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

Average Error: 5.9 → 0.1

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, -4 \cdot \left(z \cdot y\right), x \cdot x\right)\\ \end{array} \]

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) 5e+295)
   (- (* x x) (* (* y 4.0) (fma z z (- t))))
   (fma z (* -4.0 (* z y)) (* 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) <= 5e+295) {
		tmp = (x * x) - ((y * 4.0) * fma(z, z, -t));
	} else {
		tmp = fma(z, (-4.0 * (z * y)), (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) <= 5e+295)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * fma(z, z, Float64(-t))));
	else
		tmp = fma(z, Float64(-4.0 * Float64(z * y)), 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], 5e+295], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(z * z + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(-4.0 * N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Target

Original	5.9
Target	5.9
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999991e295

   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

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

   if 4.99999999999999991e295 < (*.f64 z z)

   1. Initial program 58.2

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

   2. Taylor expanded in t around 0 58.2

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

   3. Simplified 0.3

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+295}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, -4 \cdot \left(z \cdot y\right), x \cdot x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022197 
(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))))