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

Average Error: 6.0 → 0.2

Time: 13.7s

Precision: binary64

Cost: 7364

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y \cdot -4\right), z, x \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+256)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (fma (* z (* y -4.0)) z (* 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 * z) <= 2e+256) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = fma((z * (y * -4.0)), z, (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 (Float64(z * z) <= 2e+256)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = fma(Float64(z * Float64(y * -4.0)), z, 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[LessEqual[N[(z * z), $MachinePrecision], 2e+256], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision] * z + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.0
Target	5.9
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e256

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   if 2.0000000000000001e256 < (*.f64 z z)

   1. Initial program 47.5

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

   2. Taylor expanded in z around inf 48.4

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

   3. Simplified 48.4

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z\right)} \]
      Proof
      (*.f64 z z): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unpow2_binary64 (pow.f64 z 2)): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 1.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y \cdot -4\right), z, x \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7236

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right) + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(y \cdot -4\right), z, x \cdot x\right)\\ \end{array} \]

Alternative 2
Error	27.9
Cost	1748

\[\begin{array}{l} t_1 := 4 \cdot \left(t \cdot y\right)\\ \mathbf{if}\;z \cdot z \leq 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 10^{-233}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{-137}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 3
Error	19.3
Cost	1740

\[\begin{array}{l} t_1 := z \cdot z - t\\ t_2 := t_1 \cdot \left(y \cdot -4\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-129}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 4
Error	7.8
Cost	1232

\[\begin{array}{l} t_1 := x \cdot x + \left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.0648518555048894 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.422403819666622 \cdot 10^{-55}:\\ \;\;\;\;x \cdot x + y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;z \leq 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	26.3
Cost	1100

\[\begin{array}{l} t_1 := 4 \cdot \left(t \cdot y\right)\\ \mathbf{if}\;x \cdot x \leq 5.903348252639539 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 3.6616406915803755 \cdot 10^{-193}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 1.9426676544576173 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Error	8.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-82}:\\ \;\;\;\;x \cdot x + y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 7
Error	1.3
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right) + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 8
Error	26.2
Cost	848

\[\begin{array}{l} t_1 := 4 \cdot \left(t \cdot y\right)\\ \mathbf{if}\;x \leq -1.7599262833541525 \cdot 10^{-63}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.1585915846538582 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.988158621646355 \cdot 10^{-76}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.570961412167542 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 9
Error	41.1
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2022297 
(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))))