Average Error: 12.4 → 2.0
Time: 1.8s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.53410616126814706 \cdot 10^{89} \lor \neg \left(x \le 1.3615881164524897 \cdot 10^{-138}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.53410616126814706 \cdot 10^{89} \lor \neg \left(x \le 1.3615881164524897 \cdot 10^{-138}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\

\end{array}
double f(double x, double y, double z) {
        double r431521 = x;
        double r431522 = y;
        double r431523 = z;
        double r431524 = r431522 + r431523;
        double r431525 = r431521 * r431524;
        double r431526 = r431525 / r431523;
        return r431526;
}
↓
double f(double x, double y, double z) {
        double r431527 = x;
        double r431528 = -6.534106161268147e+89;
        bool r431529 = r431527 <= r431528;
        double r431530 = 1.3615881164524897e-138;
        bool r431531 = r431527 <= r431530;
        double r431532 = !r431531;
        bool r431533 = r431529 || r431532;
        double r431534 = y;
        double r431535 = z;
        double r431536 = r431534 / r431535;
        double r431537 = fma(r431536, r431527, r431527);
        double r431538 = r431527 / r431535;
        double r431539 = fma(r431538, r431534, r431527);
        double r431540 = r431533 ? r431537 : r431539;
        return r431540;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.4
Target3.1
Herbie2.0
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6.534106161268147e+89 or 1.3615881164524897e-138 < x

    1. Initial program 19.1

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

    2. Simplified0.9

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

    if -6.534106161268147e+89 < x < 1.3615881164524897e-138

    1. Initial program 6.3

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

    2. Taylor expanded around 0 3.3

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

    3. Simplified3.0

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.53410616126814706 \cdot 10^{89} \lor \neg \left(x \le 1.3615881164524897 \cdot 10^{-138}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))