| // Copyright 2020 The SwiftShader Authors. All Rights Reserved. |
| // |
| // Licensed under the Apache License, Version 2.0 (the "License"); |
| // you may not use this file except in compliance with the License. |
| // You may obtain a copy of the License at |
| // |
| // http://www.apache.org/licenses/LICENSE-2.0 |
| // |
| // Unless required by applicable law or agreed to in writing, software |
| // distributed under the License is distributed on an "AS IS" BASIS, |
| // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| // See the License for the specific language governing permissions and |
| // limitations under the License. |
| |
| #include "OptimalIntrinsics.hpp" |
| |
| namespace rr { |
| namespace { |
| Float4 Reciprocal(RValue<Float4> x, bool pp = false, bool finite = false, bool exactAtPow2 = false) |
| { |
| Float4 rcp = Rcp_pp(x, exactAtPow2); |
| |
| if(!pp) |
| { |
| rcp = (rcp + rcp) - (x * rcp * rcp); |
| } |
| |
| if(finite) |
| { |
| int big = 0x7F7FFFFF; |
| rcp = Min(rcp, Float4((float &)big)); |
| } |
| |
| return rcp; |
| } |
| |
| Float4 SinOrCos(RValue<Float4> x, bool sin) |
| { |
| // Reduce to [-0.5, 0.5] range |
| Float4 y = x * Float4(1.59154943e-1f); // 1/2pi |
| y = y - Round(y); |
| |
| // From the paper: "A Fast, Vectorizable Algorithm for Producing Single-Precision Sine-Cosine Pairs" |
| // This implementation passes OpenGL ES 3.0 precision requirements, at the cost of more operations: |
| // !pp : 17 mul, 7 add, 1 sub, 1 reciprocal |
| // pp : 4 mul, 2 add, 2 abs |
| |
| Float4 y2 = y * y; |
| Float4 c1 = y2 * (y2 * (y2 * Float4(-0.0204391631f) + Float4(0.2536086171f)) + Float4(-1.2336977925f)) + Float4(1.0f); |
| Float4 s1 = y * (y2 * (y2 * (y2 * Float4(-0.0046075748f) + Float4(0.0796819754f)) + Float4(-0.645963615f)) + Float4(1.5707963235f)); |
| Float4 c2 = (c1 * c1) - (s1 * s1); |
| Float4 s2 = Float4(2.0f) * s1 * c1; |
| Float4 r = Reciprocal(s2 * s2 + c2 * c2, false, true, false); |
| |
| if(sin) |
| { |
| return Float4(2.0f) * s2 * c2 * r; |
| } |
| else |
| { |
| return ((c2 * c2) - (s2 * s2)) * r; |
| } |
| } |
| |
| // Approximation of atan in [0..1] |
| Float4 Atan_01(Float4 x) |
| { |
| // From 4.4.49, page 81 of the Handbook of Mathematical Functions, by Milton Abramowitz and Irene Stegun |
| const Float4 a2(-0.3333314528f); |
| const Float4 a4(0.1999355085f); |
| const Float4 a6(-0.1420889944f); |
| const Float4 a8(0.1065626393f); |
| const Float4 a10(-0.0752896400f); |
| const Float4 a12(0.0429096138f); |
| const Float4 a14(-0.0161657367f); |
| const Float4 a16(0.0028662257f); |
| Float4 x2 = x * x; |
| return (x + x * (x2 * (a2 + x2 * (a4 + x2 * (a6 + x2 * (a8 + x2 * (a10 + x2 * (a12 + x2 * (a14 + x2 * a16))))))))); |
| } |
| } // namespace |
| |
| namespace optimal { |
| |
| Float4 Sin(RValue<Float4> x) |
| { |
| return SinOrCos(x, true); |
| } |
| |
| Float4 Cos(RValue<Float4> x) |
| { |
| return SinOrCos(x, false); |
| } |
| |
| Float4 Tan(RValue<Float4> x) |
| { |
| return SinOrCos(x, true) / SinOrCos(x, false); |
| } |
| |
| Float4 Asin_4_terms(RValue<Float4> x) |
| { |
| // From 4.4.45, page 81 of the Handbook of Mathematical Functions, by Milton Abramowitz and Irene Stegun |
| // |e(x)| <= 5e-8 |
| const Float4 half_pi(1.57079632f); |
| const Float4 a0(1.5707288f); |
| const Float4 a1(-0.2121144f); |
| const Float4 a2(0.0742610f); |
| const Float4 a3(-0.0187293f); |
| Float4 absx = Abs(x); |
| return As<Float4>(As<Int4>(half_pi - Sqrt(Float4(1.0f) - absx) * (a0 + absx * (a1 + absx * (a2 + absx * a3)))) ^ |
| (As<Int4>(x) & Int4(0x80000000))); |
| } |
| |
| Float4 Asin_8_terms(RValue<Float4> x) |
| { |
| // From 4.4.46, page 81 of the Handbook of Mathematical Functions, by Milton Abramowitz and Irene Stegun |
| // |e(x)| <= 0e-8 |
| const Float4 half_pi(1.5707963268f); |
| const Float4 a0(1.5707963050f); |
| const Float4 a1(-0.2145988016f); |
| const Float4 a2(0.0889789874f); |
| const Float4 a3(-0.0501743046f); |
| const Float4 a4(0.0308918810f); |
| const Float4 a5(-0.0170881256f); |
| const Float4 a6(0.006700901f); |
| const Float4 a7(-0.0012624911f); |
| Float4 absx = Abs(x); |
| return As<Float4>(As<Int4>(half_pi - Sqrt(Float4(1.0f) - absx) * (a0 + absx * (a1 + absx * (a2 + absx * (a3 + absx * (a4 + absx * (a5 + absx * (a6 + absx * a7)))))))) ^ |
| (As<Int4>(x) & Int4(0x80000000))); |
| } |
| |
| Float4 Acos_4_terms(RValue<Float4> x) |
| { |
| // pi/2 - arcsin(x) |
| return Float4(1.57079632e+0f) - Asin_4_terms(x); |
| } |
| |
| Float4 Acos_8_terms(RValue<Float4> x) |
| { |
| // pi/2 - arcsin(x) |
| return Float4(1.57079632e+0f) - Asin_8_terms(x); |
| } |
| |
| Float4 Atan(RValue<Float4> x) |
| { |
| Float4 absx = Abs(x); |
| Int4 O = CmpNLT(absx, Float4(1.0f)); |
| Float4 y = As<Float4>((O & As<Int4>(Float4(1.0f) / absx)) | (~O & As<Int4>(absx))); // FIXME: Vector select |
| |
| const Float4 half_pi(1.57079632f); |
| Float4 theta = Atan_01(y); |
| return As<Float4>(((O & As<Int4>(half_pi - theta)) | (~O & As<Int4>(theta))) ^ // FIXME: Vector select |
| (As<Int4>(x) & Int4(0x80000000))); |
| } |
| |
| Float4 Atan2(RValue<Float4> y, RValue<Float4> x) |
| { |
| const Float4 pi(3.14159265f); // pi |
| const Float4 minus_pi(-3.14159265f); // -pi |
| const Float4 half_pi(1.57079632f); // pi/2 |
| const Float4 quarter_pi(7.85398163e-1f); // pi/4 |
| |
| // Rotate to upper semicircle when in lower semicircle |
| Int4 S = CmpLT(y, Float4(0.0f)); |
| Float4 theta = As<Float4>(S & As<Int4>(minus_pi)); |
| Float4 x0 = As<Float4>((As<Int4>(y) & Int4(0x80000000)) ^ As<Int4>(x)); |
| Float4 y0 = Abs(y); |
| |
| // Rotate to right quadrant when in left quadrant |
| Int4 Q = CmpLT(x0, Float4(0.0f)); |
| theta += As<Float4>(Q & As<Int4>(half_pi)); |
| Float4 x1 = As<Float4>((Q & As<Int4>(y0)) | (~Q & As<Int4>(x0))); // FIXME: Vector select |
| Float4 y1 = As<Float4>((Q & As<Int4>(-x0)) | (~Q & As<Int4>(y0))); // FIXME: Vector select |
| |
| // Mirror to first octant when in second octant |
| Int4 O = CmpNLT(y1, x1); |
| Float4 x2 = As<Float4>((O & As<Int4>(y1)) | (~O & As<Int4>(x1))); // FIXME: Vector select |
| Float4 y2 = As<Float4>((O & As<Int4>(x1)) | (~O & As<Int4>(y1))); // FIXME: Vector select |
| |
| // Approximation of atan in [0..1] |
| Int4 zero_x = CmpEQ(x2, Float4(0.0f)); |
| Int4 inf_y = IsInf(y2); // Since x2 >= y2, this means x2 == y2 == inf, so we use 45 degrees or pi/4 |
| Float4 atan2_theta = Atan_01(y2 / x2); |
| theta += As<Float4>((~zero_x & ~inf_y & ((O & As<Int4>(half_pi - atan2_theta)) | (~O & (As<Int4>(atan2_theta))))) | // FIXME: Vector select |
| (inf_y & As<Int4>(quarter_pi))); |
| |
| // Recover loss of precision for tiny theta angles |
| // This combination results in (-pi + half_pi + half_pi - atan2_theta) which is equivalent to -atan2_theta |
| Int4 precision_loss = S & Q & O & ~inf_y; |
| |
| return As<Float4>((precision_loss & As<Int4>(-atan2_theta)) | (~precision_loss & As<Int4>(theta))); // FIXME: Vector select |
| } |
| |
| Float4 Exp2(RValue<Float4> x) |
| { |
| // This implementation is based on 2^(i + f) = 2^i * 2^f, |
| // where i is the integer part of x and f is the fraction. |
| |
| // For 2^i we can put the integer part directly in the exponent of |
| // the IEEE-754 floating-point number. Clamp to prevent overflow |
| // past the representation of infinity. |
| Float4 x0 = x; |
| x0 = Min(x0, As<Float4>(Int4(0x43010000))); // 129.00000e+0f |
| x0 = Max(x0, As<Float4>(Int4(0xC2FDFFFF))); // -126.99999e+0f |
| |
| Int4 i = RoundInt(x0 - Float4(0.5f)); |
| Float4 ii = As<Float4>((i + Int4(127)) << 23); // Add single-precision bias, and shift into exponent. |
| |
| // For the fractional part use a polynomial |
| // which approximates 2^f in the 0 to 1 range. |
| Float4 f = x0 - Float4(i); |
| Float4 ff = As<Float4>(Int4(0x3AF61905)); // 1.8775767e-3f |
| ff = ff * f + As<Float4>(Int4(0x3C134806)); // 8.9893397e-3f |
| ff = ff * f + As<Float4>(Int4(0x3D64AA23)); // 5.5826318e-2f |
| ff = ff * f + As<Float4>(Int4(0x3E75EAD4)); // 2.4015361e-1f |
| ff = ff * f + As<Float4>(Int4(0x3F31727B)); // 6.9315308e-1f |
| ff = ff * f + Float4(1.0f); |
| |
| return ii * ff; |
| } |
| |
| Float4 Log2(RValue<Float4> x) |
| { |
| Float4 x0; |
| Float4 x1; |
| Float4 x2; |
| Float4 x3; |
| |
| x0 = x; |
| |
| x1 = As<Float4>(As<Int4>(x0) & Int4(0x7F800000)); |
| x1 = As<Float4>(As<UInt4>(x1) >> 8); |
| x1 = As<Float4>(As<Int4>(x1) | As<Int4>(Float4(1.0f))); |
| x1 = (x1 - Float4(1.4960938f)) * Float4(256.0f); // FIXME: (x1 - 1.4960938f) * 256.0f; |
| x0 = As<Float4>((As<Int4>(x0) & Int4(0x007FFFFF)) | As<Int4>(Float4(1.0f))); |
| |
| x2 = (Float4(9.5428179e-2f) * x0 + Float4(4.7779095e-1f)) * x0 + Float4(1.9782813e-1f); |
| x3 = ((Float4(1.6618466e-2f) * x0 + Float4(2.0350508e-1f)) * x0 + Float4(2.7382900e-1f)) * x0 + Float4(4.0496687e-2f); |
| x2 /= x3; |
| |
| x1 += (x0 - Float4(1.0f)) * x2; |
| |
| Int4 pos_inf_x = CmpEQ(As<Int4>(x), Int4(0x7F800000)); |
| return As<Float4>((pos_inf_x & As<Int4>(x)) | (~pos_inf_x & As<Int4>(x1))); |
| } |
| |
| Float4 Exp(RValue<Float4> x) |
| { |
| // TODO: Propagate the constant |
| return optimal::Exp2(Float4(1.44269504f) * x); // 1/ln(2) |
| } |
| |
| Float4 Log(RValue<Float4> x) |
| { |
| // TODO: Propagate the constant |
| return Float4(6.93147181e-1f) * optimal::Log2(x); // ln(2) |
| } |
| |
| Float4 Pow(RValue<Float4> x, RValue<Float4> y) |
| { |
| Float4 log = optimal::Log2(x); |
| log *= y; |
| return optimal::Exp2(log); |
| } |
| |
| Float4 Sinh(RValue<Float4> x) |
| { |
| return (optimal::Exp(x) - optimal::Exp(-x)) * Float4(0.5f); |
| } |
| |
| Float4 Cosh(RValue<Float4> x) |
| { |
| return (optimal::Exp(x) + optimal::Exp(-x)) * Float4(0.5f); |
| } |
| |
| Float4 Tanh(RValue<Float4> x) |
| { |
| Float4 e_x = optimal::Exp(x); |
| Float4 e_minus_x = optimal::Exp(-x); |
| return (e_x - e_minus_x) / (e_x + e_minus_x); |
| } |
| |
| Float4 Asinh(RValue<Float4> x) |
| { |
| return optimal::Log(x + Sqrt(x * x + Float4(1.0f))); |
| } |
| |
| Float4 Acosh(RValue<Float4> x) |
| { |
| return optimal::Log(x + Sqrt(x + Float4(1.0f)) * Sqrt(x - Float4(1.0f))); |
| } |
| |
| Float4 Atanh(RValue<Float4> x) |
| { |
| return optimal::Log((Float4(1.0f) + x) / (Float4(1.0f) - x)) * Float4(0.5f); |
| } |
| |
| } // namespace optimal |
| } // namespace rr |