175     (when (> (abs power) *intexp-maximum-exponent*)
176     (cerror "Continue with calculation."
177     "The absolute value of ~S exceeds ~S."
178     power '*intexp-maximum-exponent* base power))
179     (cond ((minusp power)
180     (/ (intexp base (- power))))
181     ((eql base 2)
182     (ash 1 power))
183     (t
184     (do ((nextn (ash power -1) (ash power -1))
185     (total (if (oddp power) base 1)
186     (if (oddp power) (* base total) total)))
187     ((zerop nextn) total)
188     (setq base (* base base))
189     (setq power nextn)))))
190    
191    
192 ram 1.6 ;;; EXPT -- Public
193     ;;;
194     ;;; If an integer power of a rational, use INTEXP above. Otherwise, do
195     ;;; floating point stuff. If both args are real, we try %POW right off,
196     ;;; assuming it will return 0 if the result may be complex. If so, we call
197     ;;; COMPLEX-POW which directly computes the complex result. We also separate
198     ;;; the complex-real and real-complex cases from the general complex case.
199     ;;;
200 wlott 1.1 (defun expt (base power)
201 wlott 1.3 "Returns BASE raised to the POWER."
202 wlott 1.1 (if (zerop power)
203 ram 1.6 (1+ (* base power))
204 dtc 1.22 (labels (;; determine if the double float is an integer.
205     ;; 0 - not an integer
206     ;; 1 - an odd int
207     ;; 2 - an even int
208     (isint (ihi lo)
209     (declare (type (unsigned-byte 31) ihi)
210     (type (unsigned-byte 32) lo)
211     (optimize (speed 3) (safety 0)))
212     (let ((isint 0))
213     (declare (type fixnum isint))
214     (cond ((>= ihi #x43400000) ; exponent >= 53
215     (setq isint 2))
216     ((>= ihi #x3ff00000)
217     (let ((k (- (ash ihi -20) #x3ff))) ; exponent
218     (declare (type (mod 53) k))
219     (cond ((> k 20)
220     (let* ((shift (- 52 k))
221     (j (logand (ash lo (- shift))))
222     (j2 (ash j shift)))
223     (declare (type (mod 32) shift)
224     (type (unsigned-byte 32) j j2))
225     (when (= j2 lo)
226     (setq isint (- 2 (logand j 1))))))
227     ((= lo 0)
228     (let* ((shift (- 20 k))
229     (j (ash ihi (- shift)))
230     (j2 (ash j shift)))
231     (declare (type (mod 32) shift)
232     (type (unsigned-byte 31) j j2))
233     (when (= j2 ihi)
234     (setq isint (- 2 (logand j 1))))))))))
235     isint))
236     (real-expt (x y rtype)
237     (let ((x (coerce x 'double-float))
238     (y (coerce y 'double-float)))
239     (declare (double-float x y))
240     (let* ((x-hi (kernel:double-float-high-bits x))
241     (x-lo (kernel:double-float-low-bits x))
242     (x-ihi (logand x-hi #x7fffffff))
243     (y-hi (kernel:double-float-high-bits y))
244     (y-lo (kernel:double-float-low-bits y))
245     (y-ihi (logand y-hi #x7fffffff)))
246     (declare (type (signed-byte 32) x-hi y-hi)
247     (type (unsigned-byte 31) x-ihi y-ihi)
248     (type (unsigned-byte 32) x-lo y-lo))
249     ;; y==zero: x**0 = 1
250     (when (zerop (logior y-ihi y-lo))
251     (return-from real-expt (coerce 1d0 rtype)))
252     ;; +-NaN return x+y
253     (when (or (> x-ihi #x7ff00000)
254     (and (= x-ihi #x7ff00000) (/= x-lo 0))
255     (> y-ihi #x7ff00000)
256     (and (= y-ihi #x7ff00000) (/= y-lo 0)))
257     (return-from real-expt (coerce (+ x y) rtype)))
258     (let ((yisint (if (< x-hi 0) (isint y-ihi y-lo) 0)))
259     (declare (type fixnum yisint))
260     ;; special value of y
261     (when (and (zerop y-lo) (= y-ihi #x7ff00000))
262     ;; y is +-inf
263     (return-from real-expt
264     (cond ((and (= x-ihi #x3ff00000) (zerop x-lo))
265     ;; +-1**inf is NaN
266     (coerce (- y y) rtype))
267     ((>= x-ihi #x3ff00000)
268     ;; (|x|>1)**+-inf = inf,0
269     (if (>= y-hi 0)
270     (coerce y rtype)
271     (coerce 0 rtype)))
272     (t
273     ;; (|x|<1)**-,+inf = inf,0
274     (if (< y-hi 0)
275     (coerce (- y) rtype)
276     (coerce 0 rtype))))))
277    
278     (let ((abs-x (abs x)))
279     (declare (double-float abs-x))
280     ;; special value of x
281     (when (and (zerop x-lo)
282     (or (= x-ihi #x7ff00000) (zerop x-ihi)
283     (= x-ihi #x3ff00000)))
284     ;; x is +-0,+-inf,+-1
285     (let ((z (if (< y-hi 0)
286     (/ 1 abs-x) ; z = (1/|x|)
287     abs-x)))
288     (declare (double-float z))
289     (when (< x-hi 0)
290     (cond ((and (= x-ihi #x3ff00000) (zerop yisint))
291     ;; (-1)**non-int
292     (let ((y*pi (* y pi)))
293     (declare (double-float y*pi))
294     (return-from real-expt
295 dtc 1.24 (complex
296 dtc 1.22 (coerce (%cos y*pi) rtype)
297     (coerce (%sin y*pi) rtype)))))
298     ((= yisint 1)
299     ;; (x<0)**odd = -(|x|**odd)
300     (setq z (- z)))))
301     (return-from real-expt (coerce z rtype))))
302    
303     (if (>= x-hi 0)
304     ;; x>0
305     (coerce (kernel::%pow x y) rtype)
306     ;; x<0
307     (let ((pow (kernel::%pow abs-x y)))
308     (declare (double-float pow))
309     (case yisint
310     (1 ; Odd
311     (coerce (* -1d0 pow) rtype))
312     (2 ; Even
313     (coerce pow rtype))
314     (t ; Non-integer
315     (let ((y*pi (* y pi)))
316     (declare (double-float y*pi))
317 dtc 1.24 (complex
318 dtc 1.22 (coerce (* pow (%cos y*pi)) rtype)
319     (coerce (* pow (%sin y*pi)) rtype)))))))))))))
320     (declare (inline real-expt))
321     (number-dispatch ((base number) (power number))
322     (((foreach fixnum (or bignum ratio) (complex rational)) integer)
323     (intexp base power))
324     (((foreach single-float double-float) rational)
325     (real-expt base power '(dispatch-type base)))
326     (((foreach fixnum (or bignum ratio) single-float)
327     (foreach ratio single-float))
328     (real-expt base power 'single-float))
329     (((foreach fixnum (or bignum ratio) single-float double-float)
330     double-float)
331     (real-expt base power 'double-float))
332     ((double-float single-float)
333     (real-expt base power 'double-float))
334     (((foreach (complex rational) (complex float)) rational)
335     (* (expt (abs base) power)
336     (cis (* power (phase base)))))
337     (((foreach fixnum (or bignum ratio) single-float double-float)
338     complex)
339     (if (and (zerop base) (plusp (realpart power)))
340     (* base power)
341     (exp (* power (log base)))))
342     (((foreach (complex float) (complex rational))
343     (foreach complex double-float single-float))
344     (if (and (zerop base) (plusp (realpart power)))
345     (* base power)
346     (exp (* power (log base)))))))))
347 wlott 1.1
348     (defun log (number &optional (base nil base-p))
349 wlott 1.3 "Return the logarithm of NUMBER in the base BASE, which defaults to e."
350 wlott 1.1 (if base-p
351 pw 1.18 (if (zerop base)
352     base ; ANSI spec
353     (/ (log number) (log base)))
354 wlott 1.1 (number-dispatch ((number number))
355 ram 1.17 (((foreach fixnum bignum ratio))
356 wlott 1.3 (if (minusp number)
357     (complex (log (- number)) (coerce pi 'single-float))
358     (coerce (%log (coerce number 'double-float)) 'single-float)))
359 ram 1.17 (((foreach single-float double-float))
360     ;; Is (log -0) -infinity (libm.a) or -infinity + i*pi (Kahan)?
361     ;; Since this doesn't seem to be an implementation issue
362     ;; I (pw) take the Kahan result.
363     (if (< (float-sign number)
364     (coerce 0 '(dispatch-type number)))
365     (complex (log (- number)) (coerce pi '(dispatch-type number)))
366     (coerce (%log (coerce number 'double-float))
367     '(dispatch-type number))))
368     ((complex)
369     (complex-log number)))))
370 wlott 1.1
371     (defun sqrt (number)
372     "Return the square root of NUMBER."
373     (number-dispatch ((number number))
374 ram 1.17 (((foreach fixnum bignum ratio))
375 wlott 1.3 (if (minusp number)
376 ram 1.17 (complex-sqrt number)
377 wlott 1.3 (coerce (%sqrt (coerce number 'double-float)) 'single-float)))
378 ram 1.17 (((foreach single-float double-float))
379 dtc 1.25 (if (< (float-sign number)
380 ram 1.17 (coerce 0 '(dispatch-type number)))
381     (complex-sqrt number)
382     (coerce (%sqrt (coerce number 'double-float))
383     '(dispatch-type number))))
384     ((complex)
385     (complex-sqrt number))))
386 wlott 1.1
387    
388     ;;;; Trigonometic and Related Functions
389    
390 wlott 1.2 (defun abs (number)
391     "Returns the absolute value of the number."
392     (number-dispatch ((number number))
393     (((foreach single-float double-float fixnum rational))
394     (abs number))
395     ((complex)
396     (let ((rx (realpart number))
397     (ix (imagpart number)))
398     (etypecase rx
399     (rational
400     (sqrt (+ (* rx rx) (* ix ix))))
401     (single-float
402     (coerce (%hypot (coerce rx 'double-float)
403     (coerce ix 'double-float))
404     'single-float))
405     (double-float
406     (%hypot rx ix)))))))
407 wlott 1.1
408     (defun phase (number)
409     "Returns the angle part of the polar representation of a complex number.
410 wlott 1.3 For complex numbers, this is (atan (imagpart number) (realpart number)).
411 wlott 1.1 For non-complex positive numbers, this is 0. For non-complex negative
412     numbers this is PI."
413 wlott 1.3 (etypecase number
414 ram 1.17 (rational
415 wlott 1.3 (if (minusp number)
416     (coerce pi 'single-float)
417     0.0f0))
418 ram 1.17 (single-float
419     (if (minusp (float-sign number))
420     (coerce pi 'single-float)
421     0.0f0))
422 wlott 1.3 (double-float
423 ram 1.17 (if (minusp (float-sign number))
424 wlott 1.3 (coerce pi 'double-float)
425     0.0d0))
426     (complex
427     (atan (imagpart number) (realpart number)))))
428 wlott 1.1
429    
430     (defun sin (number)
431 wlott 1.3 "Return the sine of NUMBER."
432 wlott 1.1 (number-dispatch ((number number))
433     (handle-reals %sin number)
434     ((complex)
435     (let ((x (realpart number))
436     (y (imagpart number)))
437 ram 1.17 (complex (* (sin x) (cosh y))
438     (* (cos x) (sinh y)))))))
439 wlott 1.1
440     (defun cos (number)
441 wlott 1.3 "Return the cosine of NUMBER."
442 wlott 1.1 (number-dispatch ((number number))
443     (handle-reals %cos number)
444     ((complex)
445     (let ((x (realpart number))
446     (y (imagpart number)))
447 ram 1.17 (complex (* (cos x) (cosh y))
448     (- (* (sin x) (sinh y))))))))
449 wlott 1.1
450     (defun tan (number)
451 wlott 1.3 "Return the tangent of NUMBER."
452 wlott 1.1 (number-dispatch ((number number))
453     (handle-reals %tan number)
454     ((complex)
455 ram 1.17 (complex-tan number))))
456 wlott 1.1
457 wlott 1.2 (defun cis (theta)
458 wlott 1.3 "Return cos(Theta) + i sin(Theta), AKA exp(i Theta)."
459 wlott 1.2 (if (complexp theta)
460     (error "Argument to CIS is complex: ~S" theta)
461     (complex (cos theta) (sin theta))))
462 wlott 1.1
463     (defun asin (number)
464 wlott 1.3 "Return the arc sine of NUMBER."
465 wlott 1.1 (number-dispatch ((number number))
466 wlott 1.3 ((rational)
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))))
476 wlott 1.1 ((complex)
477 wlott 1.3 (complex-asin number))))
478 wlott 1.1
479     (defun acos (number)
480 wlott 1.3 "Return the arc cosine of NUMBER."
481 wlott 1.1 (number-dispatch ((number number))
482 wlott 1.3 ((rational)
483     (if (or (> number 1) (< number -1))
484     (complex-acos number)
485     (coerce (%acos (coerce number 'double-float)) 'single-float)))
486     (((foreach single-float double-float))
487     (if (or (> number (coerce 1 '(dispatch-type number)))
488     (< number (coerce -1 '(dispatch-type number))))
489     (complex-acos number)
490     (coerce (%acos (coerce number 'double-float))
491     '(dispatch-type number))))
492 wlott 1.1 ((complex)
493 wlott 1.3 (complex-acos number))))
494 wlott 1.1
495    
496     (defun atan (y &optional (x nil xp))
497 wlott 1.3 "Return the arc tangent of Y if X is omitted or Y/X if X is supplied."
498 wlott 1.1 (if xp
499 wlott 1.12 (flet ((atan2 (y x)
500 wlott 1.13 (declare (type double-float y x)
501     (values double-float))
502     (if (zerop x)
503     (if (zerop y)
504     (if (plusp (float-sign x))
505     y
506     (float-sign y pi))
507     (float-sign y (/ pi 2)))
508     (%atan2 y x))))
509     (number-dispatch ((y number) (x number))
510     ((double-float
511     (foreach double-float single-float fixnum bignum ratio))
512     (atan2 y (coerce x 'double-float)))
513     (((foreach single-float fixnum bignum ratio)
514     double-float)
515     (atan2 (coerce y 'double-float) x))
516     (((foreach single-float fixnum bignum ratio)
517     (foreach single-float fixnum bignum ratio))
518     (coerce (atan2 (coerce y 'double-float) (coerce x 'double-float))
519     'single-float))))
520 wlott 1.1 (number-dispatch ((y number))
521     (handle-reals %atan y)
522     ((complex)
523 ram 1.17 (complex-atan y)))))
524 wlott 1.3
525 ram 1.17 ;; It seems that everyone has a C version of sinh, cosh, and
526     ;; tanh. Let's use these for reals because the original
527     ;; implementations based on the definitions lose big in round-off
528     ;; error. These bad definitions also mean that sin and cos for
529     ;; complex numbers can also lose big.
530 wlott 1.3
531 ram 1.17 #+nil
532 wlott 1.3 (defun sinh (number)
533     "Return the hyperbolic sine of NUMBER."
534     (/ (- (exp number) (exp (- number))) 2))
535    
536 ram 1.17 (defun sinh (number)
537     "Return the hyperbolic sine of NUMBER."
538     (number-dispatch ((number number))
539     (handle-reals %sinh number)
540     ((complex)
541     (let ((x (realpart number))
542     (y (imagpart number)))
543     (complex (* (sinh x) (cos y))
544     (* (cosh x) (sin y)))))))
545    
546     #+nil
547 wlott 1.3 (defun cosh (number)
548     "Return the hyperbolic cosine of NUMBER."
549     (/ (+ (exp number) (exp (- number))) 2))
550    
551 ram 1.17 (defun cosh (number)
552     "Return the hyperbolic cosine of NUMBER."
553     (number-dispatch ((number number))
554     (handle-reals %cosh number)
555     ((complex)
556     (let ((x (realpart number))
557     (y (imagpart number)))
558     (complex (* (cosh x) (cos y))
559     (* (sinh x) (sin y)))))))
560    
561 wlott 1.3 (defun tanh (number)
562     "Return the hyperbolic tangent of NUMBER."
563 ram 1.17 (number-dispatch ((number number))
564     (handle-reals %tanh number)
565     ((complex)
566     (complex-tanh number))))
567 wlott 1.3
568     (defun asinh (number)
569     "Return the hyperbolic arc sine of NUMBER."
570 ram 1.17 (number-dispatch ((number number))
571     (handle-reals %asinh number)
572     ((complex)
573     (complex-asinh number))))
574 wlott 1.3
575     (defun acosh (number)
576     "Return the hyperbolic arc cosine of NUMBER."
577 ram 1.17 (number-dispatch ((number number))
578     ((rational)
579     ;; acosh is complex if number < 1
580     (if (< number 1)
581     (complex-acosh number)
582     (coerce (%acosh (coerce number 'double-float)) 'single-float)))
583     (((foreach single-float double-float))
584     (if (< number (coerce 1 '(dispatch-type number)))
585     (complex-acosh number)
586     (coerce (%acosh (coerce number 'double-float))
587     '(dispatch-type number))))
588     ((complex)
589     (complex-acosh number))))
590 wlott 1.3
591     (defun atanh (number)
592     "Return the hyperbolic arc tangent of NUMBER."
593 ram 1.17 (number-dispatch ((number number))
594     ((rational)
595     ;; atanh is complex if |number| > 1
596     (if (or (> number 1) (< number -1))
597     (complex-atanh number)
598     (coerce (%atanh (coerce number 'double-float)) 'single-float)))
599     (((foreach single-float double-float))
600     (if (or (> number (coerce 1 '(dispatch-type number)))
601     (< number (coerce -1 '(dispatch-type number))))
602     (complex-atanh number)
603     (coerce (%atanh (coerce number 'double-float))
604     '(dispatch-type number))))
605     ((complex)
606     (complex-atanh number))))
607 wlott 1.14
608 pw 1.18 ;;; HP-UX does not supply a C version of log1p, so
609     ;;; use the definition.
610 wlott 1.14
611     #+hpux
612 pw 1.18 (declaim (inline %log1p))
613 wlott 1.14 #+hpux
614 pw 1.18 (defun %log1p (number)
615     (declare (double-float number)
616     (optimize (speed 3) (safety 0)))
617     (the double-float (log (the (double-float 0d0) (+ number 1d0)))))
618 ram 1.17
619    
620     #+old-elfun
621     (progn
622     ;;; Here are the old definitions of the special functions, for
623     ;;; complex-valued arguments. Some of these functions suffer from
624     ;;; severe round-off error or unnecessary overflow.
625    
626     (proclaim '(inline mult-by-i))
627     (defun mult-by-i (number)
628     (complex (- (imagpart number))
629     (realpart number)))
630    
631     (defun complex-sqrt (x)
632     (exp (/ (log x) 2)))
633    
634     (defun complex-log (x)
635     (complex (log (abs x))
636     (phase x)))
637    
638     (defun complex-atanh (number)
639 ram 1.16 (/ (- (log (1+ number)) (log (- 1 number))) 2))
640 ram 1.17
641     (defun complex-tanh (number)
642     (/ (- (exp number) (exp (- number)))
643     (+ (exp number) (exp (- number)))))
644    
645     (defun complex-acos (number)
646     (* -2 (mult-by-i (log (+ (sqrt (/ (1+ number) 2))
647     (mult-by-i (sqrt (/ (- 1 number) 2))))))))
648    
649     (defun complex-acosh (number)
650     (* 2 (log (+ (sqrt (/ (1+ number) 2)) (sqrt (/ (1- number) 2))))))
651    
652     (defun complex-asin (number)
653     (- (mult-by-i (log (+ (mult-by-i number) (sqrt (- 1 (* number number))))))))
654    
655     (defun complex-asinh (number)
656     (log (+ number (sqrt (1+ (* number number))))))
657    
658     (defun complex-atan (y)
659     (let ((im (imagpart y))
660     (re (realpart y)))
661     (/ (- (log (complex (- 1 im) re))
662     (log (complex (+ 1 im) (- re))))
663     (complex 0 2))))
664    
665     (defun complex-tan (number)
666     (let* ((num (sin number))
667     (denom (cos number)))
668     (if (zerop denom) (error "~S undefined tangent." number)
669     (/ num denom))))
670     )
671    
672     #-old-specfun
673     (progn
674     ;;;;
675     ;;;; This is a set of routines that implement many elementary
676     ;;;; transcendental functions as specified by ANSI Common Lisp. The
677     ;;;; implementation is based on Kahan's paper.
678     ;;;;
679     ;;;; I believe I have accurately implemented the routines and are
680     ;;;; correct, but you may want to check for your self.
681     ;;;;
682     ;;;; These functions are written for CMU Lisp and take advantage of
683     ;;;; some of the features available there. It may be possible,
684     ;;;; however, to port this to other Lisps.
685     ;;;;
686     ;;;; Some functions are significantly more accurate than the original
687     ;;;; definitions in CMU Lisp. In fact, some functions in CMU Lisp
688     ;;;; give the wrong answer like (acos #c(-2.0 0.0)), where the true
689     ;;;; answer is pi + i*log(2-sqrt(3)).
690     ;;;;
691     ;;;; All of the implemented functions will take any number for an
692     ;;;; input, but the result will always be a either a complex
693     ;;;; single-float or a complex double-float.
694     ;;;;
695     ;;;; General functions
696     ;;;; complex-sqrt
697     ;;;; complex-log
698     ;;;; complex-atanh
699     ;;;; complex-tanh
700     ;;;; complex-acos
701     ;;;; complex-acosh
702     ;;;; complex-asin
703     ;;;; complex-asinh
704     ;;;; complex-atan
705     ;;;; complex-tan
706     ;;;;
707     ;;;; Utility functions:
708     ;;;; scalb logb
709     ;;;;
710     ;;;; Internal functions:
711     ;;;; square coerce-to-complex-type cssqs complex-log-scaled
712     ;;;;
713     ;;;;
714     ;;;; Please send any bug reports, comments, or improvements to Raymond
715     ;;;; Toy at toy@rtp.ericsson.se.
716     ;;;;
717     ;;;; References
718     ;;;;
719     ;;;; Kahan, W. "Branch Cuts for Complex Elementary Functions, or Much
720     ;;;; Ado About Nothing's Sign Bit" in Iserles and Powell (eds.) "The
721     ;;;; State of the Art in Numerical Analysis", pp. 165-211, Clarendon
722     ;;;; Press, 1987
723     ;;;;
724     (declaim (inline square))
725     (declaim (ftype (function (double-float) (double-float 0d0)) square))
726     (defun square (x)
727     (declare (double-float x)
728     (values (double-float 0d0)))
729     (* x x))
730    
731     ;; If you have these functions in libm, perhaps they should be used
732     ;; instead of these Lisp versions. These versions are probably good
733     ;; enough, especially since they are portable.
734    
735 dtc 1.19 (declaim (inline scalb))
736 ram 1.17 (defun scalb (x n)
737     "Compute 2^N * X without compute 2^N first (use properties of the
738     underlying floating-point format"
739     (declare (type double-float x)
740 dtc 1.19 (type double-float-exponent n))
741 ram 1.17 (scale-float x n))
742    
743     (defun logb (x)
744     "Compute an integer N such that 1 <= |2^N * x| < 2.
745     For the special cases, the following values are used:
746    
747     x logb
748     NaN NaN
749     +/- infinity +infinity
750     0 -infinity
751     "
752     (declare (type double-float x))
753 dtc 1.26 (cond ((float-nan-p x)
754 ram 1.17 x)
755     ((float-infinity-p x)
756     #.ext:double-float-positive-infinity)
757     ((zerop x)
758     ;; The answer is negative infinity, but we are supposed to
759     ;; signal divide-by-zero.
760     ;; (error 'division-by-zero :operation 'logb :operands (list x))
761     (/ -1.0d0 x)
762     )
763     (t
764     (multiple-value-bind (signif expon sign)
765     (decode-float x)
766     (declare (ignore signif sign))
767     ;; decode-float is almost right, except that the exponent
768     ;; is off by one
769     (1- expon)))))
770    
771     ;; This function is used to create a complex number of the appropriate
772     ;; type.
773    
774     (declaim (inline coerce-to-complex-type))
775     (defun coerce-to-complex-type (x y z)
776     "Create complex number with real part X and imaginary part Y such that
777     it has the same type as Z. If Z has type (complex rational), the X
778     and Y are coerced to single-float."
779     (declare (double-float x y)
780     (number z))
781     (if (subtypep (type-of (realpart z)) 'double-float)
782     (complex x y)
783     ;; Convert anything that's not a double-float to a single-float.
784     (complex (float x 1.0)
785     (float y 1.0))))
786    
787     (defun cssqs (z)
788 dtc 1.21 ;; Compute |(x+i*y)/2^k|^2 scaled to avoid over/underflow. The
789     ;; result is r + i*k, where k is an integer.
790 ram 1.17
791     ;; Save all FP flags
792 dtc 1.21 (let ((x (float (realpart z) 1d0))
793     (y (float (imagpart z) 1d0))
794     (k 0)
795     (rho 0d0))
796 ram 1.17 (declare (double-float x y)
797     (type (double-float 0d0) rho)
798     (fixnum k))
799 dtc 1.21 ;; Would this be better handled using an exception handler to
800     ;; catch the overflow or underflow signal? For now, we turn all
801     ;; traps off and look at the accrued exceptions to see if any
802     ;; signal would have been raised.
803     (with-float-traps-masked (:underflow :overflow)
804     (setf rho (+ (square x) (square y)))
805 dtc 1.26 (cond ((and (or (float-nan-p rho)
806 dtc 1.21 (float-infinity-p rho))
807     (or (float-infinity-p (abs x))
808     (float-infinity-p (abs y))))
809     (setf rho #.ext:double-float-positive-infinity))
810     ((let ((threshold #.(/ least-positive-double-float
811     double-float-epsilon))
812 dtc 1.23 (traps (ldb vm::float-sticky-bits
813 dtc 1.21 (vm:floating-point-modes))))
814     ;; Overflow raised or (underflow raised and rho <
815     ;; lambda/eps)
816     (or (not (zerop (logand vm:float-overflow-trap-bit traps)))
817     (and (not (zerop (logand vm:float-underflow-trap-bit traps)))
818     (< rho threshold))))
819     (setf k (logb (max (abs x) (abs y))))
820     (setf rho (+ (square (scalb x (- k)))
821     (square (scalb y (- k))))))))
822     (values rho k)))
823 ram 1.17
824     (defun complex-sqrt (z)
825     "Principle square root of Z
826    
827     Z may be any number, but the result is always a complex."
828     (declare (number z))
829     (multiple-value-bind (rho k)
830     (cssqs z)
831     (declare (type (double-float 0d0) rho)
832     (fixnum k))
833     (let ((x (float (realpart z) 1.0d0))
834     (y (float (imagpart z) 1.0d0))
835     (eta 0d0)
836     (nu 0d0))
837     (declare (double-float x y eta nu))
838    
839 dtc 1.26 (if (not (float-nan-p x))
840 ram 1.17 (setf rho (+ (scalb (abs x) (- k)) (sqrt rho))))
841    
842     (cond ((oddp k)
843     (setf k (ash k -1)))
844     (t
845     (setf k (1- (ash k -1)))
846     (setf rho (+ rho rho))))
847    
848     (setf rho (scalb (sqrt rho) k))
849    
850     (setf eta rho)
851     (setf nu y)
852    
853     (when (/= rho 0d0)
854     (when (not (float-infinity-p (abs nu)))
855     (setf nu (/ (/ nu rho) 2d0)))
856     (when (< x 0d0)
857     (setf eta (abs nu))
858     (setf nu (float-sign y rho))))
859     (coerce-to-complex-type eta nu z))))
860    
861     (defun complex-log-scaled (z j)
862     "Compute log(2^j*z).
863    
864     This is for use with J /= 0 only when |z| is huge."
865     (declare (number z)
866     (fixnum j))
867     ;; The constants t0, t1, t2 should be evaluated to machine
868     ;; precision. In addition, Kahan says the accuracy of log1p
869     ;; influences the choices of these constants but doesn't say how to
870     ;; choose them. We'll just assume his choices matches our
871     ;; implementation of log1p.
872     (let ((t0 #.(/ 1 (sqrt 2.0d0)))
873     (t1 1.2d0)
874     (t2 3d0)
875     (ln2 #.(log 2d0))
876     (x (float (realpart z) 1.0d0))
877     (y (float (imagpart z) 1.0d0)))
878     (multiple-value-bind (rho k)
879     (cssqs z)
880     (declare (type (double-float 0d0) rho)
881     (fixnum k))
882     (let ((beta (max (abs x) (abs y)))
883     (theta (min (abs x) (abs y))))
884     (declare (type (double-float 0d0) beta theta))
885     (if (and (zerop k)
886     (< t0 beta)
887     (or (<= beta t1)
888     (< rho t2)))
889     (setf rho (/ (%log1p (+ (* (- beta 1.0d0)
890     (+ beta 1.0d0))
891     (* theta theta)))
892     2d0))
893     (setf rho (+ (/ (log rho) 2d0)
894     (* (+ k j) ln2))))
895     (setf theta (atan y x))
896     (coerce-to-complex-type rho theta z)))))
897    
898     (defun complex-log (z)
899     "Log of Z = log |Z| + i * arg Z
900    
901     Z may be any number, but the result is always a complex."
902     (declare (number z))
903     (complex-log-scaled z 0))
904    
905     ;; Let us note the following "strange" behavior. atanh 1.0d0 is
906     ;; +infinity, but the following code returns approx 176 + i*pi/4. The
907     ;; reason for the imaginary part is caused by the fact that arg i*y is
908     ;; never 0 since we have positive and negative zeroes.
909    
910     (defun complex-atanh (z)
911     "Compute atanh z = (log(1+z) - log(1-z))/2"
912     (declare (number z))
913     (let* (;; Constants
914     (theta #.(/ (sqrt most-positive-double-float) 4.0d0))
915     (rho #.(/ 4.0d0 (sqrt most-positive-double-float)))
916     (half-pi #.(/ pi 2.0d0))
917     (rp (float (realpart z) 1.0d0))
918     (beta (float-sign rp 1.0d0))
919     (x (* beta rp))
920     (y (* beta (- (float (imagpart z) 1.0d0))))
921     (eta 0.0d0)
922     (nu 0.0d0))
923     (declare (double-float theta rho half-pi rp beta y eta nu)
924     (type (double-float 0d0) x))
925     (cond ((or (> x theta)
926     (> (abs y) theta))
927     ;; To avoid overflow...
928     (setf eta (float-sign y half-pi))
929     ;; nu is real part of 1/(x + iy). This is x/(x^2+y^2),
930     ;; which can cause overflow. Arrange this computation so
931     ;; that it won't overflow.
932     (setf nu (let* ((x-bigger (> x (abs y)))
933     (r (if x-bigger (/ y x) (/ x y)))
934     (d (+ 1.0d0 (* r r))))
935     (declare (double-float r d))
936     (if x-bigger
937     (/ (/ x) d)
938     (/ (/ r y) d)))))
939     ((= x 1.0d0)
940     ;; Should this be changed so that if y is zero, eta is set
941     ;; to +infinity instead of approx 176? In any case
942     ;; tanh(176) is 1.0d0 within working precision.
943     (let ((t1 (+ 4d0 (square y)))
944     (t2 (+ (abs y) rho)))
945     (declare (type (double-float 0d0) t1 t2))
946     #+nil
947     (setf eta (log (/ (sqrt (sqrt t1)))
948     (sqrt t2)))
949     (setf eta (* 0.5d0 (log (the (double-float 0.0d0)
950     (/ (sqrt t1) t2)))))
951     (setf nu (* 0.5d0
952     (float-sign y
953     (+ half-pi (atan (* 0.5d0 t2))))))))
954     (t
955     (let ((t1 (+ (abs y) rho)))
956     (declare (double-float t1))
957     ;; Normal case using log1p(x) = log(1 + x)
958     (setf eta (* 0.25d0
959     (%log1p (/ (* 4.0d0 x)
960     (+ (square (- 1.0d0 x))
961     (square t1))))))
962     (setf nu (* 0.5d0
963     (atan (* 2.0d0 y)
964     (- (* (- 1.0d0 x)
965     (+ 1.0d0 x))
966     (square t1))))))))
967     (coerce-to-complex-type (* beta eta)
968     (- (* beta nu))
969     z)))
970    
971     (defun complex-tanh (z)
972     "Compute tanh z = sinh z / cosh z"
973     (declare (number z))
974     (let ((x (float (realpart z) 1.0d0))
975     (y (float (imagpart z) 1.0d0)))
976     (declare (double-float x y))
977     (cond ((> (abs x)
978 pw 1.18 #-(or linux hpux) #.(/ (%asinh most-positive-double-float) 4d0)
979 ram 1.17 ;; This is more accurate under linux.
980 pw 1.18 #+(or linux hpux) #.(/ (+ (%log 2.0d0)
981     (%log most-positive-double-float)) 4d0))
982 ram 1.17 (complex (float-sign x)
983     (float-sign y 0.0d0)))
984     (t
985 dtc 1.20 (let* ((tv (%tan y))
986 ram 1.17 (beta (+ 1.0d0 (* tv tv)))
987     (s (sinh x))
988     (rho (sqrt (+ 1.0d0 (* s s)))))
989     (declare (double-float tv s)
990     (type (double-float 0.0d0) beta rho))
991     (if (float-infinity-p (abs tv))
992     (coerce-to-complex-type (/ rho s)
993     (/ tv)
994     z)
995     (let ((den (+ 1.0d0 (* beta s s))))
996     (coerce-to-complex-type (/ (* beta rho s)
997     den)
998     (/ tv den)
999     z))))))))
1000    
1001     ;; Kahan says we should only compute the parts needed. Thus, the
1002     ;; realpart's below should only compute the real part, not the whole
1003     ;; complex expression. Doing this can be important because we may get
1004     ;; spurious signals that occur in the part that we are not using.
1005     ;;
1006     ;; However, we take a pragmatic approach and just use the whole
1007     ;; expression.
1008    
1009     ;; NOTE: The formula given by Kahan is somewhat ambiguous in whether
1010     ;; it's the conjugate of the square root or the square root of the
1011     ;; conjugate. This needs to be checked.
1012    
1013     ;; I checked. It doesn't matter because (conjugate (sqrt z)) is the
1014     ;; same as (sqrt (conjugate z)) for all z. This follows because
1015     ;;
1016     ;; (conjugate (sqrt z)) = exp(0.5*log |z|)*exp(-0.5*j*arg z).
1017     ;;
1018     ;; (sqrt (conjugate z)) = exp(0.5*log|z|)*exp(0.5*j*arg conj z)
1019     ;;
1020     ;;.and these two expressions are equal if and only if arg conj z =
1021     ;;-arg z, which is clearly true for all z.
1022    
1023     (defun complex-acos (z)
1024     "Compute acos z = pi/2 - asin z
1025    
1026     Z may be any number, but the result is always a complex."
1027     (declare (number z))
1028     (let ((sqrt-1+z (complex-sqrt (+ 1 z)))
1029     (sqrt-1-z (complex-sqrt (- 1 z))))
1030 dtc 1.21 (with-float-traps-masked (:divide-by-zero)
1031     (complex (* 2 (atan (/ (realpart sqrt-1-z)
1032     (realpart sqrt-1+z))))
1033     (asinh (imagpart (* (conjugate sqrt-1+z)
1034     sqrt-1-z)))))))
1035 ram 1.17
1036     (defun complex-acosh (z)
1037     "Compute acosh z = 2 * log(sqrt((z+1)/2) + sqrt((z-1)/2))
1038    
1039     Z may be any number, but the result is always a complex."
1040     (declare (number z))
1041     (let ((sqrt-z-1 (complex-sqrt (- z 1)))
1042     (sqrt-z+1 (complex-sqrt (+ z 1))))
1043 dtc 1.21 (with-float-traps-masked (:divide-by-zero)
1044     (complex (asinh (realpart (* (conjugate sqrt-z-1)
1045     sqrt-z+1)))
1046     (* 2 (atan (/ (imagpart sqrt-z-1)
1047     (realpart sqrt-z+1))))))))
1048 ram 1.17
1049    
1050     (defun complex-asin (z)
1051     "Compute asin z = asinh(i*z)/i
1052    
1053     Z may be any number, but the result is always a complex."
1054     (declare (number z))
1055     (let ((sqrt-1-z (complex-sqrt (- 1 z)))
1056     (sqrt-1+z (complex-sqrt (+ 1 z))))
1057 dtc 1.21 (with-float-traps-masked (:divide-by-zero)
1058     (complex (atan (/ (realpart z)
1059     (realpart (* sqrt-1-z sqrt-1+z))))
1060     (asinh (imagpart (* (conjugate sqrt-1-z)
1061     sqrt-1+z)))))))
1062 ram 1.17
1063     (defun complex-asinh (z)
1064     "Compute asinh z = log(z + sqrt(1 + z*z))
1065    
1066     Z may be any number, but the result is always a complex."
1067     (declare (number z))
1068     ;; asinh z = -i * asin (i*z)
1069     (let* ((iz (complex (- (imagpart z)) (realpart z)))
1070     (result (complex-asin iz)))
1071     (complex (imagpart result)
1072     (- (realpart result)))))
1073    
1074     (defun complex-atan (z)
1075     "Compute atan z = atanh (i*z) / i
1076    
1077     Z may be any number, but the result is always a complex."
1078     (declare (number z))
1079     ;; atan z = -i * atanh (i*z)
1080     (let* ((iz (complex (- (imagpart z)) (realpart z)))
1081     (result (complex-atanh iz)))
1082     (complex (imagpart result)
1083     (- (realpart result)))))
1084    
1085     (defun complex-tan (z)
1086     "Compute tan z = -i * tanh(i * z)
1087    
1088     Z may be any number, but the result is always a complex."
1089     (declare (number z))
1090     ;; tan z = -i * tanh(i*z)
1091     (let* ((iz (complex (- (imagpart z)) (realpart z)))
1092     (result (complex-tanh iz)))
1093     (complex (imagpart result)
1094     (- (realpart result)))))
1095     )

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