Average Error: 12.1 → 2.3
Time: 7.6s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -9.08005395530494447 \cdot 10^{51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -9.08005395530494447 \cdot 10^{51}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r435546 = x;
        double r435547 = y;
        double r435548 = z;
        double r435549 = r435547 + r435548;
        double r435550 = r435546 * r435549;
        double r435551 = r435550 / r435548;
        return r435551;
}
↓
double f(double x, double y, double z) {
        double r435552 = y;
        double r435553 = -9.080053955304944e+51;
        bool r435554 = r435552 <= r435553;
        double r435555 = x;
        double r435556 = z;
        double r435557 = r435555 / r435556;
        double r435558 = fma(r435552, r435557, r435555);
        double r435559 = r435556 / r435552;
        double r435560 = r435555 / r435559;
        double r435561 = r435560 + r435555;
        double r435562 = r435554 ? r435558 : r435561;
        return r435562;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.1
Target3.1
Herbie2.3
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -9.080053955304944e+51

    1. Initial program 11.9

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

    2. Simplified3.7

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

    if -9.080053955304944e+51 < y

    1. Initial program 12.1

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

    2. Simplified5.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{z}, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef5.0

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

    5. Simplified4.1

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

    7. Applied associate-/l*2.0

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.08005395530494447 \cdot 10^{51}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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))