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

Average Error: 6.1 → 2.3

Time: 4.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq 6.8873471163078 \cdot 10^{-311}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(t \cdot 4\right) - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(4 \cdot \left(z \cdot \left(\sqrt[3]{y} \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(4 \cdot \left(z + \sqrt{t}\right)\right)\right) \cdot \left(\sqrt{t} - z\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((t <= 6.8873471163078e-311)) {
		VAR = ((double) (((double) (x * x)) + ((double) (((double) (y * ((double) (t * 4.0)))) - ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) (4.0 * ((double) (z * ((double) (((double) cbrt(y)) * z))))))))))));
	} else {
		VAR = ((double) (((double) (x * x)) + ((double) (((double) (y * ((double) (4.0 * ((double) (z + ((double) sqrt(t)))))))) * ((double) (((double) sqrt(t)) - z))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.1
Target	6.1
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if t < 6.88734711630777e-311

   1. Initial program 5.8

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
   2. Using strategy rm

   3. Applied sub-neg 5.8

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

   4. Applied distribute-lft-in 5.8

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

   5. Simplified 5.8

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

   6. Simplified 5.9

      \[\leadsto x \cdot x - \left(y \cdot \left(4 \cdot \left(z \cdot z\right)\right) + \color{blue}{y \cdot \left(4 \cdot \left(-t\right)\right)}\right)\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 6.1

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

   9. Applied associate-*l* 6.1

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

   10. Simplified 6.1

       \[\leadsto x \cdot x - \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \color{blue}{\left(4 \cdot \left(\left(z \cdot z\right) \cdot \sqrt[3]{y}\right)\right)} + y \cdot \left(4 \cdot \left(-t\right)\right)\right)\]
   11. Using strategy rm

   12. Applied associate-*l* 4.2

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

   13. Simplified 4.2

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

   if 6.88734711630777e-311 < t

   1. Initial program 6.3

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 6.5

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

   4. Applied difference-of-squares 6.5

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

   5. Applied associate-*r* 0.3

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

   6. Simplified 0.3

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8873471163078 \cdot 10^{-311}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(t \cdot 4\right) - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(4 \cdot \left(z \cdot \left(\sqrt[3]{y} \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot \left(4 \cdot \left(z + \sqrt{t}\right)\right)\right) \cdot \left(\sqrt{t} - z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))))