Average Error: 6.2 → 0.1
Time: 3.4s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(y \cdot 4, t, \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(y \cdot 4, t, \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right)\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) fma(x, x, ((double) fma(((double) (y * 4.0)), t, ((double) (((double) (((double) (y * 4.0)) * ((double) -(z)))) * z))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

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Target

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

Derivation

  1. Initial program 6.2

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

  2. Simplified6.2

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

  4. Applied sub-neg6.2

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

  5. Applied distribute-lft-in6.2

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

  7. Applied distribute-lft-neg-in6.2

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

  8. Applied associate-*r*0.1

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

  10. Applied fma-def0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020121 +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))))