Average Error: 6.1 → 0.1
Time: 4.7s
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 r621864 = x;
        double r621865 = r621864 * r621864;
        double r621866 = y;
        double r621867 = 4.0;
        double r621868 = r621866 * r621867;
        double r621869 = z;
        double r621870 = r621869 * r621869;
        double r621871 = t;
        double r621872 = r621870 - r621871;
        double r621873 = r621868 * r621872;
        double r621874 = r621865 - r621873;
        return r621874;
}
↓
double f(double x, double y, double z, double t) {
        double r621875 = x;
        double r621876 = y;
        double r621877 = 4.0;
        double r621878 = r621876 * r621877;
        double r621879 = t;
        double r621880 = r621878 * r621879;
        double r621881 = z;
        double r621882 = -r621881;
        double r621883 = r621878 * r621882;
        double r621884 = r621883 * r621881;
        double r621885 = r621880 + r621884;
        double r621886 = fma(r621875, r621875, r621885);
        return r621886;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

Derivation

  1. Initial program 6.1

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

  2. Simplified6.1

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

  4. Applied sub-neg6.1

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

  5. Applied distribute-lft-in6.1

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

  7. Applied distribute-lft-neg-in6.1

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

  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 2020057 +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))))