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 system dies on stack overflow or memory
160 ;;; exhaustion, it seems reasonable to have this, but its default
161 ;;; should be NIL, and when it's NIL, anything should be accepted.
162 (defparameter *intexp-maximum-exponent* 10000)
164 ;;; This function precisely calculates base raised to an integral
165 ;;; power. It separates the cases by the sign of power, for efficiency
166 ;;; reasons, as powers can be calculated more efficiently if power is
167 ;;; a positive integer. Values of power are calculated as positive
168 ;;; 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
191 ;;; off, assuming it will return 0 if the result may be complex. If
192 ;;; so, we call COMPLEX-POW which directly computes the complex
193 ;;; result. We also separate the complex-real and real-complex cases
194 ;;; from the general complex case.
195 (defun expt (base power)
197 "Returns BASE raised to the POWER."
200 (labels (;; determine if the double float is an integer.
201 ;; 0 - not an integer
205 (declare (type (unsigned-byte 31) ihi)
206 (type (unsigned-byte 32) lo)
207 (optimize (speed 3) (safety 0)))
209 (declare (type fixnum isint))
210 (cond ((>= ihi #x43400000) ; exponent >= 53
213 (let ((k (- (ash ihi -20) #x3ff))) ; exponent
214 (declare (type (mod 53) k))
216 (let* ((shift (- 52 k))
217 (j (logand (ash lo (- shift))))
219 (declare (type (mod 32) shift)
220 (type (unsigned-byte 32) j j2))
222 (setq isint (- 2 (logand j 1))))))
224 (let* ((shift (- 20 k))
225 (j (ash ihi (- shift)))
227 (declare (type (mod 32) shift)
228 (type (unsigned-byte 31) j j2))
230 (setq isint (- 2 (logand j 1))))))))))
232 (real-expt (x y rtype)
233 (let ((x (coerce x 'double-float))
234 (y (coerce y 'double-float)))
235 (declare (double-float x y))
236 (let* ((x-hi (sb!kernel:double-float-high-bits x))
237 (x-lo (sb!kernel:double-float-low-bits x))
238 (x-ihi (logand x-hi #x7fffffff))
239 (y-hi (sb!kernel:double-float-high-bits y))
240 (y-lo (sb!kernel:double-float-low-bits y))
241 (y-ihi (logand y-hi #x7fffffff)))
242 (declare (type (signed-byte 32) x-hi y-hi)
243 (type (unsigned-byte 31) x-ihi y-ihi)
244 (type (unsigned-byte 32) x-lo y-lo))
246 (when (zerop (logior y-ihi y-lo))
247 (return-from real-expt (coerce 1d0 rtype)))
249 (when (or (> x-ihi #x7ff00000)
250 (and (= x-ihi #x7ff00000) (/= x-lo 0))
252 (and (= y-ihi #x7ff00000) (/= y-lo 0)))
253 (return-from real-expt (coerce (+ x y) rtype)))
254 (let ((yisint (if (< x-hi 0) (isint y-ihi y-lo) 0)))
255 (declare (type fixnum yisint))
256 ;; special value of y
257 (when (and (zerop y-lo) (= y-ihi #x7ff00000))
259 (return-from real-expt
260 (cond ((and (= x-ihi #x3ff00000) (zerop x-lo))
262 (coerce (- y y) rtype))
263 ((>= x-ihi #x3ff00000)
264 ;; (|x|>1)**+-inf = inf,0
269 ;; (|x|<1)**-,+inf = inf,0
272 (coerce 0 rtype))))))
274 (let ((abs-x (abs x)))
275 (declare (double-float abs-x))
276 ;; special value of x
277 (when (and (zerop x-lo)
278 (or (= x-ihi #x7ff00000) (zerop x-ihi)
279 (= x-ihi #x3ff00000)))
280 ;; x is +-0,+-inf,+-1
281 (let ((z (if (< y-hi 0)
282 (/ 1 abs-x) ; z = (1/|x|)
284 (declare (double-float z))
286 (cond ((and (= x-ihi #x3ff00000) (zerop yisint))
288 (let ((y*pi (* y pi)))
289 (declare (double-float y*pi))
290 (return-from real-expt
292 (coerce (%cos y*pi) rtype)
293 (coerce (%sin y*pi) rtype)))))
295 ;; (x<0)**odd = -(|x|**odd)
297 (return-from real-expt (coerce z rtype))))
301 (coerce (sb!kernel::%pow x y) rtype)
303 (let ((pow (sb!kernel::%pow abs-x y)))
304 (declare (double-float pow))
307 (coerce (* -1d0 pow) rtype))
311 (let ((y*pi (* y pi)))
312 (declare (double-float y*pi))
314 (coerce (* pow (%cos y*pi))
316 (coerce (* pow (%sin y*pi))
318 (declare (inline real-expt))
319 (number-dispatch ((base number) (power number))
320 (((foreach fixnum (or bignum ratio) (complex rational)) integer)
322 (((foreach single-float double-float) rational)
323 (real-expt base power '(dispatch-type base)))
324 (((foreach fixnum (or bignum ratio) single-float)
325 (foreach ratio single-float))
326 (real-expt base power 'single-float))
327 (((foreach fixnum (or bignum ratio) single-float double-float)
329 (real-expt base power 'double-float))
330 ((double-float single-float)
331 (real-expt base power 'double-float))
332 (((foreach (complex rational) (complex float)) rational)
333 (* (expt (abs base) power)
334 (cis (* power (phase base)))))
335 (((foreach fixnum (or bignum ratio) single-float double-float)
337 (if (and (zerop base) (plusp (realpart power)))
339 (exp (* power (log base)))))
340 (((foreach (complex float) (complex rational))
341 (foreach complex double-float single-float))
342 (if (and (zerop base) (plusp (realpart power)))
344 (exp (* power (log base)))))))))
346 (defun log (number &optional (base nil base-p))
348 "Return the logarithm of NUMBER in the base BASE, which defaults to e."
352 (/ (log number) (log base)))
353 (number-dispatch ((number number))
354 (((foreach fixnum bignum ratio))
356 (complex (log (- number)) (coerce pi 'single-float))
357 (coerce (%log (coerce number 'double-float)) 'single-float)))
358 (((foreach single-float double-float))
359 ;; Is (log -0) -infinity (libm.a) or -infinity + i*pi (Kahan)?
360 ;; Since this doesn't seem to be an implementation issue
361 ;; I (pw) take the Kahan result.
362 (if (< (float-sign number)
363 (coerce 0 '(dispatch-type number)))
364 (complex (log (- number)) (coerce pi '(dispatch-type number)))
365 (coerce (%log (coerce number 'double-float))
366 '(dispatch-type number))))
368 (complex-log number)))))
372 "Return the square root of NUMBER."
373 (number-dispatch ((number number))
374 (((foreach fixnum bignum ratio))
376 (complex-sqrt number)
377 (coerce (%sqrt (coerce number 'double-float)) 'single-float)))
378 (((foreach single-float double-float))
380 (complex-sqrt number)
381 (coerce (%sqrt (coerce number 'double-float))
382 '(dispatch-type number))))
384 (complex-sqrt number))))
386 ;;;; trigonometic and related functions
390 "Returns the absolute value of the number."
391 (number-dispatch ((number number))
392 (((foreach single-float double-float fixnum rational))
395 (let ((rx (realpart number))
396 (ix (imagpart number)))
399 (sqrt (+ (* rx rx) (* ix ix))))
401 (coerce (%hypot (coerce rx 'double-float)
402 (coerce ix 'double-float))
407 (defun phase (number)
409 "Return the angle part of the polar representation of a complex number.
410 For complex numbers, this is (atan (imagpart number) (realpart number)).
411 For non-complex positive numbers, this is 0. For non-complex negative
416 (coerce pi 'single-float)
419 (if (minusp (float-sign number))
420 (coerce pi 'single-float)
423 (if (minusp (float-sign number))
424 (coerce pi 'double-float)
427 (atan (imagpart number) (realpart number)))))
431 "Return the sine of NUMBER."
432 (number-dispatch ((number number))
433 (handle-reals %sin number)
435 (let ((x (realpart number))
436 (y (imagpart number)))
437 (complex (* (sin x) (cosh y))
438 (* (cos x) (sinh y)))))))
442 "Return the cosine of NUMBER."
443 (number-dispatch ((number number))
444 (handle-reals %cos number)
446 (let ((x (realpart number))
447 (y (imagpart number)))
448 (complex (* (cos x) (cosh y))
449 (- (* (sin x) (sinh y))))))))
453 "Return the tangent of NUMBER."
454 (number-dispatch ((number number))
455 (handle-reals %tan number)
457 (complex-tan number))))
461 "Return cos(Theta) + i sin(Theta), AKA exp(i Theta)."
462 (declare (type real theta))
463 (complex (cos theta) (sin theta)))
467 "Return the arc sine of NUMBER."
468 (number-dispatch ((number number))
470 (if (or (> number 1) (< number -1))
471 (complex-asin number)
472 (coerce (%asin (coerce number 'double-float)) 'single-float)))
473 (((foreach single-float double-float))
474 (if (or (> number (coerce 1 '(dispatch-type number)))
475 (< number (coerce -1 '(dispatch-type number))))
476 (complex-asin number)
477 (coerce (%asin (coerce number 'double-float))
478 '(dispatch-type number))))
480 (complex-asin number))))
484 "Return the arc cosine of NUMBER."
485 (number-dispatch ((number number))
487 (if (or (> number 1) (< number -1))
488 (complex-acos number)
489 (coerce (%acos (coerce number 'double-float)) 'single-float)))
490 (((foreach single-float double-float))
491 (if (or (> number (coerce 1 '(dispatch-type number)))
492 (< number (coerce -1 '(dispatch-type number))))
493 (complex-acos number)
494 (coerce (%acos (coerce number 'double-float))
495 '(dispatch-type number))))
497 (complex-acos number))))
499 (defun atan (y &optional (x nil xp))
501 "Return the arc tangent of Y if X is omitted or Y/X if X is supplied."
504 (declare (type double-float y x)
505 (values double-float))
508 (if (plusp (float-sign x))
511 (float-sign y (/ pi 2)))
513 (number-dispatch ((y number) (x number))
515 (foreach double-float single-float fixnum bignum ratio))
516 (atan2 y (coerce x 'double-float)))
517 (((foreach single-float fixnum bignum ratio)
519 (atan2 (coerce y 'double-float) x))
520 (((foreach single-float fixnum bignum ratio)
521 (foreach single-float fixnum bignum ratio))
522 (coerce (atan2 (coerce y 'double-float) (coerce x 'double-float))
524 (number-dispatch ((y number))
525 (handle-reals %atan y)
529 ;; It seems that everyone has a C version of sinh, cosh, and
530 ;; tanh. Let's use these for reals because the original
531 ;; implementations based on the definitions lose big in round-off
532 ;; error. These bad definitions also mean that sin and cos for
533 ;; complex numbers can also lose big.
538 "Return the hyperbolic sine of NUMBER."
539 (/ (- (exp number) (exp (- number))) 2))
543 "Return the hyperbolic sine of NUMBER."
544 (number-dispatch ((number number))
545 (handle-reals %sinh number)
547 (let ((x (realpart number))
548 (y (imagpart number)))
549 (complex (* (sinh x) (cos y))
550 (* (cosh x) (sin y)))))))
555 "Return the hyperbolic cosine of NUMBER."
556 (/ (+ (exp number) (exp (- number))) 2))
560 "Return the hyperbolic cosine of NUMBER."
561 (number-dispatch ((number number))
562 (handle-reals %cosh number)
564 (let ((x (realpart number))
565 (y (imagpart number)))
566 (complex (* (cosh x) (cos y))
567 (* (sinh x) (sin y)))))))
571 "Return the hyperbolic tangent of NUMBER."
572 (number-dispatch ((number number))
573 (handle-reals %tanh number)
575 (complex-tanh number))))
577 (defun asinh (number)
579 "Return the hyperbolic arc sine of NUMBER."
580 (number-dispatch ((number number))
581 (handle-reals %asinh number)
583 (complex-asinh number))))
585 (defun acosh (number)
587 "Return the hyperbolic arc cosine of NUMBER."
588 (number-dispatch ((number number))
590 ;; acosh is complex if number < 1
592 (complex-acosh number)
593 (coerce (%acosh (coerce number 'double-float)) 'single-float)))
594 (((foreach single-float double-float))
595 (if (< number (coerce 1 '(dispatch-type number)))
596 (complex-acosh number)
597 (coerce (%acosh (coerce number 'double-float))
598 '(dispatch-type number))))
600 (complex-acosh number))))
602 (defun atanh (number)
604 "Return the hyperbolic arc tangent of NUMBER."
605 (number-dispatch ((number number))
607 ;; atanh is complex if |number| > 1
608 (if (or (> number 1) (< number -1))
609 (complex-atanh number)
610 (coerce (%atanh (coerce number 'double-float)) 'single-float)))
611 (((foreach single-float double-float))
612 (if (or (> number (coerce 1 '(dispatch-type number)))
613 (< number (coerce -1 '(dispatch-type number))))
614 (complex-atanh number)
615 (coerce (%atanh (coerce number 'double-float))
616 '(dispatch-type number))))
618 (complex-atanh number))))
620 ;;; HP-UX does not supply a C version of log1p, so
621 ;;; use the definition.
623 #!-sb-fluid (declaim (inline %log1p))
625 (defun %log1p (number)
626 (declare (double-float number)
627 (optimize (speed 3) (safety 0)))
628 (the double-float (log (the (double-float 0d0) (+ number 1d0)))))
630 ;;;; OLD-SPECFUN stuff
632 ;;;; (This was conditional on #-OLD-SPECFUN in the CMU CL sources,
633 ;;;; but OLD-SPECFUN was mentioned nowhere else, so it seems to be
634 ;;;; the standard special function system.)
636 ;;;; This is a set of routines that implement many elementary
637 ;;;; transcendental functions as specified by ANSI Common Lisp. The
638 ;;;; implementation is based on Kahan's paper.
640 ;;;; I believe I have accurately implemented the routines and are
641 ;;;; correct, but you may want to check for your self.
643 ;;;; These functions are written for CMU Lisp and take advantage of
644 ;;;; some of the features available there. It may be possible,
645 ;;;; however, to port this to other Lisps.
647 ;;;; Some functions are significantly more accurate than the original
648 ;;;; definitions in CMU Lisp. In fact, some functions in CMU Lisp
649 ;;;; give the wrong answer like (acos #c(-2.0 0.0)), where the true
650 ;;;; answer is pi + i*log(2-sqrt(3)).
652 ;;;; All of the implemented functions will take any number for an
653 ;;;; input, but the result will always be a either a complex
654 ;;;; single-float or a complex double-float.
656 ;;;; general functions:
668 ;;;; utility functions:
671 ;;;; internal functions:
672 ;;;; square coerce-to-complex-type cssqs complex-log-scaled
675 ;;;; Kahan, W. "Branch Cuts for Complex Elementary Functions, or Much
676 ;;;; Ado About Nothing's Sign Bit" in Iserles and Powell (eds.) "The
677 ;;;; State of the Art in Numerical Analysis", pp. 165-211, Clarendon
680 ;;;; The original CMU CL code requested:
681 ;;;; Please send any bug reports, comments, or improvements to
682 ;;;; Raymond Toy at toy@rtp.ericsson.se.
684 ;;; FIXME: In SBCL, the floating point infinity constants like
685 ;;; SB!EXT:DOUBLE-FLOAT-POSITIVE-INFINITY aren't available as
686 ;;; constants at cross-compile time, because the cross-compilation
687 ;;; host might not have support for floating point infinities. Thus,
688 ;;; they're effectively implemented as special variable references,
689 ;;; and the code below which uses them might be unnecessarily
690 ;;; inefficient. Perhaps some sort of MAKE-LOAD-TIME-VALUE hackery
691 ;;; should be used instead?
693 (declaim (inline square))
694 (declaim (ftype (function (double-float) (double-float 0d0)) square))
696 (declare (double-float x)
697 (values (double-float 0d0)))
700 ;;; original CMU CL comment, apparently re. SCALB and LOGB and
702 ;;; If you have these functions in libm, perhaps they should be used
703 ;;; instead of these Lisp versions. These versions are probably good
704 ;;; enough, especially since they are portable.
706 ;;; Compute 2^N * X without computing 2^N first. (Use properties of
707 ;;; the underlying floating-point format.)
708 (declaim (inline scalb))
710 (declare (type double-float x)
711 (type double-float-exponent n))
714 ;;; Compute an integer N such that 1 <= |2^N * x| < 2.
715 ;;; For the special cases, the following values are used:
718 ;;; +/- infinity +infinity
721 (declare (type double-float x))
722 (cond ((float-nan-p x)
724 ((float-infinity-p x)
725 sb!ext:double-float-positive-infinity)
727 ;; The answer is negative infinity, but we are supposed to
728 ;; signal divide-by-zero.
729 ;; (error 'division-by-zero :operation 'logb :operands (list x))
733 (multiple-value-bind (signif expon sign)
735 (declare (ignore signif sign))
736 ;; DECODE-FLOAT is almost right, except that the exponent
740 ;;; This function is used to create a complex number of the
741 ;;; appropriate type:
742 ;;; Create complex number with real part X and imaginary part Y
743 ;;; such that has the same type as Z. If Z has type (complex
744 ;;; rational), the X and Y are coerced to single-float.
745 #!+long-float (eval-when (:compile-toplevel :load-toplevel :execute)
746 (error "needs work for long float support"))
747 (declaim (inline coerce-to-complex-type))
748 (defun coerce-to-complex-type (x y z)
749 (declare (double-float x y)
751 (if (subtypep (type-of (realpart z)) 'double-float)
753 ;; Convert anything that's not a DOUBLE-FLOAT to a SINGLE-FLOAT.
754 (complex (float x 1.0)
757 ;;; Compute |(x+i*y)/2^k|^2 scaled to avoid over/underflow. The
758 ;;; result is r + i*k, where k is an integer.
759 #!+long-float (eval-when (:compile-toplevel :load-toplevel :execute)
760 (error "needs work for long float support"))
763 (let ((x (float (realpart z) 1d0))
764 (y (float (imagpart z) 1d0))
767 (declare (double-float x y)
768 (type (double-float 0d0) rho)
770 ;; Would this be better handled using an exception handler to
771 ;; catch the overflow or underflow signal? For now, we turn all
772 ;; traps off and look at the accrued exceptions to see if any
773 ;; signal would have been raised.
774 (with-float-traps-masked (:underflow :overflow)
775 (setf rho (+ (square x) (square y)))
776 (cond ((and (or (float-nan-p rho)
777 (float-infinity-p rho))
778 (or (float-infinity-p (abs x))
779 (float-infinity-p (abs y))))
780 (setf rho sb!ext:double-float-positive-infinity))
781 ((let ((threshold #.(/ least-positive-double-float
782 double-float-epsilon))
783 (traps (ldb sb!vm::float-sticky-bits
784 (sb!vm:floating-point-modes))))
785 ;; overflow raised or (underflow raised and rho < lambda/eps)
786 (or (not (zerop (logand sb!vm:float-overflow-trap-bit traps)))
787 (and (not (zerop (logand sb!vm:float-underflow-trap-bit
790 (setf k (logb (max (abs x) (abs y))))
791 (setf rho (+ (square (scalb x (- k)))
792 (square (scalb y (- k))))))))
795 ;;; principal square root of Z
797 ;;; Z may be any NUMBER, but the result is always a COMPLEX.
798 (defun complex-sqrt (z)
800 (multiple-value-bind (rho k)
802 (declare (type (double-float 0d0) rho)
804 (let ((x (float (realpart z) 1.0d0))
805 (y (float (imagpart z) 1.0d0))
808 (declare (double-float x y eta nu))
810 (if (not (float-nan-p x))
811 (setf rho (+ (scalb (abs x) (- k)) (sqrt rho))))
816 (setf k (1- (ash k -1)))
817 (setf rho (+ rho rho))))
819 (setf rho (scalb (sqrt rho) k))
825 (when (not (float-infinity-p (abs nu)))
826 (setf nu (/ (/ nu rho) 2d0)))
829 (setf nu (float-sign y rho))))
830 (coerce-to-complex-type eta nu z))))
832 ;;; Compute log(2^j*z).
834 ;;; This is for use with J /= 0 only when |z| is huge.
835 (defun complex-log-scaled (z j)
838 ;; The constants t0, t1, t2 should be evaluated to machine
839 ;; precision. In addition, Kahan says the accuracy of log1p
840 ;; influences the choices of these constants but doesn't say how to
841 ;; choose them. We'll just assume his choices matches our
842 ;; implementation of log1p.
843 (let ((t0 #.(/ 1 (sqrt 2.0d0)))
847 (x (float (realpart z) 1.0d0))
848 (y (float (imagpart z) 1.0d0)))
849 (multiple-value-bind (rho k)
851 (declare (type (double-float 0d0) rho)
853 (let ((beta (max (abs x) (abs y)))
854 (theta (min (abs x) (abs y))))
855 (declare (type (double-float 0d0) beta theta))
860 (setf rho (/ (%log1p (+ (* (- beta 1.0d0)
864 (setf rho (+ (/ (log rho) 2d0)
866 (setf theta (atan y x))
867 (coerce-to-complex-type rho theta z)))))
869 ;;; log of Z = log |Z| + i * arg Z
871 ;;; Z may be any number, but the result is always a complex.
872 (defun complex-log (z)
874 (complex-log-scaled z 0))
876 ;;; KLUDGE: Let us note the following "strange" behavior. atanh 1.0d0
877 ;;; is +infinity, but the following code returns approx 176 + i*pi/4.
878 ;;; The reason for the imaginary part is caused by the fact that arg
879 ;;; i*y is never 0 since we have positive and negative zeroes. -- rtoy
880 ;;; Compute atanh z = (log(1+z) - log(1-z))/2.
881 (defun complex-atanh (z)
884 (theta #.(/ (sqrt most-positive-double-float) 4.0d0))
885 (rho #.(/ 4.0d0 (sqrt most-positive-double-float)))
886 (half-pi #.(/ pi 2.0d0))
887 (rp (float (realpart z) 1.0d0))
888 (beta (float-sign rp 1.0d0))
890 (y (* beta (- (float (imagpart z) 1.0d0))))
893 (declare (double-float theta rho half-pi rp beta y eta nu)
894 (type (double-float 0d0) x))
895 (cond ((or (> x theta)
897 ;; to avoid overflow...
898 (setf eta (float-sign y half-pi))
899 ;; nu is real part of 1/(x + iy). This is x/(x^2+y^2),
900 ;; which can cause overflow. Arrange this computation so
901 ;; that it won't overflow.
902 (setf nu (let* ((x-bigger (> x (abs y)))
903 (r (if x-bigger (/ y x) (/ x y)))
904 (d (+ 1.0d0 (* r r))))
905 (declare (double-float r d))
910 ;; Should this be changed so that if y is zero, eta is set
911 ;; to +infinity instead of approx 176? In any case
912 ;; tanh(176) is 1.0d0 within working precision.
913 (let ((t1 (+ 4d0 (square y)))
914 (t2 (+ (abs y) rho)))
915 (declare (type (double-float 0d0) t1 t2))
917 (setf eta (log (/ (sqrt (sqrt t1)))
919 (setf eta (* 0.5d0 (log (the (double-float 0.0d0)
923 (+ half-pi (atan (* 0.5d0 t2))))))))
925 (let ((t1 (+ (abs y) rho)))
926 (declare (double-float t1))
927 ;; normal case using log1p(x) = log(1 + x)
929 (%log1p (/ (* 4.0d0 x)
930 (+ (square (- 1.0d0 x))
937 (coerce-to-complex-type (* beta eta)
941 ;;; Compute tanh z = sinh z / cosh z.
942 (defun complex-tanh (z)
944 (let ((x (float (realpart z) 1.0d0))
945 (y (float (imagpart z) 1.0d0)))
946 (declare (double-float x y))
948 #-(or linux hpux) #.(/ (asinh most-positive-double-float) 4d0)
949 ;; This is more accurate under linux.
950 #+(or linux hpux) #.(/ (+ (log 2.0d0)
951 (log most-positive-double-float))
953 (complex (float-sign x)
954 (float-sign y 0.0d0)))
957 (beta (+ 1.0d0 (* tv tv)))
959 (rho (sqrt (+ 1.0d0 (* s s)))))
960 (declare (double-float tv s)
961 (type (double-float 0.0d0) beta rho))
962 (if (float-infinity-p (abs tv))
963 (coerce-to-complex-type (/ rho s)
966 (let ((den (+ 1.0d0 (* beta s s))))
967 (coerce-to-complex-type (/ (* beta rho s)
972 ;;; Compute acos z = pi/2 - asin z.
974 ;;; Z may be any NUMBER, but the result is always a COMPLEX.
975 (defun complex-acos (z)
976 ;; Kahan says we should only compute the parts needed. Thus, the
977 ;; REALPART's below should only compute the real part, not the whole
978 ;; complex expression. Doing this can be important because we may get
979 ;; spurious signals that occur in the part that we are not using.
981 ;; However, we take a pragmatic approach and just use the whole
984 ;; NOTE: The formula given by Kahan is somewhat ambiguous in whether
985 ;; it's the conjugate of the square root or the square root of the
986 ;; conjugate. This needs to be checked.
988 ;; I checked. It doesn't matter because (conjugate (sqrt z)) is the
989 ;; same as (sqrt (conjugate z)) for all z. This follows because
991 ;; (conjugate (sqrt z)) = exp(0.5*log |z|)*exp(-0.5*j*arg z).
993 ;; (sqrt (conjugate z)) = exp(0.5*log|z|)*exp(0.5*j*arg conj z)
995 ;; and these two expressions are equal if and only if arg conj z =
996 ;; -arg z, which is clearly true for all z.
998 (let ((sqrt-1+z (complex-sqrt (+ 1 z)))
999 (sqrt-1-z (complex-sqrt (- 1 z))))
1000 (with-float-traps-masked (:divide-by-zero)
1001 (complex (* 2 (atan (/ (realpart sqrt-1-z)
1002 (realpart sqrt-1+z))))
1003 (asinh (imagpart (* (conjugate sqrt-1+z)
1006 ;;; Compute acosh z = 2 * log(sqrt((z+1)/2) + sqrt((z-1)/2))
1008 ;;; Z may be any NUMBER, but the result is always a COMPLEX.
1009 (defun complex-acosh (z)
1010 (declare (number z))
1011 (let ((sqrt-z-1 (complex-sqrt (- z 1)))
1012 (sqrt-z+1 (complex-sqrt (+ z 1))))
1013 (with-float-traps-masked (:divide-by-zero)
1014 (complex (asinh (realpart (* (conjugate sqrt-z-1)
1016 (* 2 (atan (/ (imagpart sqrt-z-1)
1017 (realpart sqrt-z+1))))))))
1019 ;;; Compute asin z = asinh(i*z)/i.
1021 ;;; Z may be any NUMBER, but the result is always a COMPLEX.
1022 (defun complex-asin (z)
1023 (declare (number z))
1024 (let ((sqrt-1-z (complex-sqrt (- 1 z)))
1025 (sqrt-1+z (complex-sqrt (+ 1 z))))
1026 (with-float-traps-masked (:divide-by-zero)
1027 (complex (atan (/ (realpart z)
1028 (realpart (* sqrt-1-z sqrt-1+z))))
1029 (asinh (imagpart (* (conjugate sqrt-1-z)
1032 ;;; Compute asinh z = log(z + sqrt(1 + z*z)).
1034 ;;; Z may be any number, but the result is always a complex.
1035 (defun complex-asinh (z)
1036 (declare (number z))
1037 ;; asinh z = -i * asin (i*z)
1038 (let* ((iz (complex (- (imagpart z)) (realpart z)))
1039 (result (complex-asin iz)))
1040 (complex (imagpart result)
1041 (- (realpart result)))))
1043 ;;; Compute atan z = atanh (i*z) / i.
1045 ;;; Z may be any number, but the result is always a complex.
1046 (defun complex-atan (z)
1047 (declare (number z))
1048 ;; atan z = -i * atanh (i*z)
1049 (let* ((iz (complex (- (imagpart z)) (realpart z)))
1050 (result (complex-atanh iz)))
1051 (complex (imagpart result)
1052 (- (realpart result)))))
1054 ;;; Compute tan z = -i * tanh(i * z)
1056 ;;; Z may be any number, but the result is always a complex.
1057 (defun complex-tan (z)
1058 (declare (number z))
1059 ;; tan z = -i * tanh(i*z)
1060 (let* ((iz (complex (- (imagpart z)) (realpart z)))
1061 (result (complex-tanh iz)))
1062 (complex (imagpart result)
1063 (- (realpart result)))))