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

Average Error: 9.27% → 0.19%

Time: 12.5s

Precision: binary64

Cost: 7364

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

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) 2e+304)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (- (* 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) <= 2e+304) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (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) <= 2e+304)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(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], 2e+304], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	9.27%
Target	9.21%
Herbie	0.19%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e304

   1. Initial program 0.17

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

   2. Simplified 0.16

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

      [Start] 0.17	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
      fma-neg [=>] 0.16	\[ \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)} \]
      distribute-lft-neg-in [=>] 0.16	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(-y \cdot 4\right) \cdot \left(z \cdot z - t\right)}\right) \]
      *-commutative [=>] 0.16	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-y \cdot 4\right)}\right) \]
      distribute-rgt-neg-in [=>] 0.16	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right) \]
      metadata-eval [=>] 0.16	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot \color{blue}{-4}\right)\right) \]

   if 1.9999999999999999e304 < (*.f64 z z)

   1. Initial program 97.34

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

   2. Applied egg-rr 97.35

      \[\leadsto x \cdot x - \color{blue}{\frac{y \cdot 4}{\frac{1}{z \cdot z - t}}} \]

   3. Taylor expanded in z around inf 97.35

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]

   4. Simplified 90.88

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{\frac{1}{z}}{z}}} \]
      Proof

      [Start] 97.35	\[ x \cdot x - \frac{y \cdot 4}{\frac{1}{{z}^{2}}} \]
      unpow2 [=>] 97.35	\[ x \cdot x - \frac{y \cdot 4}{\frac{1}{\color{blue}{z \cdot z}}} \]
      associate-/r* [=>] 90.88	\[ x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{\frac{1}{z}}{z}}} \]

   5. Applied egg-rr 0.4

      \[\leadsto x \cdot x - \color{blue}{\left(\left(y \cdot 4\right) \cdot z\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.19

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	31.09%
Cost	1878

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 9.5 \cdot 10^{-265} \lor \neg \left(x \cdot x \leq 1.7 \cdot 10^{-191}\right) \land \left(x \cdot x \leq 8.5 \cdot 10^{-119} \lor \neg \left(x \cdot x \leq 8 \cdot 10^{-88}\right) \land x \cdot x \leq 1.05 \cdot 10^{+97}\right):\\ \;\;\;\;y \cdot \left(\left(z \cdot z - t\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	11.92%
Cost	1744

\[\begin{array}{l} t_1 := x \cdot x - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-112}:\\ \;\;\;\;x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+94}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z - t\right) \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 3
Error	43.45%
Cost	1622

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.02 \cdot 10^{-268} \lor \neg \left(x \cdot x \leq 1.7 \cdot 10^{-191}\right) \land \left(x \cdot x \leq 10^{-118} \lor \neg \left(x \cdot x \leq 8 \cdot 10^{-87}\right) \land x \cdot x \leq 0.0055\right):\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4
Error	10.78%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-112}:\\ \;\;\;\;x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+66} \lor \neg \left(z \cdot z \leq 4 \cdot 10^{+94}\right):\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z \cdot z - t\right) \cdot -4\right)\\ \end{array} \]

Alternative 5
Error	0.23%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\left(x \cdot x + y \cdot \left(t \cdot 4\right)\right) - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 6
Error	14.06%
Cost	1236

\[\begin{array}{l} t_1 := y \cdot \left(\left(z \cdot z - t\right) \cdot -4\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ t_3 := x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2100000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-54}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+52}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	0.19%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+304}:\\ \;\;\;\;x \cdot x - \left(z \cdot z - t\right) \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 8
Error	41.65%
Cost	976

\[\begin{array}{l} t_1 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -1450000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-43}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-89}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-54}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	64.55%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023089 
(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))))