Average Error: 12.6 → 1.8
Time: 15.0s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.700485457152707062620180610805141877378 \cdot 10^{-101} \lor \neg \left(y \le 7.532388650323267223108156009419171807314 \cdot 10^{-81}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.700485457152707062620180610805141877378 \cdot 10^{-101} \lor \neg \left(y \le 7.532388650323267223108156009419171807314 \cdot 10^{-81}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r295277 = x;
        double r295278 = y;
        double r295279 = z;
        double r295280 = r295278 + r295279;
        double r295281 = r295277 * r295280;
        double r295282 = r295281 / r295279;
        return r295282;
}
↓
double f(double x, double y, double z) {
        double r295283 = y;
        double r295284 = -4.700485457152707e-101;
        bool r295285 = r295283 <= r295284;
        double r295286 = 7.532388650323267e-81;
        bool r295287 = r295283 <= r295286;
        double r295288 = !r295287;
        bool r295289 = r295285 || r295288;
        double r295290 = x;
        double r295291 = z;
        double r295292 = r295290 / r295291;
        double r295293 = fma(r295292, r295283, r295290);
        double r295294 = r295283 / r295291;
        double r295295 = fma(r295294, r295290, r295290);
        double r295296 = r295289 ? r295293 : r295295;
        return r295296;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.6
Target3.2
Herbie1.8
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.700485457152707e-101 or 7.532388650323267e-81 < y

    1. Initial program 11.6

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

    2. Simplified5.8

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

    3. Taylor expanded around 0 6.6

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

    4. Simplified3.1

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

    if -4.700485457152707e-101 < y < 7.532388650323267e-81

    1. Initial program 14.0

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

    2. Simplified0.0

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.700485457152707062620180610805141877378 \cdot 10^{-101} \lor \neg \left(y \le 7.532388650323267223108156009419171807314 \cdot 10^{-81}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))