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

Average Error: 6.2 → 0.1

Time: 4.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+279}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot {z}^{2}\right) + 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(-4 \cdot y\right)\right)\right)\\ \end{array} \]

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) 2e+279)
   (fma x x (+ (* -4.0 (* y (pow z 2.0))) (* 4.0 (* y t))))
   (fma x x (* z (* z (* -4.0 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) <= 2e+279) {
		tmp = fma(x, x, ((-4.0 * (y * pow(z, 2.0))) + (4.0 * (y * t))));
	} else {
		tmp = fma(x, x, (z * (z * (-4.0 * 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) <= 2e+279)
		tmp = fma(x, x, Float64(Float64(-4.0 * Float64(y * (z ^ 2.0))) + Float64(4.0 * Float64(y * t))));
	else
		tmp = fma(x, x, Float64(z * Float64(z * Float64(-4.0 * 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], 2e+279], N[(x * x + N[(N[(-4.0 * N[(y * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(z * N[(z * N[(-4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.2
Target	6.2
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) < 2.00000000000000012e279

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.2

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot z}, \sqrt[3]{z} \cdot z, -t\right)} \cdot -4, x \cdot x\right) \]

   4. Taylor expanded in y around 0 0.1

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

   5. Simplified 0.1

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

   6. Taylor expanded in t around 0 0.1

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

   if 2.00000000000000012e279 < (*.f64 z z)

   1. Initial program 52.9

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

   2. Simplified 53.1

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

   3. Applied egg-rr 53.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot z}, \sqrt[3]{z} \cdot z, -t\right)} \cdot -4, x \cdot x\right) \]

   4. Taylor expanded in t around 0 53.2

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

   5. Simplified 0.5

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+279}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot {z}^{2}\right) + 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(z \cdot \left(-4 \cdot y\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(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))))