text: // 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