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/
16 ; /* @(#)k_cos.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 ; * __kernel_cos( x, y )
30 ; * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
31 ; * Input x is assumed to be bounded by ~pi/4 in magnitude.
32 ; * Input y is the tail of x.
35 ; * 1. Since cos(-x) = cos(x), we need only to consider positive x.
36 ; * 2. if x < 2^-27 (hx<#x3e400000 0), return 1 with inexact if x!=0.
37 ; * 3. cos(x) is approximated by a polynomial of degree 14 on
40 ; * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
41 ; * where the remez error is
43 ; * | 2 4 6 8 10 12 14 | -58
44 ; * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
48 ; * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
49 ; * cos(x) = 1 - x*x/2 + r
50 ; * since cos(x+y) ~ cos(x) - sin(x)*y
52 ; * a correction term is necessary in cos(x) and hence
53 ; * cos(x+y) = 1 - (x*x/2 - (r - x*y))
54 ; * For better accuracy when x > 0.3, let qx = |x|/4 with
55 ; * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
57 ; * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
58 ; * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
59 ; * magnitude of the latter is at least a quarter of x*x/2,
60 ; * thus, reducing the rounding error in the subtraction.
63 (defun float64-kernel-cos (x y)
64 (declare (values double-float))
65 (declare (type double-float x y))
67 ((one #+nil 1.00000000000000000000e+00 #.(encode-float64 #x3FF00000 #x00000000))
68 (C1 #+nil 4.16666666666666019037e-02 #.(encode-float64 #x3FA55555 #x5555554C))
69 (C2 #+nil -1.38888888888741095749e-03 #.(encode-float64 #xBF56C16C #x16C15177))
70 (C3 #+nil 2.48015872894767294178e-05 #.(encode-float64 #x3EFA01A0 #x19CB1590))
71 (C4 #+nil -2.75573143513906633035e-07 #.(encode-float64 #xBE927E4F #x809C52AD))
72 (C5 #+nil 2.08757232129817482790e-09 #.(encode-float64 #x3E21EE9E #xBDB4B1C4))
73 (C6 #+nil -1.13596475577881948265e-11 #.(encode-float64 #xBDA8FAE9 #xBE8838D4))
75 (hx (decode-float64 x))
76 (ix (logand hx #x7fffffff)))
78 (when (< ix #x3e400000)
80 (when (eql (truncate x) 0) (return one)))
83 (r (* z (+ c1 (* z (+ c2 (* z (+ c3 (* z (+ c4 (* z (+ c5 (* z c6)))))))))))))
85 (when (< ix #x3FD33333) (return (- one (- (* 0.5d0 z) (- (* z r) (* x y))))))
88 (if (> ix #x3fe90000) ; x > 0.78125
90 (encode-float64 (- ix #x00200000) 0))) ; x/4
91 (hz (- (* 0.5d0 z) qx))
93 (return (- a (- hz (- (* z r) (* x y)))))))))