Average Error: 12.6 → 1.7
Time: 12.6s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -492310718521397431238656 \lor \neg \left(y \le 3.163987042355706814221308641241888055374 \cdot 10^{85}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -492310718521397431238656 \lor \neg \left(y \le 3.163987042355706814221308641241888055374 \cdot 10^{85}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{z}{y}} + x\\

\end{array}
double f(double x, double y, double z) {
        double r280093 = x;
        double r280094 = y;
        double r280095 = z;
        double r280096 = r280094 + r280095;
        double r280097 = r280093 * r280096;
        double r280098 = r280097 / r280095;
        return r280098;
}
↓
double f(double x, double y, double z) {
        double r280099 = y;
        double r280100 = -4.923107185213974e+23;
        bool r280101 = r280099 <= r280100;
        double r280102 = 3.163987042355707e+85;
        bool r280103 = r280099 <= r280102;
        double r280104 = !r280103;
        bool r280105 = r280101 || r280104;
        double r280106 = x;
        double r280107 = z;
        double r280108 = r280106 / r280107;
        double r280109 = fma(r280108, r280099, r280106);
        double r280110 = r280107 / r280099;
        double r280111 = r280106 / r280110;
        double r280112 = r280111 + r280106;
        double r280113 = r280105 ? r280109 : r280112;
        return r280113;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.6
Target2.9
Herbie1.7
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.923107185213974e+23 or 3.163987042355707e+85 < y

    1. Initial program 12.4

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

    2. Simplified4.5

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

    if -4.923107185213974e+23 < y < 3.163987042355707e+85

    1. Initial program 12.7

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

    2. Simplified4.8

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

    4. Applied fma-udef4.8

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

    5. Simplified2.3

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x\]
    6. Using strategy rm

    7. Applied associate-/l*0.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -492310718521397431238656 \lor \neg \left(y \le 3.163987042355706814221308641241888055374 \cdot 10^{85}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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))