Average Error: 10.4 → 0.8
Time: 12.0s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.2525042507888118 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z} + y, \frac{-1}{\frac{\frac{z}{y}}{x}}\right)\\ \mathbf{elif}\;z \le 3.640402671389334 \cdot 10^{-286}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{x}}{\sqrt[3]{z}}, y\right) - \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}}\\ \end{array}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.2525042507888118 \cdot 10^{-93}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{z} + y, \frac{-1}{\frac{\frac{z}{y}}{x}}\right)\\

\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 r778896 = x;
        double r778897 = y;
        double r778898 = z;
        double r778899 = r778898 - r778896;
        double r778900 = r778897 * r778899;
        double r778901 = r778896 + r778900;
        double r778902 = r778901 / r778898;
        return r778902;
}
↓
double f(double x, double y, double z) {
        double r778903 = z;
        double r778904 = -1.2525042507888118e-93;
        bool r778905 = r778903 <= r778904;
        double r778906 = 1.0;
        double r778907 = x;
        double r778908 = r778907 / r778903;
        double r778909 = y;
        double r778910 = r778908 + r778909;
        double r778911 = -r778906;
        double r778912 = r778903 / r778909;
        double r778913 = r778912 / r778907;
        double r778914 = r778911 / r778913;
        double r778915 = fma(r778906, r778910, r778914);
        double r778916 = 3.640402671389334e-286;
        bool r778917 = r778903 <= r778916;
        double r778918 = cbrt(r778907);
        double r778919 = r778918 * r778918;
        double r778920 = cbrt(r778903);
        double r778921 = r778920 * r778920;
        double r778922 = r778919 / r778921;
        double r778923 = r778918 / r778920;
        double r778924 = fma(r778922, r778923, r778909);
        double r778925 = r778907 * r778909;
        double r778926 = r778925 / r778903;
        double r778927 = r778924 - r778926;
        double r778928 = sqrt(r778903);
        double r778929 = r778907 / r778928;
        double r778930 = r778909 / r778928;
        double r778931 = r778929 * r778930;
        double r778932 = r778910 - r778931;
        double r778933 = r778917 ? r778927 : r778932;
        double r778934 = r778905 ? r778915 : r778933;
        return r778934;
}

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 *-un-lft-identity4.5

      \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{z} + y\right)} - \frac{x \cdot y}{z}\]

    6. Applied fma-neg4.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{x}{z} + y, -\frac{x \cdot y}{z}\right)}\]

    7. Simplified0.4

      \[\leadsto \mathsf{fma}\left(1, \frac{x}{z} + y, \color{blue}{\frac{-x}{\frac{z}{y}}}\right)\]
    8. Using strategy rm

    9. Applied *-un-lft-identity0.4

      \[\leadsto \mathsf{fma}\left(1, \frac{x}{z} + y, \frac{-\color{blue}{1 \cdot x}}{\frac{z}{y}}\right)\]

    10. Applied distribute-lft-neg-in0.4

      \[\leadsto \mathsf{fma}\left(1, \frac{x}{z} + y, \frac{\color{blue}{\left(-1\right) \cdot x}}{\frac{z}{y}}\right)\]

    11. Applied associate-/l*0.4

      \[\leadsto \mathsf{fma}\left(1, \frac{x}{z} + y, \color{blue}{\frac{-1}{\frac{\frac{z}{y}}{x}}}\right)\]

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2525042507888118 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z} + y, \frac{-1}{\frac{\frac{z}{y}}{x}}\right)\\ \mathbf{elif}\;z \le 3.640402671389334 \cdot 10^{-286}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{x}}{\sqrt[3]{z}}, y\right) - \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}}\\ \end{array}\]

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))