库函数 (libm) 是如何计算三角函数值的？

发表于 2024-02-16 更新于 2026-07-02 分类于 Math Waline：阅读次数：本文字数： 3.1k 阅读时长 ≈ 3 分钟

By Long Luo

挖坑！

/**
 * Returns the trigonometric sine of an angle.  Special cases:
 * <ul><li>If the argument is NaN or an infinity, then the
 * result is NaN.
 * <li>If the argument is zero, then the result is a zero with the
 * same sign as the argument.</ul>
 *
 * <p>The computed result must be within 1 ulp of the exact result.
 * Results must be semi-monotonic.
 *
 * @param   a   an angle, in radians.
 * @return  the sine of the argument.
 */
@HotSpotIntrinsicCandidate
public static double sin(double a) {
    return StrictMath.sin(a); // default impl. delegates to StrictMath
}

/**
 * Returns the trigonometric sine of an angle. Special cases:
 * <ul><li>If the argument is NaN or an infinity, then the
 * result is NaN.
 * <li>If the argument is zero, then the result is a zero with the
 * same sign as the argument.</ul>
 *
 * @param   a   an angle, in radians.
 * @return  the sine of the argument.
 */
public static native double sin(double a);

/* @(#)k_sin.c 1.3 95/01/18 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/* __kernel_sin( x, y, iy)
 * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). 
 *
 * Algorithm
 *	1. Since sin(-x) = -sin(x), we need only to consider positive x. 
 *	2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
 *	3. sin(x) is approximated by a polynomial of degree 13 on
 *	   [0,pi/4]
 *		  	         3            13
 *	   	sin(x) ~ x + S1*x + ... + S6*x
 *	   where
 *	
 * 	|sin(x)         2     4     6     8     10     12  |     -58
 * 	|----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x  +S6*x   )| <= 2
 * 	|  x 					           | 
 * 
 *	4. sin(x+y) = sin(x) + sin'(x')*y
 *		    ~ sin(x) + (1-x*x/2)*y
 *	   For better accuracy, let 
 *		     3      2      2      2      2
 *		r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
 *	   then                   3    2
 *		sin(x) = x + (S1*x + (x *(r-y/2)+y))
 */

#ifndef __FDLIBM_H__
#include "fdlibm.h"
#endif

double __kernel_sin(double x, double y, int iy)
{
	double z, r, v;
	int32_t ix;

	static const double half = 5.00000000000000000000e-01;	/* 0x3FE00000, 0x00000000 */
	static const double S1 = -1.66666666666666324348e-01;	/* 0xBFC55555, 0x55555549 */
	static const double S2 = 8.33333333332248946124e-03;	/* 0x3F811111, 0x1110F8A6 */
	static const double S3 = -1.98412698298579493134e-04;	/* 0xBF2A01A0, 0x19C161D5 */
	static const double S4 = 2.75573137070700676789e-06;	/* 0x3EC71DE3, 0x57B1FE7D */
	static const double S5 = -2.50507602534068634195e-08;	/* 0xBE5AE5E6, 0x8A2B9CEB */
	static const double S6 = 1.58969099521155010221e-10;	/* 0x3DE5D93A, 0x5ACFD57C */

	GET_HIGH_WORD(ix, x);
	ix &= IC(0x7fffffff);					/* high word of x */
	if (ix < IC(0x3e400000))				/* |x| < 2**-27 */
	{
		if ((int32_t) x == 0)
			return x;						/* generate inexact */
	}
	z = x * x;
	v = z * x;
	r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
	if (iy == 0)
		return x + v * (S1 + z * r);
	else
		return x - ((z * (half * y - v * r) - y) - v * S1);
}

\[ \sin (x_0 + \Delta x) \approx \sin (x_0) + \sin'(x_0) \frac {\Delta x}{1!} + \sin''(x_0) \frac { \Delta x^2}{2!} + \sin'''(x_0) \frac {\Delta x^3}{3!} + \cdots \]