1 (in-package #:strictmath)
2 ; This file is taken from part of Evita Common Lisp.
4 ; Copyright (C) 1996-2007 by Project Vogue.
5 ; Written by Yoshifumi "VOGUE" INOUE. (yosi@msn.com)
7 ; Before that, it was based off of fdlibm
9 ; See fdlibm (http://www.netlib.org/fdlibm/)
10 ; See http://sources.redhat.com/newlib/
13 ; From fdlibm (http://www.netlib.org/fdlibm/)
14 ; See http://sources.redhat.com/newlib/
16 ; /* @(#)e_rem_pio2.c 5.1 93/09/24 *
18 ; * ====================================================
19 ; * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
21 ; * Developed at SunPro, a Sun Microsystems, Inc. business.
22 ; * Permission to use, copy, modify, and distribute this
23 ; * software is freely granted, provided that this notice
25 ; * ====================================================
29 ; * __ieee754_rem_pio2(x,y)
31 ; * return the remainder of x rem pi/2 in y[0]+y[1]
32 ; * use __kernel_rem_pio2()
35 ;;; Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
36 (defparameter +two-over-pi+
37 (make-array 66 :element-type '(unsigned-byte 32)
39 #(#xA2F983 #x6E4E44 #x1529FC #x2757D1 #xF534DD #xC0DB62
40 #x95993C #x439041 #xFE5163 #xABDEBB #xC561B7 #x246E3A
41 #x424DD2 #xE00649 #x2EEA09 #xD1921C #xFE1DEB #x1CB129
42 #xA73EE8 #x8235F5 #x2EBB44 #x84E99C #x7026B4 #x5F7E41
43 #x3991D6 #x398353 #x39F49C #x845F8B #xBDF928 #x3B1FF8
44 #x97FFDE #x05980F #xEF2F11 #x8B5A0A #x6D1F6D #x367ECF
45 #x27CB09 #xB74F46 #x3F669E #x5FEA2D #x7527BA #xC7EBE5
46 #xF17B3D #x0739F7 #x8A5292 #xEA6BFB #x5FB11F #x8D5D08
47 #x560330 #x46FC7B #x6BABF0 #xCFBC20 #x9AF436 #x1DA9E3
48 #x91615E #xE61B08 #x659985 #x5F14A0 #x68408D #xFFD880
49 #x4D7327 #x310606 #x1556CA #x73A8C9 #x60E27B #xC08C6B)))
51 (defparameter +npio2-hw+
52 (make-array 32 :element-type '(unsigned-byte 32)
54 #(#x3FF921FB #x400921FB #x4012D97C #x401921FB #x401F6A7A #x4022D97C
55 #x4025FDBB #x402921FB #x402C463A #x402F6A7A #x4031475C #x4032D97C
56 #x40346B9C #x4035FDBB #x40378FDB #x403921FB #x403AB41B #x403C463A
57 #x403DD85A #x403F6A7A #x40407E4C #x4041475C #x4042106C #x4042D97C
58 #x4043A28C #x40446B9C #x404534AC #x4045FDBB #x4046C6CB #x40478FDB
59 #x404858EB #x404921FB)))
63 ; invpio2: 53 bits of 2/pi
64 ; pio2_1: first 33 bit of pi/2
65 ; pio2_1t: pi/2 - pio2_1
66 ; pio2_2: second 33 bit of pi/2
67 ; pio2_2t: pi/2 - (pio2_1+pio2_2)
68 ; pio2_3: third 33 bit of pi/2
69 ; pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
73 (defun float64-rem-pio2-small (x ix hx)
75 ((pio2_1 #+nil 1.57079632673412561417e+00 #.(encode-float64 #x3FF921FB #x54400000))
76 (pio2_1t #+nil 6.07710050650619224932e-11 #.(encode-float64 #x3DD0B461 #x1A626331))
77 (pio2_2 #+nil 6.07710050630396597660e-11 #.(encode-float64 #x3DD0B461 #x1A600000))
78 (pio2_2t #+nil 2.02226624879595063154e-21 #.(encode-float64 #x3BA3198A #x2E037073)))
80 (let ((z (- x pio2_1)))
81 (if (not (eql ix #x3ff921fb))
82 ;; 33+53 bit pi is good enough
85 (y1 (- (- z y0) pio2_1t)))
86 (return (values 1 y0 y1)))
87 ;; near pi/2, use 33+33+53 bit pi *
91 (y1 (- (- z y0) pio2_2t)))
92 (return (values 1 y0 y1)))))
96 (if (not (eql ix #x3ff921fb))
97 ;; 33+53 bit pi is good enough
100 (y1 (+ (- z y0) pio2_1t)))
101 (return (values -1 y0 y1)))
102 ;; near pi/2, use 33+33+53 bit pi
106 (y1 (+ (- z y0) pio2_2t)))
107 (return (values -1 y0 y1))))))))
109 (defun float64-rem-pio2-medium (x ix hx)
111 ((half #+nil 5.00000000000000000000e-01 #.(encode-float64 #x3FE00000 #x00000000))
112 (invpio2 #+nil 6.36619772367581382433e-01 #.(encode-float64 #x3FE45F30 #x6DC9C883))
113 (pio2_1 #+nil 1.57079632673412561417e+00 #.(encode-float64 #x3FF921FB #x54400000))
114 (pio2_1t #+nil 6.07710050650619224932e-11 #.(encode-float64 #x3DD0B461 #x1A626331))
115 (pio2_2 #+nil 6.07710050630396597660e-11 #.(encode-float64 #x3DD0B461 #x1A600000))
116 (pio2_2t #+nil 2.02226624879595063154e-21 #.(encode-float64 #x3BA3198A #x2E037073))
117 (pio2_3 #+nil 2.02226624871116645580e-21 #.(encode-float64 #x3BA3198A #x2E000000))
118 (pio2_3t #+nil 8.47842766036889956997e-32 #.(encode-float64 #x397B839A #x252049C1)))
120 ((tt (float64-abs x))
121 (n (truncate (+ (* tt invpio2) half)))
123 (r (- tt (* fn pio2_1)))
124 (w (* fn pio2_1t)) ; 1st round good to 85 bit *
126 ;; quick check no cancellation
127 (unless (and (< n 32) (not (eql ix (elt +npio2-hw+ (- n 1)))))
130 (high (decode-float64 y0))
131 (i (- j (logand (ash high -20) #x7ff))))
132 ;; 2nd iteration needed, good to 118 *
135 (setq w (* fn pio2_2))
137 (setq w (- (* fn pio2_2t) (- (- tt r) w)))
139 (setq high (decode-float64 y0))
140 (setq i (- j (logand (ash high -20) #x7ff)))
142 ;; 3rd iteration need, 151 bits acc *
144 (setq tt r) ; will cover all possible cases *
145 (setq w (* fn pio2_3))
147 (setq w (- (* fn pio2_3t) (- (- tt r) w)))
148 (setq y0 (- r w))))))
149 (let ((y1 (- (- r y0) w)))
151 (return (values (- n) (- y0) (- y1)))
152 (return (values n y0 y1)))))))
154 (defun float64-rem-pio2 (x)
155 (declare (values fixnum double-float double-float))
156 (declare (type double-float x))
158 ((two24 #+nil 1.67772160000000000000e+07 #.(encode-float64 #x41700000 #x00000000)))
159 (multiple-value-bind (hx lx) (decode-float64 x)
161 ((ix (logand hx #x7fffffff)))
163 (when (<= ix #x3fe921fb) (return (values 0 x 0d0))) ;; |x| ~<= pi/4 , no need for reduction
164 (when (< ix #x4002d97c) (return (float64-rem-pio2-small x ix hx))) ;; |x| < 3pi/4, special case with n=+-1
165 (when (<= ix #x413921fb) (return (float64-rem-pio2-medium x ix hx))) ;; |x| ~<= 2^19*(pi/2), medium size
167 ;; all other (large) arguments
168 (when (>= ix #x7ff00000) ; x is inf or NaN *
169 (let ((nan (- x x))) (return (values 0 nan nan))))
171 ;; set z = scalbn(|x|,ilogb(x)-23)
173 ((e0 (- (ash ix -20) 1046)) ;; e0 = ilogb(z)-23
174 (z0 (encode-float64 (- ix (ash e0 20)) lx))
175 (tx0 (float (truncate z0) 0d0))
176 (z1 (* (- z0 tx0) two24))
177 (tx1 (float (truncate z1) 0d0))
178 (z2 (* (- z1 tx1) two24))
180 (multiple-value-bind (n y0 y1)
183 (cond ;; skip zero term
184 ((not (zerop tx2)) 3)
185 ((not (zerop tx1)) 2)
186 ((not (zerop tx0)) 1)
189 ((tx (make-array 3 :element-type 'double-float)))
190 (setf (elt tx 0) tx0)
191 (setf (elt tx 1) tx1)
192 (setf (elt tx 2) tx2)
193 (float64-kernel-rem-pio2 tx e0 nx 2 +two-over-pi+)))
195 (return (values (- n) (- y0) (- y1)))
196 (return (values n y0 y1)))))))))