Average Error: 12.7 → 1.7
Time: 1.8s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 5.46403839941028375 \cdot 10^{54}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.4690217933528748 \cdot 10^{307}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 5.46403839941028375 \cdot 10^{54}:\\
\;\;\;\;\frac{x}{\frac{z}{y + z}}\\

\mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.4690217933528748 \cdot 10^{307}:\\
\;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r440323 = x;
        double r440324 = y;
        double r440325 = z;
        double r440326 = r440324 + r440325;
        double r440327 = r440323 * r440326;
        double r440328 = r440327 / r440325;
        return r440328;
}
↓
double f(double x, double y, double z) {
        double r440329 = x;
        double r440330 = y;
        double r440331 = z;
        double r440332 = r440330 + r440331;
        double r440333 = r440329 * r440332;
        double r440334 = r440333 / r440331;
        double r440335 = 5.464038399410284e+54;
        bool r440336 = r440334 <= r440335;
        double r440337 = r440331 / r440332;
        double r440338 = r440329 / r440337;
        double r440339 = 2.469021793352875e+307;
        bool r440340 = r440334 <= r440339;
        double r440341 = r440330 / r440331;
        double r440342 = 1.0;
        double r440343 = r440341 + r440342;
        double r440344 = r440329 * r440343;
        double r440345 = r440340 ? r440334 : r440344;
        double r440346 = r440336 ? r440338 : r440345;
        return r440346;
}

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

Original12.7
Target3.2
Herbie1.7
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* x (+ y z)) z) < 5.464038399410284e+54

    1. Initial program 10.6

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

    3. Applied associate-/l*2.2

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

    if 5.464038399410284e+54 < (/ (* x (+ y z)) z) < 2.469021793352875e+307

    1. Initial program 0.2

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

    if 2.469021793352875e+307 < (/ (* x (+ y z)) z)

    1. Initial program 63.5

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

    3. Applied *-un-lft-identity63.5

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

    4. Applied times-frac0.4

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

    5. Simplified0.4

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

    6. Taylor expanded around 0 0.4

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 5.46403839941028375 \cdot 10^{54}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.4690217933528748 \cdot 10^{307}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))