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

Average Error: 5.8 → 0.5

Time: 11.0s

Precision: binary64

Cost: 7364

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]

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+192)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (fma x x (* -4.0 (* z (* z y))))))

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+192) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = fma(x, x, (-4.0 * (z * (z * y))));
	}
	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+192)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = fma(x, x, Float64(-4.0 * Float64(z * Float64(z * y))));
	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+192], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	5.8
Target	5.8
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e192

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
      fma-neg [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)} \]
      distribute-lft-neg-in [=>] 0.1	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(-y \cdot 4\right) \cdot \left(z \cdot z - t\right)}\right) \]
      *-commutative [=>] 0.1	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-y \cdot 4\right)}\right) \]
      distribute-rgt-neg-in [=>] 0.1	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot \color{blue}{-4}\right)\right) \]

   if 1.00000000000000004e192 < (*.f64 z z)

   1. Initial program 33.6

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

   2. Simplified 33.6

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

      [Start] 33.6	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
      fma-neg [=>] 33.6	\[ \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)} \]
      distribute-lft-neg-in [=>] 33.6	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(-y \cdot 4\right) \cdot \left(z \cdot z - t\right)}\right) \]
      *-commutative [=>] 33.6	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(z \cdot z - t\right) \cdot \left(-y \cdot 4\right)}\right) \]
      distribute-rgt-neg-in [=>] 33.6	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right) \]
      metadata-eval [=>] 33.6	\[ \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot \color{blue}{-4}\right)\right) \]

   3. Taylor expanded in z around inf 36.1

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

   4. Simplified 2.7

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

      [Start] 36.1	\[ \mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot {z}^{2}\right)\right) \]
      unpow2 [=>] 36.1	\[ \mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]
      associate-*r* [=>] 2.7	\[ \mathsf{fma}\left(x, x, -4 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	7236

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+192}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right) + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	7.0
Cost	1745

\[\begin{array}{l} t_1 := x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \cdot z \leq 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 1000:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+62} \lor \neg \left(z \cdot z \leq 5 \cdot 10^{+112}\right):\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right) + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	9.6
Cost	1616

\[\begin{array}{l} t_1 := \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ t_2 := x \cdot x + t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;z \cdot z \leq 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot z \leq 50000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 1.5 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot z \leq 10^{+192}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 4
Error	16.1
Cost	1357

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.7 \cdot 10^{-73} \lor \neg \left(x \cdot x \leq 105000000000\right) \land x \cdot x \leq 5.8 \cdot 10^{+72}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5
Error	26.6
Cost	1112

\[\begin{array}{l} t_1 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ t_2 := t \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-21}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-301}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Error	0.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+192}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right) + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(z \cdot \left(z \cdot y\right)\right) + x \cdot x\\ \end{array} \]

Alternative 7
Error	24.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-19}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-46}:\\ \;\;\;\;t \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8
Error	41.5
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023083 
(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))))