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 r424739 = x;
        double r424740 = y;
        double r424741 = z;
        double r424742 = r424740 + r424741;
        double r424743 = r424739 * r424742;
        double r424744 = r424743 / r424741;
        return r424744;
}
↓
double f(double x, double y, double z) {
        double r424745 = x;
        double r424746 = -3.9747265790110112e+87;
        bool r424747 = r424745 <= r424746;
        double r424748 = 4.411147860766137e-11;
        bool r424749 = r424745 <= r424748;
        double r424750 = !r424749;
        bool r424751 = r424747 || r424750;
        double r424752 = y;
        double r424753 = z;
        double r424754 = r424752 / r424753;
        double r424755 = fma(r424754, r424745, r424745);
        double r424756 = r424745 / r424753;
        double r424757 = fma(r424756, r424752, r424745);
        double r424758 = r424751 ? r424755 : r424757;
        return r424758;
}

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))