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

Average Accuracy: 90.3% → 99.8%

Time: 13.0s

Precision: binary64

Cost: 7369

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+154} \lor \neg \left(z \leq 3.7 \cdot 10^{+126}\right):\\ \;\;\;\;x \cdot x + z \cdot \left(\left(z \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.35e+154) (not (<= z 3.7e+126)))
   (+ (* x x) (* z (* (* z y) -4.0)))
   (fma (- (* z z) t) (* 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 <= -1.35e+154) || !(z <= 3.7e+126)) {
		tmp = (x * x) + (z * ((z * y) * -4.0));
	} else {
		tmp = fma(((z * z) - t), (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 ((z <= -1.35e+154) || !(z <= 3.7e+126))
		tmp = Float64(Float64(x * x) + Float64(z * Float64(Float64(z * y) * -4.0)));
	else
		tmp = fma(Float64(Float64(z * z) - t), 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[Or[LessEqual[z, -1.35e+154], N[Not[LessEqual[z, 3.7e+126]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] + N[(z * N[(N[(z * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.3%
Target	90.3%
Herbie	99.8%

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

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000003e154 or 3.6999999999999998e126 < z

   1. Initial program 15.3%

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

   2. Taylor expanded in z around inf 15.1%

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

   3. Simplified 99.4%

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

      [Start] 15.1	\[ x \cdot x - 4 \cdot \left(y \cdot {z}^{2}\right) \]
      associate-*r* [=>] 15.1	\[ x \cdot x - \color{blue}{\left(4 \cdot y\right) \cdot {z}^{2}} \]
      *-commutative [<=] 15.1	\[ x \cdot x - \color{blue}{\left(y \cdot 4\right)} \cdot {z}^{2} \]
      unpow2 [=>] 15.1	\[ x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z\right)} \]
      associate-*r* [=>] 99.4	\[ x \cdot x - \color{blue}{\left(\left(y \cdot 4\right) \cdot z\right) \cdot z} \]
      *-commutative [=>] 99.4	\[ x \cdot x - \color{blue}{z \cdot \left(\left(y \cdot 4\right) \cdot z\right)} \]
      *-commutative [=>] 99.4	\[ x \cdot x - z \cdot \left(\color{blue}{\left(4 \cdot y\right)} \cdot z\right) \]
      associate-*l* [=>] 99.4	\[ x \cdot x - z \cdot \color{blue}{\left(4 \cdot \left(y \cdot z\right)\right)} \]

   if -1.35000000000000003e154 < z < 3.6999999999999998e126

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

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+154} \lor \neg \left(z \leq 3.7 \cdot 10^{+126}\right):\\ \;\;\;\;x \cdot x + z \cdot \left(\left(z \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+147} \lor \neg \left(z \leq 1.45 \cdot 10^{+128}\right):\\ \;\;\;\;x \cdot x + z \cdot \left(\left(z \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	56.3%
Cost	1372

\[\begin{array}{l} t_1 := t \cdot \left(4 \cdot y\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -9.6 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-34}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-211}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-96}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	87.2%
Cost	1104

\[\begin{array}{l} t_1 := \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -8.8 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-18}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	89.5%
Cost	1100

\[\begin{array}{l} t_1 := x \cdot x + z \cdot \left(\left(z \cdot y\right) \cdot -4\right)\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{+40}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-29}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+147} \lor \neg \left(z \leq 1.3 \cdot 10^{+128}\right):\\ \;\;\;\;x \cdot x + z \cdot \left(\left(z \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot z - t\right) \cdot \left(4 \cdot y\right)\\ \end{array} \]

Alternative 6
Accuracy	74.1%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+17}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-86}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-29}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7
Accuracy	58.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+17} \lor \neg \left(x \leq 1.2 \cdot 10^{-70}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \end{array} \]

Alternative 8
Accuracy	34.9%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023129 
(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))))