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

Average Accuracy: 90.9% → 99.5%

Time: 14.9s

Precision: binary64

Cost: 7364

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \]

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) 1e+222)
   (fma (- (* z z) t) (* y -4.0) (* x x))
   (fma x x (* z (* z (* y -4.0))))))

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) <= 1e+222) {
		tmp = fma(((z * z) - t), (y * -4.0), (x * x));
	} else {
		tmp = fma(x, x, (z * (z * (y * -4.0))));
	}
	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) <= 1e+222)
		tmp = fma(Float64(Float64(z * z) - t), Float64(y * -4.0), Float64(x * x));
	else
		tmp = fma(x, x, Float64(z * Float64(z * Float64(y * -4.0))));
	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], 1e+222], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.9%
Target	90.9%
Herbie	99.5%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e222

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
      cancel-sign-sub-inv [=>] 99.9	\[ \color{blue}{x \cdot x + \left(-y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]
      +-commutative [=>] 99.9	\[ \color{blue}{\left(-y \cdot 4\right) \cdot \left(z \cdot z - t\right) + x \cdot x} \]
      *-commutative [=>] 99.9	\[ \color{blue}{\left(z \cdot z - t\right) \cdot \left(-y \cdot 4\right)} + x \cdot x \]
      fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(z \cdot z - t, -y \cdot 4, x \cdot x\right)} \]
      distribute-rgt-neg-in [=>] 99.9	\[ \mathsf{fma}\left(z \cdot z - t, \color{blue}{y \cdot \left(-4\right)}, x \cdot x\right) \]
      metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(z \cdot z - t, y \cdot \color{blue}{-4}, x \cdot x\right) \]

   if 1e222 < (*.f64 z z)

   1. Initial program 40.8%

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

   2. Taylor expanded in z around inf 38.2%

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

   3. Simplified 97.1%

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

      [Start] 38.2	\[ x \cdot x - 4 \cdot \left(y \cdot {z}^{2}\right) \]
      associate-*r* [=>] 38.2	\[ x \cdot x - \color{blue}{\left(4 \cdot y\right) \cdot {z}^{2}} \]
      *-commutative [<=] 38.2	\[ x \cdot x - \color{blue}{\left(y \cdot 4\right)} \cdot {z}^{2} \]
      unpow2 [=>] 38.2	\[ x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z\right)} \]
      associate-*r* [=>] 97.1	\[ x \cdot x - \color{blue}{\left(\left(y \cdot 4\right) \cdot z\right) \cdot z} \]
      *-commutative [=>] 97.1	\[ x \cdot x - \color{blue}{z \cdot \left(\left(y \cdot 4\right) \cdot z\right)} \]
      *-commutative [=>] 97.1	\[ x \cdot x - z \cdot \left(\color{blue}{\left(4 \cdot y\right)} \cdot z\right) \]
      associate-*l* [=>] 97.1	\[ x \cdot x - z \cdot \color{blue}{\left(4 \cdot \left(y \cdot z\right)\right)} \]

   4. Applied egg-rr 97.1%

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

      [Start] 97.1	\[ x \cdot x - z \cdot \left(4 \cdot \left(y \cdot z\right)\right) \]
      fma-neg [=>] 97.1	\[ \color{blue}{\mathsf{fma}\left(x, x, -z \cdot \left(4 \cdot \left(y \cdot z\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 97.1	\[ \mathsf{fma}\left(x, x, \color{blue}{z \cdot \left(-4 \cdot \left(y \cdot z\right)\right)}\right) \]
      associate-*r* [=>] 97.1	\[ \mathsf{fma}\left(x, x, z \cdot \left(-\color{blue}{\left(4 \cdot y\right) \cdot z}\right)\right) \]
      *-commutative [=>] 97.1	\[ \mathsf{fma}\left(x, x, z \cdot \left(-\color{blue}{z \cdot \left(4 \cdot y\right)}\right)\right) \]
      distribute-rgt-neg-in [=>] 97.1	\[ \mathsf{fma}\left(x, x, z \cdot \color{blue}{\left(z \cdot \left(-4 \cdot y\right)\right)}\right) \]
      *-commutative [=>] 97.1	\[ \mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(-\color{blue}{y \cdot 4}\right)\right)\right) \]
      distribute-rgt-neg-in [=>] 97.1	\[ \mathsf{fma}\left(x, x, z \cdot \left(z \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)\right) \]
      metadata-eval [=>] 97.1	\[ \mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot \color{blue}{-4}\right)\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+222}:\\ \;\;\;\;x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + t \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	85.7%
Cost	1368

\[\begin{array}{l} t_1 := \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ t_3 := x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-69}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+96}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	85.7%
Cost	1368

\[\begin{array}{l} t_1 := \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ t_3 := x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-8}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y - t \cdot y\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-69}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+96}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+222}:\\ \;\;\;\;x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + t \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	55.9%
Cost	1241

\[\begin{array}{l} t_1 := 4 \cdot \left(t \cdot y\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-180}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+65} \lor \neg \left(z \leq 1.4 \cdot 10^{+98}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Accuracy	55.9%
Cost	1240

\[\begin{array}{l} t_1 := 4 \cdot \left(t \cdot y\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -8.8 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{-182}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+64}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+98}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	89.6%
Cost	1100

\[\begin{array}{l} t_1 := x \cdot x + z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-69}:\\ \;\;\;\;x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+64}:\\ \;\;\;\;\left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	99.5%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+222}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + z \cdot \left(-4 \cdot \left(z \cdot y\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	76.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-33} \lor \neg \left(x \leq 3.7 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \end{array} \]

Alternative 10
Accuracy	60.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-33}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-67}:\\ \;\;\;\;4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 11
Accuracy	35.5%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023138 
(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))))