Average Error: 10.4 → 0.8
Time: 14.5s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.2525042507888118 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \le 3.640402671389334 \cdot 10^{-286}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{x}}{\sqrt[3]{z}}, y\right) - \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}}\\ \end{array}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.2525042507888118 \cdot 10^{-93}:\\
\;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;z \le 3.640402671389334 \cdot 10^{-286}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{x}}{\sqrt[3]{z}}, y\right) - \frac{x \cdot y}{z}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}}\\

\end{array}
double f(double x, double y, double z) {
        double r757764 = x;
        double r757765 = y;
        double r757766 = z;
        double r757767 = r757766 - r757764;
        double r757768 = r757765 * r757767;
        double r757769 = r757764 + r757768;
        double r757770 = r757769 / r757766;
        return r757770;
}
↓
double f(double x, double y, double z) {
        double r757771 = z;
        double r757772 = -1.2525042507888118e-93;
        bool r757773 = r757771 <= r757772;
        double r757774 = x;
        double r757775 = r757774 / r757771;
        double r757776 = y;
        double r757777 = r757775 + r757776;
        double r757778 = r757771 / r757776;
        double r757779 = r757774 / r757778;
        double r757780 = r757777 - r757779;
        double r757781 = 3.640402671389334e-286;
        bool r757782 = r757771 <= r757781;
        double r757783 = cbrt(r757774);
        double r757784 = r757783 * r757783;
        double r757785 = cbrt(r757771);
        double r757786 = r757785 * r757785;
        double r757787 = r757784 / r757786;
        double r757788 = r757783 / r757785;
        double r757789 = fma(r757787, r757788, r757776);
        double r757790 = r757774 * r757776;
        double r757791 = r757790 / r757771;
        double r757792 = r757789 - r757791;
        double r757793 = sqrt(r757771);
        double r757794 = r757774 / r757793;
        double r757795 = r757776 / r757793;
        double r757796 = r757794 * r757795;
        double r757797 = r757777 - r757796;
        double r757798 = r757782 ? r757792 : r757797;
        double r757799 = r757773 ? r757780 : r757798;
        return r757799;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.4
Target0.0
Herbie0.8
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.2525042507888118e-93

    1. Initial program 14.0

      \[\frac{x + y \cdot \left(z - x\right)}{z}\]

    2. Simplified14.0

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

    3. Taylor expanded around 0 4.5

      \[\leadsto \color{blue}{\left(\frac{x}{z} + y\right) - \frac{x \cdot y}{z}}\]
    4. Using strategy rm

    5. Applied associate-/l*0.4

      \[\leadsto \left(\frac{x}{z} + y\right) - \color{blue}{\frac{x}{\frac{z}{y}}}\]

    if -1.2525042507888118e-93 < z < 3.640402671389334e-286

    1. Initial program 0.1

      \[\frac{x + y \cdot \left(z - x\right)}{z}\]

    2. Simplified0.1

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

    3. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(\frac{x}{z} + y\right) - \frac{x \cdot y}{z}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt0.9

      \[\leadsto \left(\frac{x}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} + y\right) - \frac{x \cdot y}{z}\]

    6. Applied add-cube-cbrt1.0

      \[\leadsto \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} + y\right) - \frac{x \cdot y}{z}\]

    7. Applied times-frac1.0

      \[\leadsto \left(\color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}} + y\right) - \frac{x \cdot y}{z}\]

    8. Applied fma-def1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{x}}{\sqrt[3]{z}}, y\right)} - \frac{x \cdot y}{z}\]

    if 3.640402671389334e-286 < z

    1. Initial program 10.4

      \[\frac{x + y \cdot \left(z - x\right)}{z}\]

    2. Simplified10.4

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

    3. Taylor expanded around 0 3.3

      \[\leadsto \color{blue}{\left(\frac{x}{z} + y\right) - \frac{x \cdot y}{z}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt3.4

      \[\leadsto \left(\frac{x}{z} + y\right) - \frac{x \cdot y}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\]

    6. Applied times-frac1.1

      \[\leadsto \left(\frac{x}{z} + y\right) - \color{blue}{\frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2525042507888118 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \le 3.640402671389334 \cdot 10^{-286}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{x}}{\sqrt[3]{z}}, y\right) - \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))