Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B

Average Error: 8.93% → 0.51%

Time: 10.8s

Precision: binary64

Cost: 7364

Math

\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \left(y \cdot -4\right), x \cdot x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+233)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (fma z (* z (* y -4.0)) (* x x))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+233) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = fma(z, (z * (y * -4.0)), (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+233)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = fma(z, Float64(z * Float64(y * -4.0)), Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+233], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Target

Original	8.93%
Target	8.91%
Herbie	0.51%

\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000009e233

   1. Initial program 0.12

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Simplified 0.12

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)} \]
      Proof

      [Start] 0.12	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
      fma-neg [=>] 0.12	\[ \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)} \]
      distribute-lft-neg-in [=>] 0.12	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(-y \cdot 4\right) \cdot \left(z \cdot z - t\right)}\right) \]
      *-commutative [=>] 0.12	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-y \cdot 4\right)}\right) \]
      distribute-rgt-neg-in [=>] 0.12	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right) \]
      metadata-eval [=>] 0.12	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot \color{blue}{-4}\right)\right) \]

   if 5.00000000000000009e233 < (*.f64 z z)

   1. Initial program 66.74

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in t around 0 69.37

      \[\leadsto \color{blue}{{x}^{2} - 4 \cdot \left(y \cdot {z}^{2}\right)} \]

   3. Simplified 3.06

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(y \cdot -4\right), x \cdot x\right)} \]
      Proof

      [Start] 69.37	\[ {x}^{2} - 4 \cdot \left(y \cdot {z}^{2}\right) \]
      cancel-sign-sub-inv [=>] 69.37	\[ \color{blue}{{x}^{2} + \left(-4\right) \cdot \left(y \cdot {z}^{2}\right)} \]
      metadata-eval [=>] 69.37	\[ {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot {z}^{2}\right) \]
      unpow2 [=>] 69.37	\[ \color{blue}{x \cdot x} + -4 \cdot \left(y \cdot {z}^{2}\right) \]
      associate-*r* [=>] 69.37	\[ x \cdot x + \color{blue}{\left(-4 \cdot y\right) \cdot {z}^{2}} \]
      *-commutative [<=] 69.37	\[ x \cdot x + \color{blue}{\left(y \cdot -4\right)} \cdot {z}^{2} \]
      unpow2 [=>] 69.36	\[ x \cdot x + \left(y \cdot -4\right) \cdot \color{blue}{\left(z \cdot z\right)} \]
      +-commutative [=>] 69.36	\[ \color{blue}{\left(y \cdot -4\right) \cdot \left(z \cdot z\right) + x \cdot x} \]
      associate-*r* [=>] 3.07	\[ \color{blue}{\left(\left(y \cdot -4\right) \cdot z\right) \cdot z} + x \cdot x \]
      *-commutative [<=] 3.07	\[ \color{blue}{z \cdot \left(\left(y \cdot -4\right) \cdot z\right)} + x \cdot x \]
      fma-def [=>] 3.06	\[ \color{blue}{\mathsf{fma}\left(z, \left(y \cdot -4\right) \cdot z, x \cdot x\right)} \]
      *-commutative [=>] 3.06	\[ \mathsf{fma}\left(z, \color{blue}{z \cdot \left(y \cdot -4\right)}, x \cdot x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.51

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \left(y \cdot -4\right), x \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.51%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right) + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \left(y \cdot -4\right), x \cdot x\right)\\ \end{array} \]

Alternative 2
Error	28.84%
Cost	1740

\[\begin{array}{l} t_1 := z \cdot z - t\\ t_2 := t_1 \cdot \left(y \cdot -4\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-165}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 3
Error	42.54%
Cost	1240

\[\begin{array}{l} t_1 := 4 \cdot \left(t \cdot y\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -3.55 \cdot 10^{+34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-212}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-297}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	10.91%
Cost	1232

\[\begin{array}{l} t_1 := \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ t_2 := x \cdot x + -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \mathbf{if}\;z \leq -9.4 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-61}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	13.33%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.5 \cdot 10^{-121}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 6
Error	0.51%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right) + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 7
Error	41.24%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+14}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+58}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8
Error	64.78%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))