Average Error: 5.9 → 3.1
Time: 3.5s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 2.3243118769397114 \cdot 10^{304}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 2.3243118769397114 \cdot 10^{304}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r719789 = x;
        double r719790 = r719789 * r719789;
        double r719791 = y;
        double r719792 = 4.0;
        double r719793 = r719791 * r719792;
        double r719794 = z;
        double r719795 = r719794 * r719794;
        double r719796 = t;
        double r719797 = r719795 - r719796;
        double r719798 = r719793 * r719797;
        double r719799 = r719790 - r719798;
        return r719799;
}
↓
double f(double x, double y, double z, double t) {
        double r719800 = z;
        double r719801 = r719800 * r719800;
        double r719802 = 2.3243118769397114e+304;
        bool r719803 = r719801 <= r719802;
        double r719804 = x;
        double r719805 = y;
        double r719806 = 4.0;
        double r719807 = r719805 * r719806;
        double r719808 = t;
        double r719809 = r719808 - r719801;
        double r719810 = r719807 * r719809;
        double r719811 = fma(r719804, r719804, r719810);
        double r719812 = sqrt(r719808);
        double r719813 = r719812 + r719800;
        double r719814 = r719807 * r719813;
        double r719815 = r719812 - r719800;
        double r719816 = r719814 * r719815;
        double r719817 = fma(r719804, r719804, r719816);
        double r719818 = r719803 ? r719811 : r719817;
        return r719818;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original5.9
Target5.9
Herbie3.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 2.3243118769397114e+304

    1. Initial program 0.1

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

    2. Simplified0.1

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

    if 2.3243118769397114e+304 < (* z z)

    1. Initial program 62.9

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

    2. Simplified62.9

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

    4. Applied add-sqr-sqrt63.2

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

    5. Applied difference-of-squares63.2

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

    6. Applied associate-*r*32.4

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 2.3243118769397114 \cdot 10^{304}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))))