Zig 0.17.0-dev (Split by item)

This is an example of documentation generated by ZigDoc, an alternative to Zig's built-in Auto Doc feature. See also examples in other modes/formats. The project being documented here (as the example) is the Zig library itself.

tan

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 + T1x + ... + T13x where

    |tan(x) 2 4 26 | -59.2 |----- - (1+T1x +T2x +.... +T13*x )| <= 2

    x

    Note: tan(x+y) = tan(x) + tan'(x)y ~ tan(x) + (1+xx)*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 + (T1x + (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)))

trig.tan
pub fn tan(x_: f64, y_: f64, odd: bool) f64

File

lib/compiler_rt/trig.zig:262

Code

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(@as(u64, @bitCast(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 = @floatFromInt(1 - 2 * @as(i3, @intFromBool(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(@as(u64, @bitCast(w0)) & 0xffffffff00000000);
    v = r - (w0 - x); // w0+v = r+x
    a = -1.0 / w;
    a0 = a;
    a0 = @bitCast(@as(u64, @bitCast(a0)) & 0xffffffff00000000);
    return a0 + a * (1.0 + a0 * w0 + a0 * v);
}