Average Error: 12.5 → 1.6
Time: 10.4s
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 r717374 = x;
        double r717375 = y;
        double r717376 = z;
        double r717377 = r717375 + r717376;
        double r717378 = r717374 * r717377;
        double r717379 = r717378 / r717376;
        return r717379;
}
↓
double f(double x, double y, double z) {
        double r717380 = z;
        double r717381 = -0.0008530053330769346;
        bool r717382 = r717380 <= r717381;
        double r717383 = x;
        double r717384 = y;
        double r717385 = r717384 / r717380;
        double r717386 = 1.0;
        double r717387 = r717385 + r717386;
        double r717388 = r717383 * r717387;
        double r717389 = 0.09682424582880349;
        bool r717390 = r717380 <= r717389;
        double r717391 = r717383 * r717384;
        double r717392 = r717391 / r717380;
        double r717393 = r717392 + r717383;
        double r717394 = r717384 + r717380;
        double r717395 = r717380 / r717394;
        double r717396 = r717383 / r717395;
        double r717397 = r717390 ? r717393 : r717396;
        double r717398 = r717382 ? r717388 : r717397;
        return r717398;
}

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))