mirror of https://gitlab.com/rnger/amath
164 lines
5.5 KiB
C
164 lines
5.5 KiB
C
/*-
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* Copyright (c) 2014-2017 Carsten Sonne Larsen <cs@innolan.net>
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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*
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* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*
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* Project homepage:
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* https://amath.innolan.net
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*
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* The original source code can be obtained from:
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* http://www.netlib.org/fdlibm/e_asin.c
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*
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* =================================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* =================================================================
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*/
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/**
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* @file asin.c
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* @brief Inverse sine function
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*/
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#include "prim.h"
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static const double
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one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
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huge = 1.000e+300,
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pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
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pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
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pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
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/* coefficient for R(x^2) */
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pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
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pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
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pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
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pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
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pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
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pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
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qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
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qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
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qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
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qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
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/**
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* @brief Inverse sine function
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* @details
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* <pre>
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* Method
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*
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* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
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* we approximate asin(x) on [0,0.5] by
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* asin(x) = x + x*x^2*R(x^2)
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* where
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* R(x^2) is a rational approximation of (asin(x)-x)/x^3
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* and its remez error is bounded by
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* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
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*
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* For x in [0.5,1]
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* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
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* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
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* then for x>0.98
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
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*
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* For x<=0.98, let pio4_hi = pio2_hi/2, then
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* f = hi part of s;
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* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
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* and
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
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* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
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*
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* Special cases
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* if x is NaN, return NaN
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* if |x|>1, return NaN
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* </pre>
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*/
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double asin(double x)
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{
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double t, w, p, q, c, r, s;
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int32_t hx, ix;
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GET_HIGH_WORD(hx, x);
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ix = hx & 0x7fffffff;
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if (ix >= 0x3ff00000)
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{ /* |x|>= 1 */
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uint32_t lx;
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GET_LOW_WORD(lx, x);
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if (((ix - 0x3ff00000) | lx) == 0)
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/* asin(1)=+-pi/2 with inexact */
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return x * pio2_hi + x * pio2_lo;
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return NAN; /* asin(|x|>1) is NaN */
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}
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else if (ix < 0x3fe00000)
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{ /* |x|<0.5 */
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if (ix < 0x3e400000)
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{ /* if |x| < 2**-27 */
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if (huge + x > one)
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{
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return x; /* return x with inexact if x!=0*/
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}
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else
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{
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t = 0;
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}
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}
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else
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{
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t = x * x;
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}
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p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
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q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
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w = p / q;
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return x + x * w;
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}
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/* 1> |x|>= 0.5 */
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w = one - fabs(x);
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t = w * 0.5;
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p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
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q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
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s = sqrt(t);
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if (ix >= 0x3FEF3333)
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{ /* if |x| > 0.975 */
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w = p / q;
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t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
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}
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else
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{
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w = s;
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SET_LOW_WORD(w, 0);
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c = (t - w * w) / (s + w);
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r = p / q;
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p = 2.0 * s * r - (pio2_lo - 2.0 * c);
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q = pio4_hi - 2.0 * w;
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t = pio4_hi - (p - q);
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}
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if (hx > 0)
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return t;
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else
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return -t;
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}
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