Average Error: 12.5 → 1.6
Time: 9.9s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.53005333076934627 \cdot 10^{-4}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{elif}\;z \le 0.0968242458288034874:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.53005333076934627 \cdot 10^{-4}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\

\mathbf{elif}\;z \le 0.0968242458288034874:\\
\;\;\;\;\frac{x \cdot y}{z} + x\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{z}{y + z}}\\

\end{array}
double f(double x, double y, double z) {
        double r514845 = x;
        double r514846 = y;
        double r514847 = z;
        double r514848 = r514846 + r514847;
        double r514849 = r514845 * r514848;
        double r514850 = r514849 / r514847;
        return r514850;
}
↓
double f(double x, double y, double z) {
        double r514851 = z;
        double r514852 = -0.0008530053330769346;
        bool r514853 = r514851 <= r514852;
        double r514854 = x;
        double r514855 = y;
        double r514856 = r514855 / r514851;
        double r514857 = 1.0;
        double r514858 = r514856 + r514857;
        double r514859 = r514854 * r514858;
        double r514860 = 0.09682424582880349;
        bool r514861 = r514851 <= r514860;
        double r514862 = r514854 * r514855;
        double r514863 = r514862 / r514851;
        double r514864 = r514863 + r514854;
        double r514865 = r514855 + r514851;
        double r514866 = r514851 / r514865;
        double r514867 = r514854 / r514866;
        double r514868 = r514861 ? r514864 : r514867;
        double r514869 = r514853 ? r514859 : r514868;
        return r514869;
}

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

Original12.5
Target3.2
Herbie1.6
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -0.0008530053330769346

    1. Initial program 16.1

      \[\frac{x \cdot \left(y + z\right)}{z}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity16.1

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

    4. Applied times-frac0.1

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y + z}{z}}\]

    5. Simplified0.1

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

    6. Taylor expanded around 0 0.1

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

    if -0.0008530053330769346 < z < 0.09682424582880349

    1. Initial program 7.0

      \[\frac{x \cdot \left(y + z\right)}{z}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity7.0

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

    4. Applied times-frac8.0

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y + z}{z}}\]

    5. Simplified8.0

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

    6. Taylor expanded around 0 8.0

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

    8. Applied *-un-lft-identity8.0

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

    9. Applied associate-*l*8.0

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

    10. Simplified3.7

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

    if 0.09682424582880349 < z

    1. Initial program 17.1

      \[\frac{x \cdot \left(y + z\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y + z}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.53005333076934627 \cdot 10^{-4}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{elif}\;z \le 0.0968242458288034874:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

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

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