Average Error: 12.9 → 1.7
Time: 21.1s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -9.980656591458488676125808816729354026739 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\

\mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -9.980656591458488676125808816729354026739 \cdot 10^{-68}:\\
\;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r70840832 = x;
        double r70840833 = y;
        double r70840834 = z;
        double r70840835 = r70840833 + r70840834;
        double r70840836 = r70840832 * r70840835;
        double r70840837 = r70840836 / r70840834;
        return r70840837;
}
↓
double f(double x, double y, double z) {
        double r70840838 = x;
        double r70840839 = y;
        double r70840840 = z;
        double r70840841 = r70840839 + r70840840;
        double r70840842 = r70840838 * r70840841;
        double r70840843 = r70840842 / r70840840;
        double r70840844 = -inf.0;
        bool r70840845 = r70840843 <= r70840844;
        double r70840846 = r70840838 / r70840840;
        double r70840847 = fma(r70840839, r70840846, r70840838);
        double r70840848 = -9.980656591458489e-68;
        bool r70840849 = r70840843 <= r70840848;
        double r70840850 = r70840841 / r70840840;
        double r70840851 = r70840838 * r70840850;
        double r70840852 = r70840849 ? r70840843 : r70840851;
        double r70840853 = r70840845 ? r70840847 : r70840852;
        return r70840853;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.9
Target2.9
Herbie1.7
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* x (+ y z)) z) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{z}, x\right)}\]

    if -inf.0 < (/ (* x (+ y z)) z) < -9.980656591458489e-68

    1. Initial program 0.2

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

    if -9.980656591458489e-68 < (/ (* x (+ y z)) z)

    1. Initial program 12.0

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

    3. Applied *-un-lft-identity12.0

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

    4. Applied times-frac2.5

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

    5. Simplified2.5

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -9.980656591458488676125808816729354026739 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(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))