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1// Ported from musl, which is licensed under the MIT license:
2// https://git.musl-libc.org/cgit/musl/tree/COPYRIGHT
3//
4// https://git.musl-libc.org/cgit/musl/tree/src/math/__cos.c
5// https://git.musl-libc.org/cgit/musl/tree/src/math/__cosdf.c
6// https://git.musl-libc.org/cgit/musl/tree/src/math/__sin.c
7// https://git.musl-libc.org/cgit/musl/tree/src/math/__sindf.c
8// https://git.musl-libc.org/cgit/musl/tree/src/math/__tand.c
9// https://git.musl-libc.org/cgit/musl/tree/src/math/__tandf.c
10
11/// kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
12/// Input x is assumed to be bounded by ~pi/4 in magnitude.
13/// Input y is the tail of x.
14///
15/// Algorithm
16/// 1. Since cos(-x) = cos(x), we need only to consider positive x.
17/// 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
18/// 3. cos(x) is approximated by a polynomial of degree 14 on
19/// [0,pi/4]
20/// 4 14
21/// cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
22/// where the remez error is
23///
24/// | 2 4 6 8 10 12 14 | -58
25/// |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
26/// | |
27///
28/// 4 6 8 10 12 14
29/// 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
30/// cos(x) ~ 1 - x*x/2 + r
31/// since cos(x+y) ~ cos(x) - sin(x)*y
32/// ~ cos(x) - x*y,
33/// a correction term is necessary in cos(x) and hence
34/// cos(x+y) = 1 - (x*x/2 - (r - x*y))
35/// For better accuracy, rearrange to
36/// cos(x+y) ~ w + (tmp + (r-x*y))
37/// where w = 1 - x*x/2 and tmp is a tiny correction term
38/// (1 - x*x/2 == w + tmp exactly in infinite precision).
39/// The exactness of w + tmp in infinite precision depends on w
40/// and tmp having the same precision as x. If they have extra
41/// precision due to compiler bugs, then the extra precision is
42/// only good provided it is retained in all terms of the final
43/// expression for cos(). Retention happens in all cases tested
44/// under FreeBSD, so don't pessimize things by forcibly clipping
45/// any extra precision in w.
46pub fn __cos(x: f64, y: f64) f64 {
47 const C1 = 4.16666666666666019037e-02; // 0x3FA55555, 0x5555554C
48 const C2 = -1.38888888888741095749e-03; // 0xBF56C16C, 0x16C15177
49 const C3 = 2.48015872894767294178e-05; // 0x3EFA01A0, 0x19CB1590
50 const C4 = -2.75573143513906633035e-07; // 0xBE927E4F, 0x809C52AD
51 const C5 = 2.08757232129817482790e-09; // 0x3E21EE9E, 0xBDB4B1C4
52 const C6 = -1.13596475577881948265e-11; // 0xBDA8FAE9, 0xBE8838D4
53
54 const z = x * x;
55 const zs = z * z;
56 const r = z * (C1 + z * (C2 + z * C3)) + zs * zs * (C4 + z * (C5 + z * C6));
57 const hz = 0.5 * z;
58 const w = 1.0 - hz;
59 return w + (((1.0 - w) - hz) + (z * r - x * y));
60}
61
62pub fn __cosdf(x: f64) f32 {
63 // |cos(x) - c(x)| < 2**-34.1 (~[-5.37e-11, 5.295e-11]).
64 const C0 = -0x1ffffffd0c5e81.0p-54; // -0.499999997251031003120
65 const C1 = 0x155553e1053a42.0p-57; // 0.0416666233237390631894
66 const C2 = -0x16c087e80f1e27.0p-62; // -0.00138867637746099294692
67 const C3 = 0x199342e0ee5069.0p-68; // 0.0000243904487962774090654
68
69 // Try to optimize for parallel evaluation as in __tandf.c.
70 const z = x * x;
71 const w = z * z;
72 const r = C2 + z * C3;
73 return @floatCast(((1.0 + z * C0) + w * C1) + (w * z) * r);
74}
75
76/// kernel sin function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
77/// Input x is assumed to be bounded by ~pi/4 in magnitude.
78/// Input y is the tail of x.
79/// Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
80///
81/// Algorithm
82/// 1. Since sin(-x) = -sin(x), we need only to consider positive x.
83/// 2. Callers must return sin(-0) = -0 without calling here since our
84/// odd polynomial is not evaluated in a way that preserves -0.
85/// Callers may do the optimization sin(x) ~ x for tiny x.
86/// 3. sin(x) is approximated by a polynomial of degree 13 on
87/// [0,pi/4]
88/// 3 13
89/// sin(x) ~ x + S1*x + ... + S6*x
90/// where
91///
92/// |sin(x) 2 4 6 8 10 12 | -58
93/// |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
94/// | x |
95///
96/// 4. sin(x+y) = sin(x) + sin'(x')*y
97/// ~ sin(x) + (1-x*x/2)*y
98/// For better accuracy, let
99/// 3 2 2 2 2
100/// r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
101/// then 3 2
102/// sin(x) = x + (S1*x + (x *(r-y/2)+y))
103pub fn __sin(x: f64, y: f64, iy: i32) f64 {
104 const S1 = -1.66666666666666324348e-01; // 0xBFC55555, 0x55555549
105 const S2 = 8.33333333332248946124e-03; // 0x3F811111, 0x1110F8A6
106 const S3 = -1.98412698298579493134e-04; // 0xBF2A01A0, 0x19C161D5
107 const S4 = 2.75573137070700676789e-06; // 0x3EC71DE3, 0x57B1FE7D
108 const S5 = -2.50507602534068634195e-08; // 0xBE5AE5E6, 0x8A2B9CEB
109 const S6 = 1.58969099521155010221e-10; // 0x3DE5D93A, 0x5ACFD57C
110
111 const z = x * x;
112 const w = z * z;
113 const r = S2 + z * (S3 + z * S4) + z * w * (S5 + z * S6);
114 const v = z * x;
115 if (iy == 0) {
116 return x + v * (S1 + z * r);
117 } else {
118 return x - ((z * (0.5 * y - v * r) - y) - v * S1);
119 }
120}
121
122pub fn __sindf(x: f64) f32 {
123 // |sin(x)/x - s(x)| < 2**-37.5 (~[-4.89e-12, 4.824e-12]).
124 const S1 = -0x15555554cbac77.0p-55; // -0.166666666416265235595
125 const S2 = 0x111110896efbb2.0p-59; // 0.0083333293858894631756
126 const S3 = -0x1a00f9e2cae774.0p-65; // -0.000198393348360966317347
127 const S4 = 0x16cd878c3b46a7.0p-71; // 0.0000027183114939898219064
128
129 // Try to optimize for parallel evaluation as in __tandf.c.
130 const z = x * x;
131 const w = z * z;
132 const r = S3 + z * S4;
133 const s = z * x;
134 return @floatCast((x + s * (S1 + z * S2)) + s * w * r);
135}
136
137/// kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
138/// Input x is assumed to be bounded by ~pi/4 in magnitude.
139/// Input y is the tail of x.
140/// Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned.
141///
142/// Algorithm
143/// 1. Since tan(-x) = -tan(x), we need only to consider positive x.
144/// 2. Callers must return tan(-0) = -0 without calling here since our
145/// odd polynomial is not evaluated in a way that preserves -0.
146/// Callers may do the optimization tan(x) ~ x for tiny x.
147/// 3. tan(x) is approximated by a odd polynomial of degree 27 on
148/// [0,0.67434]
149/// 3 27
150/// tan(x) ~ x + T1*x + ... + T13*x
151/// where
152///
153/// |tan(x) 2 4 26 | -59.2
154/// |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
155/// | x |
156///
157/// Note: tan(x+y) = tan(x) + tan'(x)*y
158/// ~ tan(x) + (1+x*x)*y
159/// Therefore, for better accuracy in computing tan(x+y), let
160/// 3 2 2 2 2
161/// r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
162/// then
163/// 3 2
164/// tan(x+y) = x + (T1*x + (x *(r+y)+y))
165///
166/// 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
167/// tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
168/// = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
169pub fn __tan(x_: f64, y_: f64, odd: bool) f64 {
170 var x = x_;
171 var y = y_;
172
173 const T = [_]f64{
174 3.33333333333334091986e-01, // 3FD55555, 55555563
175 1.33333333333201242699e-01, // 3FC11111, 1110FE7A
176 5.39682539762260521377e-02, // 3FABA1BA, 1BB341FE
177 2.18694882948595424599e-02, // 3F9664F4, 8406D637
178 8.86323982359930005737e-03, // 3F8226E3, E96E8493
179 3.59207910759131235356e-03, // 3F6D6D22, C9560328
180 1.45620945432529025516e-03, // 3F57DBC8, FEE08315
181 5.88041240820264096874e-04, // 3F4344D8, F2F26501
182 2.46463134818469906812e-04, // 3F3026F7, 1A8D1068
183 7.81794442939557092300e-05, // 3F147E88, A03792A6
184 7.14072491382608190305e-05, // 3F12B80F, 32F0A7E9
185 -1.85586374855275456654e-05, // BEF375CB, DB605373
186 2.59073051863633712884e-05, // 3EFB2A70, 74BF7AD4
187 };
188 const pio4 = 7.85398163397448278999e-01; // 3FE921FB, 54442D18
189 const pio4lo = 3.06161699786838301793e-17; // 3C81A626, 33145C07
190
191 var z: f64 = undefined;
192 var r: f64 = undefined;
193 var v: f64 = undefined;
194 var w: f64 = undefined;
195 var s: f64 = undefined;
196 var a: f64 = undefined;
197 var w0: f64 = undefined;
198 var a0: f64 = undefined;
199 var hx: u32 = undefined;
200 var sign: bool = undefined;
201
202 hx = @intCast(@as(u64, @bitCast(x)) >> 32);
203 const big = (hx & 0x7fffffff) >= 0x3FE59428; // |x| >= 0.6744
204 if (big) {
205 sign = hx >> 31 != 0;
206 if (sign) {
207 x = -x;
208 y = -y;
209 }
210 x = (pio4 - x) + (pio4lo - y);
211 y = 0.0;
212 }
213 z = x * x;
214 w = z * z;
215
216 // Break x^5*(T[1]+x^2*T[2]+...) into
217 // x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
218 // x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
219 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
220 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
221 s = z * x;
222 r = y + z * (s * (r + v) + y) + s * T[0];
223 w = x + r;
224 if (big) {
225 s = @floatFromInt(1 - 2 * @as(i3, @intFromBool(odd)));
226 v = s - 2.0 * (x + (r - w * w / (w + s)));
227 return if (sign) -v else v;
228 }
229 if (!odd) {
230 return w;
231 }
232 // -1.0/(x+r) has up to 2ulp error, so compute it accurately
233 w0 = w;
234 w0 = @bitCast(@as(u64, @bitCast(w0)) & 0xffffffff00000000);
235 v = r - (w0 - x); // w0+v = r+x
236 a = -1.0 / w;
237 a0 = a;
238 a0 = @bitCast(@as(u64, @bitCast(a0)) & 0xffffffff00000000);
239 return a0 + a * (1.0 + a0 * w0 + a0 * v);
240}
241
242pub fn __tandf(x: f64, odd: bool) f32 {
243 // |tan(x)/x - t(x)| < 2**-25.5 (~[-2e-08, 2e-08]).
244 const T = [_]f64{
245 0x15554d3418c99f.0p-54, // 0.333331395030791399758
246 0x1112fd38999f72.0p-55, // 0.133392002712976742718
247 0x1b54c91d865afe.0p-57, // 0.0533812378445670393523
248 0x191df3908c33ce.0p-58, // 0.0245283181166547278873
249 0x185dadfcecf44e.0p-61, // 0.00297435743359967304927
250 0x1362b9bf971bcd.0p-59, // 0.00946564784943673166728
251 };
252
253 const z = x * x;
254 // Split up the polynomial into small independent terms to give
255 // opportunities for parallel evaluation. The chosen splitting is
256 // micro-optimized for Athlons (XP, X64). It costs 2 multiplications
257 // relative to Horner's method on sequential machines.
258 //
259 // We add the small terms from lowest degree up for efficiency on
260 // non-sequential machines (the lowest degree terms tend to be ready
261 // earlier). Apart from this, we don't care about order of
262 // operations, and don't need to to care since we have precision to
263 // spare. However, the chosen splitting is good for accuracy too,
264 // and would give results as accurate as Horner's method if the
265 // small terms were added from highest degree down.
266 const r = T[4] + z * T[5];
267 const t = T[2] + z * T[3];
268 const w = z * z;
269 const s = z * x;
270 const u = T[0] + z * T[1];
271 const r0 = (x + s * u) + (s * w) * (t + w * r);
272 return @floatCast(if (odd) -1.0 / r0 else r0);
273}