atanBinary64

fn atanBinary64(x: f64) f64 {
    const atanhi: []const f64 = &.{
        4.63647609000806093515e-01, // atan(0.5)hi 0x3FDDAC67, 0x0561BB4F
        7.85398163397448278999e-01, // atan(1.0)hi 0x3FE921FB, 0x54442D18
        9.82793723247329054082e-01, // atan(1.5)hi 0x3FEF730B, 0xD281F69B
        1.57079632679489655800e+00, // atan(inf)hi 0x3FF921FB, 0x54442D18
    };
    const atanlo: []const f64 = &.{
        2.26987774529616870924e-17, // atan(0.5)lo 0x3C7A2B7F, 0x222F65E2
        3.06161699786838301793e-17, // atan(1.0)lo 0x3C81A626, 0x33145C07
        1.39033110312309984516e-17, // atan(1.5)lo 0x3C700788, 0x7AF0CBBD
        6.12323399573676603587e-17, // atan(inf)lo 0x3C91A626, 0x33145C07
    };
    const aT: []const f64 = &.{
        3.33333333333329318027e-01, // 0x3FD55555, 0x5555550D
        -1.99999999998764832476e-01, // 0xBFC99999, 0x9998EBC4
        1.42857142725034663711e-01, // 0x3FC24924, 0x920083FF
        -1.11111104054623557880e-01, // 0xBFBC71C6, 0xFE231671
        9.09088713343650656196e-02, // 0x3FB745CD, 0xC54C206E
        -7.69187620504482999495e-02, // 0xBFB3B0F2, 0xAF749A6D
        6.66107313738753120669e-02, // 0x3FB10D66, 0xA0D03D51
        -5.83357013379057348645e-02, // 0xBFADDE2D, 0x52DEFD9A
        4.97687799461593236017e-02, // 0x3FA97B4B, 0x24760DEB
        -3.65315727442169155270e-02, // 0xBFA2B444, 0x2C6A6C2F
        1.62858201153657823623e-02, // 0x3F90AD3A, 0xE322DA11
    };

    const hx: u64 = @bitCast(x);
    const ix: u32 = @truncate((hx >> 32) & 0x7fffffff);
    const sign = (hx >> 63) != 0;
    // if |x| >= 2^66
    if (ix >= 0x44100000) {
        if (math.isNan(x)) {
            return x;
        }
        const z = atanhi[3] + 0x1p-120;
        return if (sign) -z else z;
    }
    const x_, const id: ?usize = blk: {
        // |x| < 0.4375
        if (ix < 0x3fdc_0000) {
            // |x| < 2^(-27)
            if (ix < 0x3e40_0000) {
                if (ix < 0x0010_0000) {
                    // raise underflow for subnormal x
                    mem.doNotOptimizeAway(@as(f32, @floatCast(x)));
                }
                return x;
            }
            break :blk .{ x, null };
        } else {
            const x_ = @abs(x);
            // |x| < 1.1875
            if (ix < 0x3ff3_0000) {
                // 7/16 <= |x| < 11/16
                if (ix < 0x3fe6_0000) {
                    break :blk .{ (2.0 * x_ - 1.0) / (2.0 + x_), 0 };
                }
                // 11/16 <= |x| < 19/16
                else {
                    break :blk .{ (x_ - 1.0) / (x_ + 1.0), 1 };
                }
            } else {
                // |x| < 2.4375
                if (ix < 0x4003_8000) {
                    break :blk .{ (x_ - 1.5) / (1.0 + 1.5 * x_), 2 };
                }
                // 2.4375 <= |x| < 2^66
                else {
                    break :blk .{ -1.0 / x_, 3 };
                }
            }
        }
    };
    // end of argument reduction
    const z = x_ * x_;
    const w = z * z;
    // break sum from i=0 to 10 aT[i]z^(i+1) into odd and even poly
    const s1 = z * (aT[0] + w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
    const s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
    if (id) |id_| {
        const z_ = atanhi[id_] - (x_ * (s1 + s2) - atanlo[id_] - x_);
        return if (sign) -z_ else z_;
    } else {
        return x_ - x_ * (s1 + s2);
    }
}