Average Error: 6.3 → 0.1
Time: 4.7s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\left(x \cdot x - z \cdot \left(\left(z \cdot y\right) \cdot 4\right)\right) - \left(-t\right) \cdot \left(y \cdot 4\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\left(x \cdot x - z \cdot \left(\left(z \cdot y\right) \cdot 4\right)\right) - \left(-t\right) \cdot \left(y \cdot 4\right)
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) {
	return ((double) (((double) (((double) (x * x)) - ((double) (z * ((double) (((double) (z * y)) * 4.0)))))) - ((double) (((double) -(t)) * ((double) (y * 4.0))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Results

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Target

Original6.3
Target6.2
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  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 sub-neg6.3

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

  4. Applied distribute-rgt-in6.3

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

  5. Applied associate--r+6.3

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

  7. Applied associate-*l*0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020114 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4) (- (* z z) t))))