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/sincos.as
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/********************************* sin.as *********************************** |
* Author: Agner Fog |
* date created: 2018-03-29 |
* Last modified: 2021-04-25 |
* Version: 1.11 |
* Project: ForwardCom library math.li |
* Description: sin, cos, and tan functions. Calculate in radians, double precision |
* The argument x can be a scalar or a vector |
* The return value will be a vector with the same length |
* C declaration: double sin(double x); |
* C declaration: double cos(double x); |
* C declaration: double tan(double x); |
* C declaration: struct {double s; double c;} sincos(double x); |
* |
* This code is adapted from C++ vector class library www.github.com/vectorclass |
* Copyright 2018-2021 GNU General Public License http://www.gnu.org/licenses |
*****************************************************************************/ |
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// define constants |
% M_2_PI = 0.636619772367581343076 // 2./pi |
% P0sin = -1.66666666666666307295E-1 // polynomial coefficients for sin |
% P1sin = 8.33333333332211858878E-3 |
% P2sin = -1.98412698295895385996E-4 |
% P3sin = 2.75573136213857245213E-6 |
% P4sin = -2.50507477628578072866E-8 |
% P5sin = 1.58962301576546568060E-10 |
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% P0cos = 4.16666666666665929218E-2 // polynomial coefficients for cos |
% P1cos = -1.38888888888730564116E-3 |
% P2cos = 2.48015872888517045348E-5 |
% P3cos = -2.75573141792967388112E-7 |
% P4cos = 2.08757008419747316778E-9 |
% P5cos = -1.13585365213876817300E-11 |
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% DP1 = 7.853981554508209228515625E-1 // modulo pi/2 for extended precision |
% DP2 = 7.94662735614792836714E-9 // correction for extended precision modular artithmetic |
% DP3 = 3.06161699786838294307E-17 // correction for extended precision modular artithmetic |
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code section execute align = 4 |
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public _sin: function, reguse = 0, 0x7BF |
public _cos: function, reguse = 0, 0x7BF |
public _sincos: function, reguse = 0, 0x7BF |
public _tan: function, reguse = 0, 0x7BF |
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// common entry for sin and sincos functions |
_sin function |
_sincos: |
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/* registers: |
v0 = x |
v1 = abs(x) |
v1 = quadrant |
v10 = abs(x) reduced modulo pi/2 |
v2 = v10^2 |
v3 = v10^4 |
v4 = v10^8 |
v5 = v10^3 |
v6 = unused (vacant flag for calling function) |
v7 = temp |
v8 = sin |
v9 = cos |
*/ |
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// Find quadrant: |
// 0 - pi/4 => 0 |
// pi/4 - 3*pi/4 => 1 |
// 3*pi/4 - 5*pi/4 => 2 |
// 5*pi/4 - 7*pi/4 => 3 |
// 7*pi/4 - 8*pi/4 => 4 |
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double v1 = clear_bit(v0, 63) // abs(x) |
double v4 = v1 * M_2_PI |
double v4 = round(v4, 0) // round to integer |
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// reduce modulo pi/2, with extended precision |
// x = ((xa - y * DP1) - y * DP2) - y * DP3; |
double v10 = v4 * (-DP1*2) + v1 |
double v10 = v4 * (-DP2*2) + v10 |
double v10 = v4 * (-DP3*2) + v10 |
double v5 = !(v4 > ((1 << 51) + 0.0)) // check for loss of precision and overflow, but not NAN |
double v1 = v4 + ((1 << 52) + 0.0) // add magic number 2^52 to get integer into lowest bit |
double v10 = v5 ? v10 : 0 // zero if out of range. result will be -1, 0, or 1 |
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// Expansion of sin and cos, valid for -pi/4 <= x <= pi/4 |
double v2 = v10 * v10 // x^2 |
double v3 = v2 * v2 // x^4 |
double v4 = v3 * v3 // x^8 |
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// calculate polynomial P5sin*x2^5 + P4sin*x2^4 + P3sin*x2^3 + P2sin*x2^2 + P1sin*x2 + P0sin |
// = (p2+p3*x2)*x4 + ((p4+p5*x2)*x8 + (p0+p1*x2)); |
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double v5 = replace(v10, P0sin) // broadcast to same length as x |
double v5 = v2 * P1sin + v5 |
double v7 = replace(v10, P4sin) |
double v7 = v2 * P5sin + v7 |
double v8 = replace(v10, P2sin) |
double v8 = v2 * P3sin + v8 |
double v7 = v7 * v4 + v5 |
double v8 = v8 * v3 + v7 |
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// calculate polynomial P5cos*x2^5 + P4cos*x2^4 + P3cos*x2^3 + P2cos*x2^2 + P1cos*x2 + P0cos |
// = (p2+p3*x2)*x4 + ((p4+p5*x2)*x8 + (p0+p1*x2)); |
double v5 = replace(v10, P0cos) |
double v5 = v2 * P1cos + v5 |
double v7 = replace(v10, P4cos) |
double v7 = v2 * P5cos + v7 |
double v9 = replace(v10, P2cos) |
double v9 = v2 * P3cos + v9 |
double v7 = v7 * v4 + v5 |
double v9 = v9 * v3 + v7 |
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// s = x + (x * x2) * s; |
double v5 = v10 * v2 |
double v8 = v8 * v5 + v10 |
// c = 1.0 - x2 * 0.5 + (x2 * x2) * c; |
double v9 = v9 * v3 |
double v9 = v2 * (-0.5) + v9 |
double v9 = 1.0 + v9 |
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// swap sin and cos if odd quadrant |
double v3 = v1 ? v9 : v8 // sin |
double v4 = v1 ? v8 : v9 // cos |
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// get sign of sin |
int64 v5 = v1 << 62 // get bit 1 into sign bit, x modulo pi/2 = 2 or 3 |
int64 v5 ^= v0 // toggle with sign of original x |
int64 v5 = and(v5, 1 << 63) // isolate sign bit |
double v0 = v3 ^ v5 // apply sign bit to sin |
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// get sign of cos |
int64 v1 = v1 + 1 // change sign when x modulo pi/2 = 1 or 2 |
int64 v1 = v1 << 62 // get bit 1 into sign bit |
int64 v1 = and(v1, 1 << 63) // isolate sign bit |
double v1 = v4 ^ v1 // apply sign bit to cos |
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// return sin in v0, cos in v1 |
return |
_sin end |
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// cosine function |
_cos function |
call _sincos |
double v0 = v1 // cos is in v1 |
return |
_cos end |
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// tangent function |
_tan function |
call _sincos |
double v0 = v0 / v1 // tan(x) = sin(x)/cos(x) |
return |
_tan end |
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code end |