Average Error: 6.0 → 0.1
Time: 5.2s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[x \cdot x - \left(\left(y \cdot \left(4 \cdot z\right)\right) \cdot z + \left(y \cdot 4\right) \cdot \left(-t\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
x \cdot x - \left(\left(y \cdot \left(4 \cdot z\right)\right) \cdot z + \left(y \cdot 4\right) \cdot \left(-t\right)\right)
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) {
	return ((x * x) - (((y * (4.0 * z)) * z) + ((y * 4.0) * -t)));
}

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

Original6.0
Target6.0
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Initial program 6.0

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

  3. Applied sub-neg6.0

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

  4. Applied distribute-lft-in6.0

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

  6. Applied associate-*r*0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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))))