Average Error: 12.7 → 1.9
Time: 2.1s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.97472657901101124 \cdot 10^{87} \lor \neg \left(x \le 4.411147860766137 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.97472657901101124 \cdot 10^{87} \lor \neg \left(x \le 4.411147860766137 \cdot 10^{-11}\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 r343638 = x;
        double r343639 = y;
        double r343640 = z;
        double r343641 = r343639 + r343640;
        double r343642 = r343638 * r343641;
        double r343643 = r343642 / r343640;
        return r343643;
}
↓
double f(double x, double y, double z) {
        double r343644 = x;
        double r343645 = -3.9747265790110112e+87;
        bool r343646 = r343644 <= r343645;
        double r343647 = 4.411147860766137e-11;
        bool r343648 = r343644 <= r343647;
        double r343649 = !r343648;
        bool r343650 = r343646 || r343649;
        double r343651 = y;
        double r343652 = z;
        double r343653 = r343651 / r343652;
        double r343654 = fma(r343653, r343644, r343644);
        double r343655 = r343644 / r343652;
        double r343656 = fma(r343655, r343651, r343644);
        double r343657 = r343650 ? r343654 : r343656;
        return r343657;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.7
Target3.2
Herbie1.9
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.9747265790110112e+87 or 4.411147860766137e-11 < x

    1. Initial program 25.3

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

    2. Simplified0.1

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

    if -3.9747265790110112e+87 < x < 4.411147860766137e-11

    1. Initial program 5.6

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

    2. Taylor expanded around 0 2.9

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

    3. Simplified2.9

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.97472657901101124 \cdot 10^{87} \lor \neg \left(x \le 4.411147860766137 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))