Average Error: 5.7 → 0.1
Time: 2.5s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\mathsf{fma}\left(x, x, y \cdot \left(4 \cdot t\right) + \left(y \cdot z\right) \cdot \left(\left(-z\right) \cdot 4\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\mathsf{fma}\left(x, x, y \cdot \left(4 \cdot t\right) + \left(y \cdot z\right) \cdot \left(\left(-z\right) \cdot 4\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 fma(x, x, ((y * (4.0 * t)) + ((y * z) * (-z * 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

Original5.7
Target5.7
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Initial program 5.7

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

  2. Simplified5.7

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

  4. Applied associate-*l*5.7

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

  6. Applied sub-neg5.7

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

  7. Applied distribute-lft-in5.7

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

  8. Applied distribute-lft-in5.7

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

  9. Simplified5.7

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

  11. Applied distribute-rgt-neg-in5.7

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

  12. Applied associate-*l*5.7

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

  13. Applied associate-*r*0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020078 +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))))