1 ; Copyright 2022 Frank Duncan (frank@consxy.com) under AGPL3. See distributed LICENSE.txt.
2 (in-package #:strictmath)
3 ; This file is taken from part of Evita Common Lisp.
5 ; It has been updated to match the rest of the project's documentation and style
6 ; standards. But otherwise, the following copyright supersedes the above AGPL copyright.
8 ; Copyright (C) 1996-2007 by Project Vogue.
9 ; Written by Yoshifumi "VOGUE" INOUE. (yosi@msn.com)
11 ; Before that, it was based off of fdlibm
13 ; See fdlibm (http://www.netlib.org/fdlibm/)
14 ; See http://sources.redhat.com/newlib/
17 ; From fdlibm (http://www.netlib.org/fdlibm/)
18 ; See http://sources.redhat.com/newlib/
20 ; /* @(#)e_rem_pio2.c 5.1 93/09/24 *
22 ; * ====================================================
23 ; * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
25 ; * Developed at SunPro, a Sun Microsystems, Inc. business.
26 ; * Permission to use, copy, modify, and distribute this
27 ; * software is freely granted, provided that this notice
29 ; * ====================================================
33 ; * __ieee754_rem_pio2(x,y)
35 ; * return the remainder of x rem pi/2 in y[0]+y[1]
36 ; * use __kernel_rem_pio2()
39 ;;; Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
40 (defparameter +two-over-pi+
41 (make-array 66 :element-type '(unsigned-byte 32)
43 #(#xA2F983 #x6E4E44 #x1529FC #x2757D1 #xF534DD #xC0DB62
44 #x95993C #x439041 #xFE5163 #xABDEBB #xC561B7 #x246E3A
45 #x424DD2 #xE00649 #x2EEA09 #xD1921C #xFE1DEB #x1CB129
46 #xA73EE8 #x8235F5 #x2EBB44 #x84E99C #x7026B4 #x5F7E41
47 #x3991D6 #x398353 #x39F49C #x845F8B #xBDF928 #x3B1FF8
48 #x97FFDE #x05980F #xEF2F11 #x8B5A0A #x6D1F6D #x367ECF
49 #x27CB09 #xB74F46 #x3F669E #x5FEA2D #x7527BA #xC7EBE5
50 #xF17B3D #x0739F7 #x8A5292 #xEA6BFB #x5FB11F #x8D5D08
51 #x560330 #x46FC7B #x6BABF0 #xCFBC20 #x9AF436 #x1DA9E3
52 #x91615E #xE61B08 #x659985 #x5F14A0 #x68408D #xFFD880
53 #x4D7327 #x310606 #x1556CA #x73A8C9 #x60E27B #xC08C6B)))
55 (defparameter +npio2-hw+
56 (make-array 32 :element-type '(unsigned-byte 32)
58 #(#x3FF921FB #x400921FB #x4012D97C #x401921FB #x401F6A7A #x4022D97C
59 #x4025FDBB #x402921FB #x402C463A #x402F6A7A #x4031475C #x4032D97C
60 #x40346B9C #x4035FDBB #x40378FDB #x403921FB #x403AB41B #x403C463A
61 #x403DD85A #x403F6A7A #x40407E4C #x4041475C #x4042106C #x4042D97C
62 #x4043A28C #x40446B9C #x404534AC #x4045FDBB #x4046C6CB #x40478FDB
63 #x404858EB #x404921FB)))
67 ; invpio2: 53 bits of 2/pi
68 ; pio2_1: first 33 bit of pi/2
69 ; pio2_1t: pi/2 - pio2_1
70 ; pio2_2: second 33 bit of pi/2
71 ; pio2_2t: pi/2 - (pio2_1+pio2_2)
72 ; pio2_3: third 33 bit of pi/2
73 ; pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
77 (defun float64-rem-pio2-small (x ix hx)
79 ((pio2_1 #+nil 1.57079632673412561417e+00 #.(encode-float64 #x3FF921FB #x54400000))
80 (pio2_1t #+nil 6.07710050650619224932e-11 #.(encode-float64 #x3DD0B461 #x1A626331))
81 (pio2_2 #+nil 6.07710050630396597660e-11 #.(encode-float64 #x3DD0B461 #x1A600000))
82 (pio2_2t #+nil 2.02226624879595063154e-21 #.(encode-float64 #x3BA3198A #x2E037073)))
84 (let ((z (- x pio2_1)))
85 (if (not (eql ix #x3ff921fb))
86 ;; 33+53 bit pi is good enough
89 (y1 (- (- z y0) pio2_1t)))
90 (return (values 1 y0 y1)))
91 ;; near pi/2, use 33+33+53 bit pi *
95 (y1 (- (- z y0) pio2_2t)))
96 (return (values 1 y0 y1)))))
100 (if (not (eql ix #x3ff921fb))
101 ;; 33+53 bit pi is good enough
104 (y1 (+ (- z y0) pio2_1t)))
105 (return (values -1 y0 y1)))
106 ;; near pi/2, use 33+33+53 bit pi
110 (y1 (+ (- z y0) pio2_2t)))
111 (return (values -1 y0 y1))))))))
113 (defun float64-rem-pio2-medium (x ix hx)
115 ((half #+nil 5.00000000000000000000e-01 #.(encode-float64 #x3FE00000 #x00000000))
116 (invpio2 #+nil 6.36619772367581382433e-01 #.(encode-float64 #x3FE45F30 #x6DC9C883))
117 (pio2_1 #+nil 1.57079632673412561417e+00 #.(encode-float64 #x3FF921FB #x54400000))
118 (pio2_1t #+nil 6.07710050650619224932e-11 #.(encode-float64 #x3DD0B461 #x1A626331))
119 (pio2_2 #+nil 6.07710050630396597660e-11 #.(encode-float64 #x3DD0B461 #x1A600000))
120 (pio2_2t #+nil 2.02226624879595063154e-21 #.(encode-float64 #x3BA3198A #x2E037073))
121 (pio2_3 #+nil 2.02226624871116645580e-21 #.(encode-float64 #x3BA3198A #x2E000000))
122 (pio2_3t #+nil 8.47842766036889956997e-32 #.(encode-float64 #x397B839A #x252049C1)))
124 ((tt (float64-abs x))
125 (n (truncate (+ (* tt invpio2) half)))
127 (r (- tt (* fn pio2_1)))
128 (w (* fn pio2_1t)) ; 1st round good to 85 bit *
130 ;; quick check no cancellation
131 (unless (and (< n 32) (not (eql ix (elt +npio2-hw+ (- n 1)))))
134 (high (decode-float64 y0))
135 (i (- j (logand (ash high -20) #x7ff))))
136 ;; 2nd iteration needed, good to 118 *
139 (setq w (* fn pio2_2))
141 (setq w (- (* fn pio2_2t) (- (- tt r) w)))
143 (setq high (decode-float64 y0))
144 (setq i (- j (logand (ash high -20) #x7ff)))
146 ;; 3rd iteration need, 151 bits acc *
148 (setq tt r) ; will cover all possible cases *
149 (setq w (* fn pio2_3))
151 (setq w (- (* fn pio2_3t) (- (- tt r) w)))
152 (setq y0 (- r w))))))
153 (let ((y1 (- (- r y0) w)))
155 (return (values (- n) (- y0) (- y1)))
156 (return (values n y0 y1)))))))
158 (defun float64-rem-pio2 (x)
159 (declare (values fixnum double-float double-float))
160 (declare (type double-float x))
162 ((two24 #+nil 1.67772160000000000000e+07 #.(encode-float64 #x41700000 #x00000000)))
163 (multiple-value-bind (hx lx) (decode-float64 x)
165 ((ix (logand hx #x7fffffff)))
167 (when (<= ix #x3fe921fb) (return (values 0 x 0d0))) ;; |x| ~<= pi/4 , no need for reduction
168 (when (< ix #x4002d97c) (return (float64-rem-pio2-small x ix hx))) ;; |x| < 3pi/4, special case with n=+-1
169 (when (<= ix #x413921fb) (return (float64-rem-pio2-medium x ix hx))) ;; |x| ~<= 2^19*(pi/2), medium size
171 ;; all other (large) arguments
172 (when (>= ix #x7ff00000) ; x is inf or NaN *
173 (let ((nan (- x x))) (return (values 0 nan nan))))
175 ;; set z = scalbn(|x|,ilogb(x)-23)
177 ((e0 (- (ash ix -20) 1046)) ;; e0 = ilogb(z)-23
178 (z0 (encode-float64 (- ix (ash e0 20)) lx))
179 (tx0 (float (truncate z0) 0d0))
180 (z1 (* (- z0 tx0) two24))
181 (tx1 (float (truncate z1) 0d0))
182 (z2 (* (- z1 tx1) two24))
184 (multiple-value-bind (n y0 y1)
187 (cond ;; skip zero term
188 ((not (zerop tx2)) 3)
189 ((not (zerop tx1)) 2)
190 ((not (zerop tx0)) 1)
193 ((tx (make-array 3 :element-type 'double-float)))
194 (setf (elt tx 0) tx0)
195 (setf (elt tx 1) tx1)
196 (setf (elt tx 2) tx2)
197 (float64-kernel-rem-pio2 tx e0 nx 2 +two-over-pi+)))
199 (return (values (- n) (- y0) (- y1)))
200 (return (values n y0 y1)))))))))