Average Error: 12.5 → 3.1
Time: 9.6s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.70668106132631082765348526669112469905 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \mathbf{elif}\;z \le 1.558938646687399852113316757428115787716 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(z + y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + y}{z} \cdot x\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.70668106132631082765348526669112469905 \cdot 10^{-75}:\\
\;\;\;\;\frac{x}{\frac{z}{z + y}}\\

\mathbf{elif}\;z \le 1.558938646687399852113316757428115787716 \cdot 10^{-151}:\\
\;\;\;\;\frac{1}{\frac{z}{x \cdot \left(z + y\right)}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r21395058 = x;
        double r21395059 = y;
        double r21395060 = z;
        double r21395061 = r21395059 + r21395060;
        double r21395062 = r21395058 * r21395061;
        double r21395063 = r21395062 / r21395060;
        return r21395063;
}
↓
double f(double x, double y, double z) {
        double r21395064 = z;
        double r21395065 = -1.7066810613263108e-75;
        bool r21395066 = r21395064 <= r21395065;
        double r21395067 = x;
        double r21395068 = y;
        double r21395069 = r21395064 + r21395068;
        double r21395070 = r21395064 / r21395069;
        double r21395071 = r21395067 / r21395070;
        double r21395072 = 1.5589386466874e-151;
        bool r21395073 = r21395064 <= r21395072;
        double r21395074 = 1.0;
        double r21395075 = r21395067 * r21395069;
        double r21395076 = r21395064 / r21395075;
        double r21395077 = r21395074 / r21395076;
        double r21395078 = r21395069 / r21395064;
        double r21395079 = r21395078 * r21395067;
        double r21395080 = r21395073 ? r21395077 : r21395079;
        double r21395081 = r21395066 ? r21395071 : r21395080;
        return r21395081;
}

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
Target2.9
Herbie3.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.7066810613263108e-75

    1. Initial program 14.1

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

    3. Applied associate-/l*0.4

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

    if -1.7066810613263108e-75 < z < 1.5589386466874e-151

    1. Initial program 9.9

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

    3. Applied clear-num9.9

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

    if 1.5589386466874e-151 < z

    1. Initial program 12.8

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

    3. Applied *-un-lft-identity12.8

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

    4. Applied times-frac1.3

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

    5. Simplified1.3

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.70668106132631082765348526669112469905 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \mathbf{elif}\;z \le 1.558938646687399852113316757428115787716 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(z + y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + y}{z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"

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

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