Average Error: 12.3 → 1.8
Time: 2.2s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.12348942251738224 \cdot 10^{95}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.28734637678198 \cdot 10^{305}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.12348942251738224 \cdot 10^{95}:\\
\;\;\;\;\frac{x}{\frac{z}{y + z}}\\

\mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.28734637678198 \cdot 10^{305}:\\
\;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r426137 = x;
        double r426138 = y;
        double r426139 = z;
        double r426140 = r426138 + r426139;
        double r426141 = r426137 * r426140;
        double r426142 = r426141 / r426139;
        return r426142;
}
↓
double f(double x, double y, double z) {
        double r426143 = x;
        double r426144 = y;
        double r426145 = z;
        double r426146 = r426144 + r426145;
        double r426147 = r426143 * r426146;
        double r426148 = r426147 / r426145;
        double r426149 = 1.1234894225173822e+95;
        bool r426150 = r426148 <= r426149;
        double r426151 = r426145 / r426146;
        double r426152 = r426143 / r426151;
        double r426153 = 2.2873463767819766e+305;
        bool r426154 = r426148 <= r426153;
        double r426155 = r426143 / r426145;
        double r426156 = fma(r426155, r426144, r426143);
        double r426157 = r426154 ? r426148 : r426156;
        double r426158 = r426150 ? r426152 : r426157;
        return r426158;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.3
Target2.8
Herbie1.8
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* x (+ y z)) z) < 1.1234894225173822e+95

    1. Initial program 9.4

      \[\frac{x \cdot \left(y + z\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*2.2

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

    if 1.1234894225173822e+95 < (/ (* x (+ y z)) z) < 2.2873463767819766e+305

    1. Initial program 0.2

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

    if 2.2873463767819766e+305 < (/ (* x (+ y z)) z)

    1. Initial program 62.6

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

    2. Taylor expanded around 0 20.2

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

    3. Simplified0.6

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.12348942251738224 \cdot 10^{95}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.28734637678198 \cdot 10^{305}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))