Average Error: 12.4 → 1.7
Time: 17.3s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.77033482903865029679843273380425534796 \cdot 10^{-37} \lor \neg \left(x \le 3.269193493716526616235123635664496656065 \cdot 10^{-74}\right):\\ \;\;\;\;x \cdot \frac{y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.77033482903865029679843273380425534796 \cdot 10^{-37} \lor \neg \left(x \le 3.269193493716526616235123635664496656065 \cdot 10^{-74}\right):\\
\;\;\;\;x \cdot \frac{y}{z} + x\\

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

\end{array}
double f(double x, double y, double z) {
        double r250151 = x;
        double r250152 = y;
        double r250153 = z;
        double r250154 = r250152 + r250153;
        double r250155 = r250151 * r250154;
        double r250156 = r250155 / r250153;
        return r250156;
}
↓
double f(double x, double y, double z) {
        double r250157 = x;
        double r250158 = -8.77033482903865e-37;
        bool r250159 = r250157 <= r250158;
        double r250160 = 3.2691934937165266e-74;
        bool r250161 = r250157 <= r250160;
        double r250162 = !r250161;
        bool r250163 = r250159 || r250162;
        double r250164 = y;
        double r250165 = z;
        double r250166 = r250164 / r250165;
        double r250167 = r250157 * r250166;
        double r250168 = r250167 + r250157;
        double r250169 = r250157 / r250165;
        double r250170 = fma(r250169, r250164, r250157);
        double r250171 = r250163 ? r250168 : r250170;
        return r250171;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.4
Target3.1
Herbie1.7
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -8.77033482903865e-37 or 3.2691934937165266e-74 < x

    1. Initial program 18.3

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

    2. Simplified0.5

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

    4. Applied fma-udef0.5

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

    5. Simplified0.5

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

    if -8.77033482903865e-37 < x < 3.2691934937165266e-74

    1. Initial program 5.9

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

    2. Simplified6.6

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

    3. Taylor expanded around 0 3.4

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

    4. Simplified3.0

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.77033482903865029679843273380425534796 \cdot 10^{-37} \lor \neg \left(x \le 3.269193493716526616235123635664496656065 \cdot 10^{-74}\right):\\ \;\;\;\;x \cdot \frac{y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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))