Long Luo's Life Notes

每一天都是奇迹

By Long Luo

挖坑

\[ \begin{aligned} f(x) & = f(a) + f(x) - f(a) \\ & = f(a) + \int_a^x f'(t) \mathrm{d}t \\ & = f(a) - \int_a^x f'(t) \mathrm{d}(x-t) \\ & = f(a) - f'(t)(x - t)|_a^x - \frac12 \int_a^x f''(t) \mathrm{d}[(x-t)^2] \\ & = f(a) + f'(a)(x - a) + {f''(a) \over 2}(x - a)^2 - {1 \over 3!} \int_a^x f'''(t) \mathrm{d}[(x-t)]^3 \end{aligned} \]

不断重复这一过程,可知当 f 的 n+1 阶导数连续时,有:

\[ f(x) = \sum_{k=0}^n {f^{(k)}(a) \over k!}(x-a)^k + R_n(x) \]

其中积分余项 为:

\[ R_n(x) = {1 \over n!} \int_a^x f^{(n + 1)}(t)(x - t)^n \mathrm{d}t \]

参考文献

  1. Taylor's theorem
  2. Taylor Series
  3. Intuition for Taylor Series

By Long Luo

挖坑!

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/**
* 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
}
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/**
* 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);
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/* @(#)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);
}

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

参考文献

  1. C mathematical functions
  2. Sine and cosine
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