1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
|
/**
* @license Apache-2.0
*
* Copyright (c) 2018 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
*
* ## Notice
*
* The original C code, long comment, copyright, license, and constants are from [Cephes]{@link http://www.netlib.org/cephes}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* Copyright 1984, 1995, 2000 by Stephen L. Moshier
*
* Some software in this archive may be from the book _Methods and Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster International, 1989) or from the Cephes Mathematical Library, a commercial product. In either event, it is copyrighted by the author. What you see here may be used freely but it comes with no support or guarantee.
*
* Stephen L. Moshier
* moshier@na-net.ornl.gov
* ```
*/
'use strict';
// MODULES //
var isnan = require( './../../../../base/assert/is-nan' );
var asin = require( './../../../../base/special/asin' );
var sqrt = require( './../../../../base/special/sqrt' );
var PIO4 = require( '@stdlib/constants/float64/fourth-pi' );
// VARIABLES //
var MOREBITS = 6.123233995736765886130e-17; // pi/2 = PIO2 + MOREBITS.
// MAIN //
/**
* Computes the arccosine of a number.
*
* ## Method
*
* - Analytically,
*
* ```tex
* \operatorname{acos}(x) = \frac{\pi}{2} - \operatorname{asin}(x)
* ```
*
* However, if \\(\|x\|\\) is near \\(1\\), there is cancellation error in subtracting \\(\opertorname{asin}(x)\\) from \\(\pi/2\\). Hence, if \\(x < -0.5\\),
*
* ```tex
* \operatorname{acos}(x) = \pi - 2.0 \cdot \operatorname{asin}(\sqrt{(1+x)/2})
* ```
*
* or, if \\(x > +0.5\\),
*
* ```tex
* \operatorname{acos}(x) = 2.0 \cdot \operatorname{asin}( \sqrt{(1-x)/2} )}
* ```
*
* ## Notes
*
* - Relative error:
*
* | arithmetic | domain | # trials | peak | rms |
* |:-----------|:------:|:---------|:--------|:--------|
* | DEC | -1, 1 | 50000 | 3.3e-17 | 8.2e-18 |
* | IEEE | -1, 1 | 10^6 | 2.2e-16 | 6.5e-17 |
*
*
* @param {number} x - input value
* @returns {number} arccosine (in radians)
*
* @example
* var v = acos( 1.0 );
* // returns 0.0
*
* @example
* var v = acos( 0.707 ); // ~pi/4
* // returns ~0.7855
*
* @example
* var v = acos( NaN );
* // returns NaN
*/
function acos( x ) {
var z;
if ( isnan( x ) ) {
return NaN;
}
if ( x < -1.0 || x > 1.0 ) {
return NaN;
}
if ( x > 0.5 ) {
return 2.0 * asin( sqrt( 0.5 - (0.5*x) ) );
}
z = PIO4 - asin( x );
z += MOREBITS;
z += PIO4;
return z;
}
// EXPORTS //
module.exports = acos;
|
