Average Error: 10.7 → 1.9
Time: 17.1s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 1.010367153808741264986405239084380831559 \cdot 10^{46}:\\ \;\;\;\;y + \frac{x - y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{z}} \cdot \left(\frac{1}{\sqrt{z}} - \frac{1}{\frac{\sqrt{z}}{y}}\right) + y\\ \end{array}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le 1.010367153808741264986405239084380831559 \cdot 10^{46}:\\
\;\;\;\;y + \frac{x - y \cdot x}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\sqrt{z}} \cdot \left(\frac{1}{\sqrt{z}} - \frac{1}{\frac{\sqrt{z}}{y}}\right) + y\\

\end{array}
double f(double x, double y, double z) {
        double r586077 = x;
        double r586078 = y;
        double r586079 = z;
        double r586080 = r586079 - r586077;
        double r586081 = r586078 * r586080;
        double r586082 = r586077 + r586081;
        double r586083 = r586082 / r586079;
        return r586083;
}
↓
double f(double x, double y, double z) {
        double r586084 = z;
        double r586085 = 1.0103671538087413e+46;
        bool r586086 = r586084 <= r586085;
        double r586087 = y;
        double r586088 = x;
        double r586089 = r586087 * r586088;
        double r586090 = r586088 - r586089;
        double r586091 = r586090 / r586084;
        double r586092 = r586087 + r586091;
        double r586093 = sqrt(r586084);
        double r586094 = r586088 / r586093;
        double r586095 = 1.0;
        double r586096 = r586095 / r586093;
        double r586097 = r586093 / r586087;
        double r586098 = r586095 / r586097;
        double r586099 = r586096 - r586098;
        double r586100 = r586094 * r586099;
        double r586101 = r586100 + r586087;
        double r586102 = r586086 ? r586092 : r586101;
        return r586102;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.7
Target0.0
Herbie1.9
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 1.0103671538087413e+46

    1. Initial program 7.5

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

    2. Simplified2.5

      \[\leadsto \color{blue}{y + \frac{x - x \cdot y}{z}}\]

    if 1.0103671538087413e+46 < z

    1. Initial program 19.8

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

    2. Simplified6.2

      \[\leadsto \color{blue}{y + \frac{x - x \cdot y}{z}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt6.3

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

    5. Applied *-un-lft-identity6.3

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

    6. Applied times-frac6.3

      \[\leadsto y + \color{blue}{\frac{1}{\sqrt{z}} \cdot \frac{x - x \cdot y}{\sqrt{z}}}\]
    7. Using strategy rm

    8. Applied div-sub6.3

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

    9. Simplified2.2

      \[\leadsto y + \frac{1}{\sqrt{z}} \cdot \left(\frac{x}{\sqrt{z}} - \color{blue}{\frac{x}{\frac{\sqrt{z}}{y}}}\right)\]
    10. Using strategy rm

    11. Applied div-inv2.2

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

    12. Applied div-inv2.2

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

    13. Applied distribute-lft-out--2.2

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

    14. Applied associate-*r*0.2

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

    15. Simplified0.2

      \[\leadsto y + \color{blue}{\frac{x}{\sqrt{z}}} \cdot \left(\frac{1}{\sqrt{z}} - \frac{1}{\frac{\sqrt{z}}{y}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 1.010367153808741264986405239084380831559 \cdot 10^{46}:\\ \;\;\;\;y + \frac{x - y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{z}} \cdot \left(\frac{1}{\sqrt{z}} - \frac{1}{\frac{\sqrt{z}}{y}}\right) + y\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"

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

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