250 lines
6.7 KiB
Zig
250 lines
6.7 KiB
Zig
// Special Cases:
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//
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// - atan(+-0) = +-0
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// - atan(+-inf) = +-pi/2
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const std = @import("../index.zig");
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const math = std.math;
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const assert = std.debug.assert;
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pub fn atan(x: var) @typeOf(x) {
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const T = @typeOf(x);
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return switch (T) {
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f32 => atan32(x),
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f64 => atan64(x),
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else => @compileError("atan not implemented for " ++ @typeName(T)),
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};
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}
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fn atan32(x_: f32) f32 {
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const atanhi = []const f32{
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4.6364760399e-01, // atan(0.5)hi
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7.8539812565e-01, // atan(1.0)hi
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9.8279368877e-01, // atan(1.5)hi
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1.5707962513e+00, // atan(inf)hi
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};
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const atanlo = []const f32{
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5.0121582440e-09, // atan(0.5)lo
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3.7748947079e-08, // atan(1.0)lo
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3.4473217170e-08, // atan(1.5)lo
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7.5497894159e-08, // atan(inf)lo
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};
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const aT = []const f32{
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3.3333328366e-01,
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-1.9999158382e-01,
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1.4253635705e-01,
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-1.0648017377e-01,
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6.1687607318e-02,
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};
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var x = x_;
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var ix: u32 = @bitCast(u32, x);
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const sign = ix >> 31;
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ix &= 0x7FFFFFFF;
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// |x| >= 2^26
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if (ix >= 0x4C800000) {
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if (math.isNan(x)) {
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return x;
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} else {
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const z = atanhi[3] + 0x1.0p-120;
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return if (sign != 0) -z else z;
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}
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}
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var id: ?usize = undefined;
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// |x| < 0.4375
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if (ix < 0x3EE00000) {
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// |x| < 2^(-12)
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if (ix < 0x39800000) {
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if (ix < 0x00800000) {
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math.forceEval(x * x);
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}
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return x;
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}
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id = null;
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} else {
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x = math.fabs(x);
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// |x| < 1.1875
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if (ix < 0x3F980000) {
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// 7/16 <= |x| < 11/16
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if (ix < 0x3F300000) {
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id = 0;
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x = (2.0 * x - 1.0) / (2.0 + x);
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}
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// 11/16 <= |x| < 19/16
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else {
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id = 1;
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x = (x - 1.0) / (x + 1.0);
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}
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} else {
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// |x| < 2.4375
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if (ix < 0x401C0000) {
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id = 2;
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x = (x - 1.5) / (1.0 + 1.5 * x);
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}
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// 2.4375 <= |x| < 2^26
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else {
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id = 3;
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x = -1.0 / x;
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}
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}
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}
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const z = x * x;
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const w = z * z;
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const s1 = z * (aT[0] + w * (aT[2] + w * aT[4]));
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const s2 = w * (aT[1] + w * aT[3]);
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if (id) |id_value| {
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const zz = atanhi[id_value] - ((x * (s1 + s2) - atanlo[id_value]) - x);
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return if (sign != 0) -zz else zz;
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} else {
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return x - x * (s1 + s2);
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}
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}
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fn atan64(x_: f64) f64 {
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const atanhi = []const f64{
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4.63647609000806093515e-01, // atan(0.5)hi
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7.85398163397448278999e-01, // atan(1.0)hi
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9.82793723247329054082e-01, // atan(1.5)hi
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1.57079632679489655800e+00, // atan(inf)hi
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};
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const atanlo = []const f64{
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2.26987774529616870924e-17, // atan(0.5)lo
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3.06161699786838301793e-17, // atan(1.0)lo
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1.39033110312309984516e-17, // atan(1.5)lo
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6.12323399573676603587e-17, // atan(inf)lo
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};
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const aT = []const f64{
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3.33333333333329318027e-01,
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-1.99999999998764832476e-01,
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1.42857142725034663711e-01,
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-1.11111104054623557880e-01,
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9.09088713343650656196e-02,
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-7.69187620504482999495e-02,
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6.66107313738753120669e-02,
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-5.83357013379057348645e-02,
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4.97687799461593236017e-02,
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-3.65315727442169155270e-02,
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1.62858201153657823623e-02,
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};
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var x = x_;
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var ux = @bitCast(u64, x);
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var ix = @intCast(u32, ux >> 32);
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const sign = ix >> 31;
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ix &= 0x7FFFFFFF;
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// |x| >= 2^66
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if (ix >= 0x44100000) {
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if (math.isNan(x)) {
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return x;
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} else {
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const z = atanhi[3] + 0x1.0p-120;
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return if (sign != 0) -z else z;
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}
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}
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var id: ?usize = undefined;
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// |x| < 0.4375
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if (ix < 0x3DFC0000) {
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// |x| < 2^(-27)
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if (ix < 0x3E400000) {
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if (ix < 0x00100000) {
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math.forceEval(@floatCast(f32, x));
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}
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return x;
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}
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id = null;
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} else {
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x = math.fabs(x);
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// |x| < 1.1875
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if (ix < 0x3FF30000) {
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// 7/16 <= |x| < 11/16
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if (ix < 0x3FE60000) {
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id = 0;
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x = (2.0 * x - 1.0) / (2.0 + x);
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}
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// 11/16 <= |x| < 19/16
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else {
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id = 1;
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x = (x - 1.0) / (x + 1.0);
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}
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} else {
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// |x| < 2.4375
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if (ix < 0x40038000) {
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id = 2;
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x = (x - 1.5) / (1.0 + 1.5 * x);
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}
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// 2.4375 <= |x| < 2^66
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else {
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id = 3;
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x = -1.0 / x;
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}
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}
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}
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const z = x * x;
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const w = z * z;
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const s1 = z * (aT[0] + w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
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const s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
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if (id) |id_value| {
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const zz = atanhi[id_value] - ((x * (s1 + s2) - atanlo[id_value]) - x);
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return if (sign != 0) -zz else zz;
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} else {
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return x - x * (s1 + s2);
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}
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}
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test "math.atan" {
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assert(@bitCast(u32, atan(f32(0.2))) == @bitCast(u32, atan32(0.2)));
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assert(atan(f64(0.2)) == atan64(0.2));
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}
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test "math.atan32" {
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const epsilon = 0.000001;
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assert(math.approxEq(f32, atan32(0.2), 0.197396, epsilon));
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assert(math.approxEq(f32, atan32(-0.2), -0.197396, epsilon));
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assert(math.approxEq(f32, atan32(0.3434), 0.330783, epsilon));
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assert(math.approxEq(f32, atan32(0.8923), 0.728545, epsilon));
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assert(math.approxEq(f32, atan32(1.5), 0.982794, epsilon));
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}
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test "math.atan64" {
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const epsilon = 0.000001;
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assert(math.approxEq(f64, atan64(0.2), 0.197396, epsilon));
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assert(math.approxEq(f64, atan64(-0.2), -0.197396, epsilon));
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assert(math.approxEq(f64, atan64(0.3434), 0.330783, epsilon));
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assert(math.approxEq(f64, atan64(0.8923), 0.728545, epsilon));
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assert(math.approxEq(f64, atan64(1.5), 0.982794, epsilon));
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}
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test "math.atan32.special" {
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const epsilon = 0.000001;
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assert(atan32(0.0) == 0.0);
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assert(atan32(-0.0) == -0.0);
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assert(math.approxEq(f32, atan32(math.inf(f32)), math.pi / 2.0, epsilon));
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assert(math.approxEq(f32, atan32(-math.inf(f32)), -math.pi / 2.0, epsilon));
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}
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test "math.atan64.special" {
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const epsilon = 0.000001;
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assert(atan64(0.0) == 0.0);
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assert(atan64(-0.0) == -0.0);
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assert(math.approxEq(f64, atan64(math.inf(f64)), math.pi / 2.0, epsilon));
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assert(math.approxEq(f64, atan64(-math.inf(f64)), -math.pi / 2.0, epsilon));
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}
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