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

Percentage Accurate: 91.0% → 97.7%

Time: 11.2s

Precision: binary64

Cost: 7625

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -2 \cdot 10^{+132} \lor \neg \left(y \cdot 4 \leq 10^{-113}\right):\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - 4 \cdot \left(z \cdot \left(y \cdot z\right) - y \cdot t\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y 4.0) -2e+132) (not (<= (* y 4.0) 1e-113)))
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (- (* x x) (* 4.0 (- (* z (* y z)) (* y t))))))

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 (((y * 4.0) <= -2e+132) || !((y * 4.0) <= 1e-113)) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (x * x) - (4.0 * ((z * (y * z)) - (y * t)));
	}
	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(y * 4.0) <= -2e+132) || !(Float64(y * 4.0) <= 1e-113))
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(4.0 * Float64(Float64(z * Float64(y * z)) - Float64(y * t))));
	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[Or[LessEqual[N[(y * 4.0), $MachinePrecision], -2e+132], N[Not[LessEqual[N[(y * 4.0), $MachinePrecision], 1e-113]], $MachinePrecision]], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(4.0 * N[(N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	91.0%
Target	91.0%
Herbie	97.7%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y 4) < -1.99999999999999998e132 or 9.99999999999999979e-114 < (*.f64 y 4)

   1. Initial program 91.8%

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

   2. Simplified 97.5%

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

      [Start] 91.8%	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
      fma-neg [=>] 97.5%	\[ \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)} \]
      *-commutative [=>] 97.5%	\[ \mathsf{fma}\left(x, x, -\color{blue}{\left(z \cdot z - t\right) \cdot \left(y \cdot 4\right)}\right) \]
      distribute-rgt-neg-in [=>] 97.5%	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-y \cdot 4\right)}\right) \]
      distribute-rgt-neg-in [=>] 97.5%	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right) \]
      metadata-eval [=>] 97.5%	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot \color{blue}{-4}\right)\right) \]

   if -1.99999999999999998e132 < (*.f64 y 4) < 9.99999999999999979e-114

   1. Initial program 86.4%

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

   2. Applied egg-rr 86.4%

      \[\leadsto x \cdot x - \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-t\right) \cdot \left(y \cdot 4\right)\right)} \]
      Step-by-step derivation

      [Start] 86.4%	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
      sub-neg [=>] 86.4%	\[ x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z + \left(-t\right)\right)} \]
      distribute-rgt-in [=>] 86.4%	\[ x \cdot x - \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-t\right) \cdot \left(y \cdot 4\right)\right)} \]

   3. Applied egg-rr 99.2%

      \[\leadsto x \cdot x - \color{blue}{4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot t\right)} \]
      Step-by-step derivation

      [Start] 86.4%	\[ x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-t\right) \cdot \left(y \cdot 4\right)\right) \]
      distribute-rgt-out [=>] 86.4%	\[ x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z + \left(-t\right)\right)} \]
      unsub-neg [=>] 86.4%	\[ x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z - t\right)} \]
      add-sqr-sqrt [=>] 48.6%	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - \color{blue}{\sqrt{t} \cdot \sqrt{t}}\right) \]
      sqrt-unprod [=>] 62.0%	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - \color{blue}{\sqrt{t \cdot t}}\right) \]
      sqr-neg [<=] 62.0%	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - \sqrt{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \]
      sqrt-unprod [<=] 24.4%	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - \color{blue}{\sqrt{-t} \cdot \sqrt{-t}}\right) \]
      add-sqr-sqrt [<=] 57.2%	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - \color{blue}{\left(-t\right)}\right) \]
      distribute-rgt-out-- [<=] 57.2%	\[ x \cdot x - \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) - \left(-t\right) \cdot \left(y \cdot 4\right)\right)} \]
      associate-*r* [=>] 57.2%	\[ x \cdot x - \left(\color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot 4} - \left(-t\right) \cdot \left(y \cdot 4\right)\right) \]
      *-commutative [<=] 57.2%	\[ x \cdot x - \left(\color{blue}{\left(y \cdot \left(z \cdot z\right)\right)} \cdot 4 - \left(-t\right) \cdot \left(y \cdot 4\right)\right) \]
      associate-*r* [=>] 57.2%	\[ x \cdot x - \left(\left(y \cdot \left(z \cdot z\right)\right) \cdot 4 - \color{blue}{\left(\left(-t\right) \cdot y\right) \cdot 4}\right) \]
      distribute-rgt-out-- [=>] 57.2%	\[ x \cdot x - \color{blue}{4 \cdot \left(y \cdot \left(z \cdot z\right) - \left(-t\right) \cdot y\right)} \]
      *-commutative [=>] 57.2%	\[ x \cdot x - 4 \cdot \left(\color{blue}{\left(z \cdot z\right) \cdot y} - \left(-t\right) \cdot y\right) \]
      associate-*l* [=>] 70.0%	\[ x \cdot x - 4 \cdot \left(\color{blue}{z \cdot \left(z \cdot y\right)} - \left(-t\right) \cdot y\right) \]
      *-commutative [=>] 70.0%	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - \color{blue}{y \cdot \left(-t\right)}\right) \]
      add-sqr-sqrt [=>] 28.0%	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \color{blue}{\left(\sqrt{-t} \cdot \sqrt{-t}\right)}\right) \]
      sqrt-unprod [=>] 72.7%	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}}\right) \]
      sqr-neg [=>] 72.7%	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \sqrt{\color{blue}{t \cdot t}}\right) \]
      sqrt-unprod [<=] 57.8%	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}\right) \]
      add-sqr-sqrt [<=] 99.2%	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \color{blue}{t}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -2 \cdot 10^{+132} \lor \neg \left(y \cdot 4 \leq 10^{-113}\right):\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - 4 \cdot \left(z \cdot \left(y \cdot z\right) - y \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -2 \cdot 10^{+132} \lor \neg \left(y \cdot 4 \leq 10^{-113}\right):\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - 4 \cdot \left(z \cdot \left(y \cdot z\right) - y \cdot t\right)\\ \end{array} \]

Alternative 2
Accuracy	56.9%
Cost	1240

\[\begin{array}{l} t_1 := \left(y \cdot 4\right) \cdot t\\ t_2 := -4 \cdot \left(y \cdot \left(z \cdot z\right)\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+74}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -3.85 \cdot 10^{-187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+134}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3
Accuracy	58.4%
Cost	1240

\[\begin{array}{l} t_1 := \left(y \cdot 4\right) \cdot t\\ t_2 := -4 \cdot \left(z \cdot \left(y \cdot z\right)\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+109}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-188}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-203}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3300:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4
Accuracy	86.4%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(y \cdot z\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	95.6%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(y \cdot z\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	79.5%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+76}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+141}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7
Accuracy	58.3%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+73}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\left(y \cdot 4\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8
Accuracy	41.9%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023263 
(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))))