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

Average Error: 5.7 → 0.1

Time: 11.1s

Precision: binary64

Cost: 7360

Math

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

↓

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

FPCore

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

↓

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

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

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)
	return Float64(Float64(x * x) - fma(Float64(y * -4.0), t, Float64(4.0 * Float64(z * Float64(y * z)))))
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_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * -4.0), $MachinePrecision] * t + N[(4.0 * N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	5.7
Target	5.7
Herbie	0.1

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

Derivation

1. Initial program 5.7

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

2. Applied egg-rr 5.8

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

3. Applied egg-rr 6.4

   \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{{\left({\left(\sqrt[3]{z \cdot z - t}\right)}^{2}\right)}^{-1} \cdot {\left(\sqrt[3]{z \cdot z - t}\right)}^{-1}}} \]

4. Simplified 6.4

   \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{\frac{1}{\sqrt[3]{z \cdot z - t}} \cdot 1}{{\left(\sqrt[3]{z \cdot z - t}\right)}^{2}}}} \]
   Proof

   [Start] 6.4	\[ x \cdot x - \frac{y \cdot 4}{{\left({\left(\sqrt[3]{z \cdot z - t}\right)}^{2}\right)}^{-1} \cdot {\left(\sqrt[3]{z \cdot z - t}\right)}^{-1}} \]
   *-commutative [=>] 6.4	\[ x \cdot x - \frac{y \cdot 4}{\color{blue}{{\left(\sqrt[3]{z \cdot z - t}\right)}^{-1} \cdot {\left({\left(\sqrt[3]{z \cdot z - t}\right)}^{2}\right)}^{-1}}} \]
   unpow-1 [=>] 6.4	\[ x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{\sqrt[3]{z \cdot z - t}}} \cdot {\left({\left(\sqrt[3]{z \cdot z - t}\right)}^{2}\right)}^{-1}} \]
   unpow-1 [=>] 6.4	\[ x \cdot x - \frac{y \cdot 4}{\frac{1}{\sqrt[3]{z \cdot z - t}} \cdot \color{blue}{\frac{1}{{\left(\sqrt[3]{z \cdot z - t}\right)}^{2}}}} \]
   associate-*r/ [=>] 6.4	\[ x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{\frac{1}{\sqrt[3]{z \cdot z - t}} \cdot 1}{{\left(\sqrt[3]{z \cdot z - t}\right)}^{2}}}} \]

5. Taylor expanded in z around 0 5.7

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

6. Simplified 0.1

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

   [Start] 5.7	\[ x \cdot x - \left(4 \cdot \left(y \cdot {z}^{2}\right) + -4 \cdot \left(y \cdot t\right)\right) \]
   +-commutative [=>] 5.7	\[ x \cdot x - \color{blue}{\left(-4 \cdot \left(y \cdot t\right) + 4 \cdot \left(y \cdot {z}^{2}\right)\right)} \]
   associate-*r* [=>] 5.7	\[ x \cdot x - \left(\color{blue}{\left(-4 \cdot y\right) \cdot t} + 4 \cdot \left(y \cdot {z}^{2}\right)\right) \]
   *-commutative [<=] 5.7	\[ x \cdot x - \left(\color{blue}{\left(y \cdot -4\right)} \cdot t + 4 \cdot \left(y \cdot {z}^{2}\right)\right) \]
   associate-*r* [=>] 5.7	\[ x \cdot x - \left(\left(y \cdot -4\right) \cdot t + \color{blue}{\left(4 \cdot y\right) \cdot {z}^{2}}\right) \]
   *-commutative [<=] 5.7	\[ x \cdot x - \left(\left(y \cdot -4\right) \cdot t + \color{blue}{\left(y \cdot 4\right)} \cdot {z}^{2}\right) \]
   /-rgt-identity [<=] 5.7	\[ x \cdot x - \left(\left(y \cdot -4\right) \cdot t + \color{blue}{\frac{y \cdot 4}{1}} \cdot {z}^{2}\right) \]
   unpow2 [=>] 5.7	\[ x \cdot x - \left(\left(y \cdot -4\right) \cdot t + \frac{y \cdot 4}{1} \cdot \color{blue}{\left(z \cdot z\right)}\right) \]
   fma-def [=>] 5.7	\[ x \cdot x - \color{blue}{\mathsf{fma}\left(y \cdot -4, t, \frac{y \cdot 4}{1} \cdot \left(z \cdot z\right)\right)} \]
   /-rgt-identity [=>] 5.7	\[ x \cdot x - \mathsf{fma}\left(y \cdot -4, t, \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z\right)\right) \]
   *-commutative [=>] 5.7	\[ x \cdot x - \mathsf{fma}\left(y \cdot -4, t, \color{blue}{\left(4 \cdot y\right)} \cdot \left(z \cdot z\right)\right) \]
   unpow2 [<=] 5.7	\[ x \cdot x - \mathsf{fma}\left(y \cdot -4, t, \left(4 \cdot y\right) \cdot \color{blue}{{z}^{2}}\right) \]
   associate-*r* [<=] 5.7	\[ x \cdot x - \mathsf{fma}\left(y \cdot -4, t, \color{blue}{4 \cdot \left(y \cdot {z}^{2}\right)}\right) \]
   *-commutative [=>] 5.7	\[ x \cdot x - \mathsf{fma}\left(y \cdot -4, t, 4 \cdot \color{blue}{\left({z}^{2} \cdot y\right)}\right) \]
   unpow2 [=>] 5.7	\[ x \cdot x - \mathsf{fma}\left(y \cdot -4, t, 4 \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right)\right) \]
   associate-*l* [=>] 0.1	\[ x \cdot x - \mathsf{fma}\left(y \cdot -4, t, 4 \cdot \color{blue}{\left(z \cdot \left(z \cdot y\right)\right)}\right) \]

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.0
Cost	8392

\[\begin{array}{l} t_1 := \left(y \cdot -4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y \cdot -4}{\frac{1}{z}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \end{array} \]

Alternative 2
Error	1.0
Cost	2120

\[\begin{array}{l} t_1 := \left(y \cdot -4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y \cdot -4}{\frac{1}{z}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;x \cdot x + \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \end{array} \]

Alternative 3
Error	8.2
Cost	1744

\[\begin{array}{l} t_1 := x \cdot x + \left(y \cdot -4\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-91}:\\ \;\;\;\;x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 10^{+192}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+281}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y \cdot -4}{\frac{1}{z}}\\ \end{array} \]

Alternative 4
Error	27.6
Cost	1377

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ t_2 := z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+35}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -280:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-62}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-63} \lor \neg \left(x \leq 4.5 \cdot 10^{-23}\right) \land x \leq 2 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5
Error	27.6
Cost	1376

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot 4\right)\\ t_2 := z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+35}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -380:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-62}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{y \cdot -4}{\frac{-1}{t}}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-23}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Error	17.2
Cost	1105

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+79}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-63} \lor \neg \left(x \leq 4.1 \cdot 10^{-22}\right) \land x \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7
Error	8.1
Cost	1104

\[\begin{array}{l} t_1 := \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ t_2 := z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-14}:\\ \;\;\;\;x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	8.1
Cost	1104

\[\begin{array}{l} t_1 := \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;z \cdot \left(\left(y \cdot -4\right) \cdot z\right)\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y \cdot -4}{\frac{1}{z}}\\ \end{array} \]

Alternative 9
Error	26.4
Cost	1101

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.05 \cdot 10^{-125} \lor \neg \left(x \cdot x \leq 4.9 \cdot 10^{-72}\right) \land x \cdot x \leq 5.4 \cdot 10^{+79}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 10
Error	40.7
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023054 
(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))))