Average Error: 12.4 → 3.1
Time: 1.9s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.0872601480684541 \cdot 10^{-218} \lor \neg \left(x \le 1.24228458220251889 \cdot 10^{-142}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le 3.0872601480684541 \cdot 10^{-218} \lor \neg \left(x \le 1.24228458220251889 \cdot 10^{-142}\right):\\
\;\;\;\;x \cdot \frac{y + z}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r513309 = x;
        double r513310 = y;
        double r513311 = z;
        double r513312 = r513310 + r513311;
        double r513313 = r513309 * r513312;
        double r513314 = r513313 / r513311;
        return r513314;
}
↓
double f(double x, double y, double z) {
        double r513315 = x;
        double r513316 = 3.087260148068454e-218;
        bool r513317 = r513315 <= r513316;
        double r513318 = 1.2422845822025189e-142;
        bool r513319 = r513315 <= r513318;
        double r513320 = !r513319;
        bool r513321 = r513317 || r513320;
        double r513322 = y;
        double r513323 = z;
        double r513324 = r513322 + r513323;
        double r513325 = r513324 / r513323;
        double r513326 = r513315 * r513325;
        double r513327 = r513315 * r513322;
        double r513328 = 1.0;
        double r513329 = r513328 / r513323;
        double r513330 = r513327 * r513329;
        double r513331 = r513330 + r513315;
        double r513332 = r513321 ? r513326 : r513331;
        return r513332;
}

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

Original12.4
Target3.1
Herbie3.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 3.087260148068454e-218 or 1.2422845822025189e-142 < x

    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-frac3.1

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

    5. Simplified3.1

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

    if 3.087260148068454e-218 < x < 1.2422845822025189e-142

    1. Initial program 6.2

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

    2. Taylor expanded around 0 3.7

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

    4. Applied div-inv3.7

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.0872601480684541 \cdot 10^{-218} \lor \neg \left(x \le 1.24228458220251889 \cdot 10^{-142}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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))