ec95e00e28
stage2: change logic for detecting whether the main package is inside
the std package. Previously it relied on realpath() which is not portable.
This uses resolve() which is how imports already work.
* stage2: fix cleanup bug when creating Module
* flatten lib/std/special/* to lib/*
- this was motivated by making main_pkg_is_inside_std false for
compiler_rt & friends.
* rename "mini libc" to "universal libc"
274 lines
12 KiB
Zig
274 lines
12 KiB
Zig
// Ported from musl, which is licensed under the MIT license:
|
|
// https://git.musl-libc.org/cgit/musl/tree/COPYRIGHT
|
|
//
|
|
// https://git.musl-libc.org/cgit/musl/tree/src/math/__cos.c
|
|
// https://git.musl-libc.org/cgit/musl/tree/src/math/__cosdf.c
|
|
// https://git.musl-libc.org/cgit/musl/tree/src/math/__sin.c
|
|
// https://git.musl-libc.org/cgit/musl/tree/src/math/__sindf.c
|
|
// https://git.musl-libc.org/cgit/musl/tree/src/math/__tand.c
|
|
// https://git.musl-libc.org/cgit/musl/tree/src/math/__tandf.c
|
|
|
|
/// kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
|
|
/// Input x is assumed to be bounded by ~pi/4 in magnitude.
|
|
/// Input y is the tail of x.
|
|
///
|
|
/// Algorithm
|
|
/// 1. Since cos(-x) = cos(x), we need only to consider positive x.
|
|
/// 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
|
|
/// 3. cos(x) is approximated by a polynomial of degree 14 on
|
|
/// [0,pi/4]
|
|
/// 4 14
|
|
/// cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
|
|
/// where the remez error is
|
|
///
|
|
/// | 2 4 6 8 10 12 14 | -58
|
|
/// |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
|
|
/// | |
|
|
///
|
|
/// 4 6 8 10 12 14
|
|
/// 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
|
|
/// cos(x) ~ 1 - x*x/2 + r
|
|
/// since cos(x+y) ~ cos(x) - sin(x)*y
|
|
/// ~ cos(x) - x*y,
|
|
/// a correction term is necessary in cos(x) and hence
|
|
/// cos(x+y) = 1 - (x*x/2 - (r - x*y))
|
|
/// For better accuracy, rearrange to
|
|
/// cos(x+y) ~ w + (tmp + (r-x*y))
|
|
/// where w = 1 - x*x/2 and tmp is a tiny correction term
|
|
/// (1 - x*x/2 == w + tmp exactly in infinite precision).
|
|
/// The exactness of w + tmp in infinite precision depends on w
|
|
/// and tmp having the same precision as x. If they have extra
|
|
/// precision due to compiler bugs, then the extra precision is
|
|
/// only good provided it is retained in all terms of the final
|
|
/// expression for cos(). Retention happens in all cases tested
|
|
/// under FreeBSD, so don't pessimize things by forcibly clipping
|
|
/// any extra precision in w.
|
|
pub fn __cos(x: f64, y: f64) f64 {
|
|
const C1 = 4.16666666666666019037e-02; // 0x3FA55555, 0x5555554C
|
|
const C2 = -1.38888888888741095749e-03; // 0xBF56C16C, 0x16C15177
|
|
const C3 = 2.48015872894767294178e-05; // 0x3EFA01A0, 0x19CB1590
|
|
const C4 = -2.75573143513906633035e-07; // 0xBE927E4F, 0x809C52AD
|
|
const C5 = 2.08757232129817482790e-09; // 0x3E21EE9E, 0xBDB4B1C4
|
|
const C6 = -1.13596475577881948265e-11; // 0xBDA8FAE9, 0xBE8838D4
|
|
|
|
const z = x * x;
|
|
const zs = z * z;
|
|
const r = z * (C1 + z * (C2 + z * C3)) + zs * zs * (C4 + z * (C5 + z * C6));
|
|
const hz = 0.5 * z;
|
|
const w = 1.0 - hz;
|
|
return w + (((1.0 - w) - hz) + (z * r - x * y));
|
|
}
|
|
|
|
pub fn __cosdf(x: f64) f32 {
|
|
// |cos(x) - c(x)| < 2**-34.1 (~[-5.37e-11, 5.295e-11]).
|
|
const C0 = -0x1ffffffd0c5e81.0p-54; // -0.499999997251031003120
|
|
const C1 = 0x155553e1053a42.0p-57; // 0.0416666233237390631894
|
|
const C2 = -0x16c087e80f1e27.0p-62; // -0.00138867637746099294692
|
|
const C3 = 0x199342e0ee5069.0p-68; // 0.0000243904487962774090654
|
|
|
|
// Try to optimize for parallel evaluation as in __tandf.c.
|
|
const z = x * x;
|
|
const w = z * z;
|
|
const r = C2 + z * C3;
|
|
return @floatCast(f32, ((1.0 + z * C0) + w * C1) + (w * z) * r);
|
|
}
|
|
|
|
/// kernel sin function on ~[-pi/4, pi/4] (except on -0), 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. Callers must return sin(-0) = -0 without calling here since our
|
|
/// odd polynomial is not evaluated in a way that preserves -0.
|
|
/// Callers may do the optimization sin(x) ~ x for tiny x.
|
|
/// 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))
|
|
pub fn __sin(x: f64, y: f64, iy: i32) f64 {
|
|
const S1 = -1.66666666666666324348e-01; // 0xBFC55555, 0x55555549
|
|
const S2 = 8.33333333332248946124e-03; // 0x3F811111, 0x1110F8A6
|
|
const S3 = -1.98412698298579493134e-04; // 0xBF2A01A0, 0x19C161D5
|
|
const S4 = 2.75573137070700676789e-06; // 0x3EC71DE3, 0x57B1FE7D
|
|
const S5 = -2.50507602534068634195e-08; // 0xBE5AE5E6, 0x8A2B9CEB
|
|
const S6 = 1.58969099521155010221e-10; // 0x3DE5D93A, 0x5ACFD57C
|
|
|
|
const z = x * x;
|
|
const w = z * z;
|
|
const r = S2 + z * (S3 + z * S4) + z * w * (S5 + z * S6);
|
|
const v = z * x;
|
|
if (iy == 0) {
|
|
return x + v * (S1 + z * r);
|
|
} else {
|
|
return x - ((z * (0.5 * y - v * r) - y) - v * S1);
|
|
}
|
|
}
|
|
|
|
pub fn __sindf(x: f64) f32 {
|
|
// |sin(x)/x - s(x)| < 2**-37.5 (~[-4.89e-12, 4.824e-12]).
|
|
const S1 = -0x15555554cbac77.0p-55; // -0.166666666416265235595
|
|
const S2 = 0x111110896efbb2.0p-59; // 0.0083333293858894631756
|
|
const S3 = -0x1a00f9e2cae774.0p-65; // -0.000198393348360966317347
|
|
const S4 = 0x16cd878c3b46a7.0p-71; // 0.0000027183114939898219064
|
|
|
|
// Try to optimize for parallel evaluation as in __tandf.c.
|
|
const z = x * x;
|
|
const w = z * z;
|
|
const r = S3 + z * S4;
|
|
const s = z * x;
|
|
return @floatCast(f32, (x + s * (S1 + z * S2)) + s * w * r);
|
|
}
|
|
|
|
/// kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
|
|
/// Input x is assumed to be bounded by ~pi/4 in magnitude.
|
|
/// Input y is the tail of x.
|
|
/// Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned.
|
|
///
|
|
/// Algorithm
|
|
/// 1. Since tan(-x) = -tan(x), we need only to consider positive x.
|
|
/// 2. Callers must return tan(-0) = -0 without calling here since our
|
|
/// odd polynomial is not evaluated in a way that preserves -0.
|
|
/// Callers may do the optimization tan(x) ~ x for tiny x.
|
|
/// 3. tan(x) is approximated by a odd polynomial of degree 27 on
|
|
/// [0,0.67434]
|
|
/// 3 27
|
|
/// tan(x) ~ x + T1*x + ... + T13*x
|
|
/// where
|
|
///
|
|
/// |tan(x) 2 4 26 | -59.2
|
|
/// |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
|
|
/// | x |
|
|
///
|
|
/// Note: tan(x+y) = tan(x) + tan'(x)*y
|
|
/// ~ tan(x) + (1+x*x)*y
|
|
/// Therefore, for better accuracy in computing tan(x+y), let
|
|
/// 3 2 2 2 2
|
|
/// r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
|
|
/// then
|
|
/// 3 2
|
|
/// tan(x+y) = x + (T1*x + (x *(r+y)+y))
|
|
///
|
|
/// 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
|
|
/// tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
|
|
/// = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
|
|
pub fn __tan(x_: f64, y_: f64, odd: bool) f64 {
|
|
var x = x_;
|
|
var y = y_;
|
|
|
|
const T = [_]f64{
|
|
3.33333333333334091986e-01, // 3FD55555, 55555563
|
|
1.33333333333201242699e-01, // 3FC11111, 1110FE7A
|
|
5.39682539762260521377e-02, // 3FABA1BA, 1BB341FE
|
|
2.18694882948595424599e-02, // 3F9664F4, 8406D637
|
|
8.86323982359930005737e-03, // 3F8226E3, E96E8493
|
|
3.59207910759131235356e-03, // 3F6D6D22, C9560328
|
|
1.45620945432529025516e-03, // 3F57DBC8, FEE08315
|
|
5.88041240820264096874e-04, // 3F4344D8, F2F26501
|
|
2.46463134818469906812e-04, // 3F3026F7, 1A8D1068
|
|
7.81794442939557092300e-05, // 3F147E88, A03792A6
|
|
7.14072491382608190305e-05, // 3F12B80F, 32F0A7E9
|
|
-1.85586374855275456654e-05, // BEF375CB, DB605373
|
|
2.59073051863633712884e-05, // 3EFB2A70, 74BF7AD4
|
|
};
|
|
const pio4 = 7.85398163397448278999e-01; // 3FE921FB, 54442D18
|
|
const pio4lo = 3.06161699786838301793e-17; // 3C81A626, 33145C07
|
|
|
|
var z: f64 = undefined;
|
|
var r: f64 = undefined;
|
|
var v: f64 = undefined;
|
|
var w: f64 = undefined;
|
|
var s: f64 = undefined;
|
|
var a: f64 = undefined;
|
|
var w0: f64 = undefined;
|
|
var a0: f64 = undefined;
|
|
var hx: u32 = undefined;
|
|
var sign: bool = undefined;
|
|
|
|
hx = @intCast(u32, @bitCast(u64, x) >> 32);
|
|
const big = (hx & 0x7fffffff) >= 0x3FE59428; // |x| >= 0.6744
|
|
if (big) {
|
|
sign = hx >> 31 != 0;
|
|
if (sign) {
|
|
x = -x;
|
|
y = -y;
|
|
}
|
|
x = (pio4 - x) + (pio4lo - y);
|
|
y = 0.0;
|
|
}
|
|
z = x * x;
|
|
w = z * z;
|
|
|
|
// Break x^5*(T[1]+x^2*T[2]+...) into
|
|
// x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
|
|
// x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
|
|
r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
|
|
v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
|
|
s = z * x;
|
|
r = y + z * (s * (r + v) + y) + s * T[0];
|
|
w = x + r;
|
|
if (big) {
|
|
s = 1 - 2 * @intToFloat(f64, @boolToInt(odd));
|
|
v = s - 2.0 * (x + (r - w * w / (w + s)));
|
|
return if (sign) -v else v;
|
|
}
|
|
if (!odd) {
|
|
return w;
|
|
}
|
|
// -1.0/(x+r) has up to 2ulp error, so compute it accurately
|
|
w0 = w;
|
|
w0 = @bitCast(f64, @bitCast(u64, w0) & 0xffffffff00000000);
|
|
v = r - (w0 - x); // w0+v = r+x
|
|
a = -1.0 / w;
|
|
a0 = a;
|
|
a0 = @bitCast(f64, @bitCast(u64, a0) & 0xffffffff00000000);
|
|
return a0 + a * (1.0 + a0 * w0 + a0 * v);
|
|
}
|
|
|
|
pub fn __tandf(x: f64, odd: bool) f32 {
|
|
// |tan(x)/x - t(x)| < 2**-25.5 (~[-2e-08, 2e-08]).
|
|
const T = [_]f64{
|
|
0x15554d3418c99f.0p-54, // 0.333331395030791399758
|
|
0x1112fd38999f72.0p-55, // 0.133392002712976742718
|
|
0x1b54c91d865afe.0p-57, // 0.0533812378445670393523
|
|
0x191df3908c33ce.0p-58, // 0.0245283181166547278873
|
|
0x185dadfcecf44e.0p-61, // 0.00297435743359967304927
|
|
0x1362b9bf971bcd.0p-59, // 0.00946564784943673166728
|
|
};
|
|
|
|
const z = x * x;
|
|
// Split up the polynomial into small independent terms to give
|
|
// opportunities for parallel evaluation. The chosen splitting is
|
|
// micro-optimized for Athlons (XP, X64). It costs 2 multiplications
|
|
// relative to Horner's method on sequential machines.
|
|
//
|
|
// We add the small terms from lowest degree up for efficiency on
|
|
// non-sequential machines (the lowest degree terms tend to be ready
|
|
// earlier). Apart from this, we don't care about order of
|
|
// operations, and don't need to to care since we have precision to
|
|
// spare. However, the chosen splitting is good for accuracy too,
|
|
// and would give results as accurate as Horner's method if the
|
|
// small terms were added from highest degree down.
|
|
const r = T[4] + z * T[5];
|
|
const t = T[2] + z * T[3];
|
|
const w = z * z;
|
|
const s = z * x;
|
|
const u = T[0] + z * T[1];
|
|
const r0 = (x + s * u) + (s * w) * (t + w * r);
|
|
return @floatCast(f32, if (odd) -1.0 / r0 else r0);
|
|
}
|