Average Error: 6.2 → 0.1
Time: 4.4s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right)
double f(double x, double y, double z, double t) {
        double r640309 = x;
        double r640310 = r640309 * r640309;
        double r640311 = y;
        double r640312 = 4.0;
        double r640313 = r640311 * r640312;
        double r640314 = z;
        double r640315 = r640314 * r640314;
        double r640316 = t;
        double r640317 = r640315 - r640316;
        double r640318 = r640313 * r640317;
        double r640319 = r640310 - r640318;
        return r640319;
}
↓
double f(double x, double y, double z, double t) {
        double r640320 = x;
        double r640321 = y;
        double r640322 = 4.0;
        double r640323 = r640321 * r640322;
        double r640324 = t;
        double r640325 = r640323 * r640324;
        double r640326 = z;
        double r640327 = -r640326;
        double r640328 = r640323 * r640327;
        double r640329 = r640328 * r640326;
        double r640330 = r640325 + r640329;
        double r640331 = fma(r640320, r640320, r640330);
        return r640331;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 +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))))