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), i.e. 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.
537 "Return the hyperbolic sine of NUMBER."
538 (number-dispatch ((number number))
539 (handle-reals %sinh number)
541 (let ((x (realpart number))
542 (y (imagpart number)))
543 (complex (* (sinh x) (cos y))
544 (* (cosh x) (sin y)))))))
548 "Return the hyperbolic cosine of NUMBER."
549 (number-dispatch ((number number))
550 (handle-reals %cosh number)
552 (let ((x (realpart number))
553 (y (imagpart number)))
554 (complex (* (cosh x) (cos y))
555 (* (sinh x) (sin y)))))))
559 "Return the hyperbolic tangent of NUMBER."
560 (number-dispatch ((number number))
561 (handle-reals %tanh number)
563 (complex-tanh number))))
565 (defun asinh (number)
567 "Return the hyperbolic arc sine of NUMBER."
568 (number-dispatch ((number number))
569 (handle-reals %asinh number)
571 (complex-asinh number))))
573 (defun acosh (number)
575 "Return the hyperbolic arc cosine of NUMBER."
576 (number-dispatch ((number number))
578 ;; acosh is complex if number < 1
580 (complex-acosh number)
581 (coerce (%acosh (coerce number 'double-float)) 'single-float)))
582 (((foreach single-float double-float))
583 (if (< number (coerce 1 '(dispatch-type number)))
584 (complex-acosh number)
585 (coerce (%acosh (coerce number 'double-float))
586 '(dispatch-type number))))
588 (complex-acosh number))))
590 (defun atanh (number)
592 "Return the hyperbolic arc tangent of NUMBER."
593 (number-dispatch ((number number))
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))))
606 (complex-atanh number))))
608 ;;; HP-UX does not supply a C version of log1p, so use the definition.
610 ;;; FIXME: This is really not a good definition. As per Raymond Toy
611 ;;; working on CMU CL, "The definition really loses big-time in
612 ;;; roundoff as x gets small."
614 #!-sb-fluid (declaim (inline %log1p))
616 (defun %log1p (number)
617 (declare (double-float number)
618 (optimize (speed 3) (safety 0)))
619 (the double-float (log (the (double-float 0d0) (+ number 1d0)))))
621 ;;;; not-OLD-SPECFUN stuff
623 ;;;; (This was conditional on #-OLD-SPECFUN in the CMU CL sources,
624 ;;;; but OLD-SPECFUN was mentioned nowhere else, so it seems to be
625 ;;;; the standard special function system.)
627 ;;;; This is a set of routines that implement many elementary
628 ;;;; transcendental functions as specified by ANSI Common Lisp. The
629 ;;;; implementation is based on Kahan's paper.
631 ;;;; I believe I have accurately implemented the routines and are
632 ;;;; correct, but you may want to check for your self.
634 ;;;; These functions are written for CMU Lisp and take advantage of
635 ;;;; some of the features available there. It may be possible,
636 ;;;; however, to port this to other Lisps.
638 ;;;; Some functions are significantly more accurate than the original
639 ;;;; definitions in CMU Lisp. In fact, some functions in CMU Lisp
640 ;;;; give the wrong answer like (acos #c(-2.0 0.0)), where the true
641 ;;;; answer is pi + i*log(2-sqrt(3)).
643 ;;;; All of the implemented functions will take any number for an
644 ;;;; input, but the result will always be a either a complex
645 ;;;; single-float or a complex double-float.
647 ;;;; general functions:
659 ;;;; utility functions:
662 ;;;; internal functions:
663 ;;;; square coerce-to-complex-type cssqs complex-log-scaled
666 ;;;; Kahan, W. "Branch Cuts for Complex Elementary Functions, or Much
667 ;;;; Ado About Nothing's Sign Bit" in Iserles and Powell (eds.) "The
668 ;;;; State of the Art in Numerical Analysis", pp. 165-211, Clarendon
671 ;;;; The original CMU CL code requested:
672 ;;;; Please send any bug reports, comments, or improvements to
673 ;;;; Raymond Toy at toy@rtp.ericsson.se.
675 ;;; FIXME: In SBCL, the floating point infinity constants like
676 ;;; SB!EXT:DOUBLE-FLOAT-POSITIVE-INFINITY aren't available as
677 ;;; constants at cross-compile time, because the cross-compilation
678 ;;; host might not have support for floating point infinities. Thus,
679 ;;; they're effectively implemented as special variable references,
680 ;;; and the code below which uses them might be unnecessarily
681 ;;; inefficient. Perhaps some sort of MAKE-LOAD-TIME-VALUE hackery
682 ;;; should be used instead?
684 (declaim (inline square))
686 (declare (double-float x))
689 ;;; original CMU CL comment, apparently re. SCALB and LOGB and
691 ;;; If you have these functions in libm, perhaps they should be used
692 ;;; instead of these Lisp versions. These versions are probably good
693 ;;; enough, especially since they are portable.
695 ;;; Compute 2^N * X without computing 2^N first. (Use properties of
696 ;;; the underlying floating-point format.)
697 (declaim (inline scalb))
699 (declare (type double-float x)
700 (type double-float-exponent n))
703 ;;; This is like LOGB, but X is not infinity and non-zero and not a
704 ;;; NaN, so we can always return an integer.
705 (declaim (inline logb-finite))
706 (defun logb-finite (x)
707 (declare (type double-float x))
708 (multiple-value-bind (signif exponent sign)
710 (declare (ignore signif sign))
711 ;; DECODE-FLOAT is almost right, except that the exponent is off
715 ;;; Compute an integer N such that 1 <= |2^N * x| < 2.
716 ;;; For the special cases, the following values are used:
719 ;;; +/- infinity +infinity
722 (declare (type double-float x))
723 (cond ((float-nan-p x)
725 ((float-infinity-p x)
726 sb!ext:double-float-positive-infinity)
728 ;; The answer is negative infinity, but we are supposed to
729 ;; signal divide-by-zero, so do the actual division
735 ;;; This function is used to create a complex number of the
736 ;;; appropriate type:
737 ;;; Create complex number with real part X and imaginary part Y
738 ;;; such that has the same type as Z. If Z has type (complex
739 ;;; rational), the X and Y are coerced to single-float.
740 #!+long-float (eval-when (:compile-toplevel :load-toplevel :execute)
741 (error "needs work for long float support"))
742 (declaim (inline coerce-to-complex-type))
743 (defun coerce-to-complex-type (x y z)
744 (declare (double-float x y)
746 (if (subtypep (type-of (realpart z)) 'double-float)
748 ;; Convert anything that's not a DOUBLE-FLOAT to a SINGLE-FLOAT.
749 (complex (float x 1f0)
752 ;;; Compute |(x+i*y)/2^k|^2 scaled to avoid over/underflow. The
753 ;;; result is r + i*k, where k is an integer.
754 #!+long-float (eval-when (:compile-toplevel :load-toplevel :execute)
755 (error "needs work for long float support"))
757 (let ((x (float (realpart z) 1d0))
758 (y (float (imagpart z) 1d0)))
759 ;; Would this be better handled using an exception handler to
760 ;; catch the overflow or underflow signal? For now, we turn all
761 ;; traps off and look at the accrued exceptions to see if any
762 ;; signal would have been raised.
763 (with-float-traps-masked (:underflow :overflow)
764 (let ((rho (+ (square x) (square y))))
765 (declare (optimize (speed 3) (space 0)))
766 (cond ((and (or (float-nan-p rho)
767 (float-infinity-p rho))
768 (or (float-infinity-p (abs x))
769 (float-infinity-p (abs y))))
770 (values sb!ext:double-float-positive-infinity 0))
771 ((let ((threshold #.(/ least-positive-double-float
772 double-float-epsilon))
773 (traps (ldb sb!vm::float-sticky-bits
774 (sb!vm:floating-point-modes))))
775 ;; Overflow raised or (underflow raised and rho <
777 (or (not (zerop (logand sb!vm:float-overflow-trap-bit traps)))
778 (and (not (zerop (logand sb!vm:float-underflow-trap-bit
781 ;; If we're here, neither x nor y are infinity and at
782 ;; least one is non-zero.. Thus logb returns a nice
784 (let ((k (- (logb-finite (max (abs x) (abs y))))))
785 (values (+ (square (scalb x k))
786 (square (scalb y k)))
791 ;;; principal square root of Z
793 ;;; Z may be any NUMBER, but the result is always a COMPLEX.
794 (defun complex-sqrt (z)
796 (multiple-value-bind (rho k)
798 (declare (type (or (member 0d0) (double-float 0d0)) rho)
800 (let ((x (float (realpart z) 1.0d0))
801 (y (float (imagpart z) 1.0d0))
804 (declare (double-float x y eta nu))
807 ;; space 0 to get maybe-inline functions inlined.
808 (declare (optimize (speed 3) (space 0)))
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 (optimize (speed 3)))
852 (let ((beta (max (abs x) (abs y)))
853 (theta (min (abs x) (abs y))))
854 (coerce-to-complex-type (if (and (zerop k)
858 (/ (%log1p (+ (* (- beta 1.0d0)
867 ;;; log of Z = log |Z| + i * arg Z
869 ;;; Z may be any number, but the result is always a complex.
870 (defun complex-log (z)
872 (complex-log-scaled z 0))
874 ;;; KLUDGE: Let us note the following "strange" behavior. atanh 1.0d0
875 ;;; is +infinity, but the following code returns approx 176 + i*pi/4.
876 ;;; The reason for the imaginary part is caused by the fact that arg
877 ;;; i*y is never 0 since we have positive and negative zeroes. -- rtoy
878 ;;; Compute atanh z = (log(1+z) - log(1-z))/2.
879 (defun complex-atanh (z)
882 (theta (/ (sqrt most-positive-double-float) 4.0d0))
883 (rho (/ 4.0d0 (sqrt most-positive-double-float)))
884 (half-pi (/ pi 2.0d0))
885 (rp (float (realpart z) 1.0d0))
886 (beta (float-sign rp 1.0d0))
888 (y (* beta (- (float (imagpart z) 1.0d0))))
891 ;; Shouldn't need this declare.
892 (declare (double-float x y))
894 (declare (optimize (speed 3)))
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))))
909 ;; Should this be changed so that if y is zero, eta is set
910 ;; to +infinity instead of approx 176? In any case
911 ;; tanh(176) is 1.0d0 within working precision.
912 (let ((t1 (+ 4d0 (square y)))
913 (t2 (+ (abs y) rho)))
914 (setf eta (log (/ (sqrt (sqrt t1)))
918 (+ half-pi (atan (* 0.5d0 t2))))))))
920 (let ((t1 (+ (abs y) rho)))
921 ;; Normal case using log1p(x) = log(1 + x)
923 (%log1p (/ (* 4.0d0 x)
924 (+ (square (- 1.0d0 x))
931 (coerce-to-complex-type (* beta eta)
935 ;;; Compute tanh z = sinh z / cosh z.
936 (defun complex-tanh (z)
938 (let ((x (float (realpart z) 1.0d0))
939 (y (float (imagpart z) 1.0d0)))
941 ;; space 0 to get maybe-inline functions inlined
942 (declare (optimize (speed 3) (space 0)))
944 #-(or linux hpux) #.(/ (asinh most-positive-double-float) 4d0)
945 ;; This is more accurate under linux.
946 #+(or linux hpux) #.(/ (+ (log 2.0d0)
947 (log most-positive-double-float)) 4d0))
948 (coerce-to-complex-type (float-sign x)
952 (beta (+ 1.0d0 (* tv tv)))
954 (rho (sqrt (+ 1.0d0 (* s s)))))
955 (if (float-infinity-p (abs tv))
956 (coerce-to-complex-type (/ rho s)
959 (let ((den (+ 1.0d0 (* beta s s))))
960 (coerce-to-complex-type (/ (* beta rho s)
965 ;;; Compute acos z = pi/2 - asin z.
967 ;;; Z may be any NUMBER, but the result is always a COMPLEX.
968 (defun complex-acos (z)
969 ;; Kahan says we should only compute the parts needed. Thus, the
970 ;; REALPART's below should only compute the real part, not the whole
971 ;; complex expression. Doing this can be important because we may get
972 ;; spurious signals that occur in the part that we are not using.
974 ;; However, we take a pragmatic approach and just use the whole
977 ;; NOTE: The formula given by Kahan is somewhat ambiguous in whether
978 ;; it's the conjugate of the square root or the square root of the
979 ;; conjugate. This needs to be checked.
981 ;; I checked. It doesn't matter because (conjugate (sqrt z)) is the
982 ;; same as (sqrt (conjugate z)) for all z. This follows because
984 ;; (conjugate (sqrt z)) = exp(0.5*log |z|)*exp(-0.5*j*arg z).
986 ;; (sqrt (conjugate z)) = exp(0.5*log|z|)*exp(0.5*j*arg conj z)
988 ;; and these two expressions are equal if and only if arg conj z =
989 ;; -arg z, which is clearly true for all z.
991 (let ((sqrt-1+z (complex-sqrt (+ 1 z)))
992 (sqrt-1-z (complex-sqrt (- 1 z))))
993 (with-float-traps-masked (:divide-by-zero)
994 (complex (* 2 (atan (/ (realpart sqrt-1-z)
995 (realpart sqrt-1+z))))
996 (asinh (imagpart (* (conjugate sqrt-1+z)
999 ;;; Compute acosh z = 2 * log(sqrt((z+1)/2) + sqrt((z-1)/2))
1001 ;;; Z may be any NUMBER, but the result is always a COMPLEX.
1002 (defun complex-acosh (z)
1003 (declare (number z))
1004 (let ((sqrt-z-1 (complex-sqrt (- z 1)))
1005 (sqrt-z+1 (complex-sqrt (+ z 1))))
1006 (with-float-traps-masked (:divide-by-zero)
1007 (complex (asinh (realpart (* (conjugate sqrt-z-1)
1009 (* 2 (atan (/ (imagpart sqrt-z-1)
1010 (realpart sqrt-z+1))))))))
1012 ;;; Compute asin z = asinh(i*z)/i.
1014 ;;; Z may be any NUMBER, but the result is always a COMPLEX.
1015 (defun complex-asin (z)
1016 (declare (number z))
1017 (let ((sqrt-1-z (complex-sqrt (- 1 z)))
1018 (sqrt-1+z (complex-sqrt (+ 1 z))))
1019 (with-float-traps-masked (:divide-by-zero)
1020 (complex (atan (/ (realpart z)
1021 (realpart (* sqrt-1-z sqrt-1+z))))
1022 (asinh (imagpart (* (conjugate sqrt-1-z)
1025 ;;; Compute asinh z = log(z + sqrt(1 + z*z)).
1027 ;;; Z may be any number, but the result is always a complex.
1028 (defun complex-asinh (z)
1029 (declare (number z))
1030 ;; asinh z = -i * asin (i*z)
1031 (let* ((iz (complex (- (imagpart z)) (realpart z)))
1032 (result (complex-asin iz)))
1033 (complex (imagpart result)
1034 (- (realpart result)))))
1036 ;;; Compute atan z = atanh (i*z) / i.
1038 ;;; Z may be any number, but the result is always a complex.
1039 (defun complex-atan (z)
1040 (declare (number z))
1041 ;; atan z = -i * atanh (i*z)
1042 (let* ((iz (complex (- (imagpart z)) (realpart z)))
1043 (result (complex-atanh iz)))
1044 (complex (imagpart result)
1045 (- (realpart result)))))
1047 ;;; Compute tan z = -i * tanh(i * z)
1049 ;;; Z may be any number, but the result is always a complex.
1050 (defun complex-tan (z)
1051 (declare (number z))
1052 ;; tan z = -i * tanh(i*z)
1053 (let* ((iz (complex (- (imagpart z)) (realpart z)))
1054 (result (complex-tanh iz)))
1055 (complex (imagpart result)
1056 (- (realpart result)))))