Average Error: 13.2 → 2.3
Time: 2.0s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.356674207855083259835513572804868041722 \cdot 10^{54} \lor \neg \left(z \le 8.945393456276617422560346465194992080086 \cdot 10^{-306}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.356674207855083259835513572804868041722 \cdot 10^{54} \lor \neg \left(z \le 8.945393456276617422560346465194992080086 \cdot 10^{-306}\right):\\
\;\;\;\;\frac{x}{\frac{z}{y + z}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r476232 = x;
        double r476233 = y;
        double r476234 = z;
        double r476235 = r476233 + r476234;
        double r476236 = r476232 * r476235;
        double r476237 = r476236 / r476234;
        return r476237;
}
↓
double f(double x, double y, double z) {
        double r476238 = z;
        double r476239 = -8.356674207855083e+54;
        bool r476240 = r476238 <= r476239;
        double r476241 = 8.945393456276617e-306;
        bool r476242 = r476238 <= r476241;
        double r476243 = !r476242;
        bool r476244 = r476240 || r476243;
        double r476245 = x;
        double r476246 = y;
        double r476247 = r476246 + r476238;
        double r476248 = r476238 / r476247;
        double r476249 = r476245 / r476248;
        double r476250 = r476245 * r476246;
        double r476251 = r476250 / r476238;
        double r476252 = r476251 + r476245;
        double r476253 = r476244 ? r476249 : r476252;
        return r476253;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.2
Target2.9
Herbie2.3
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -8.356674207855083e+54 or 8.945393456276617e-306 < z

    1. Initial program 15.6

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

    3. Applied associate-/l*1.9

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

    if -8.356674207855083e+54 < z < 8.945393456276617e-306

    1. Initial program 6.2

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

    2. Taylor expanded around 0 3.2

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.356674207855083259835513572804868041722 \cdot 10^{54} \lor \neg \left(z \le 8.945393456276617422560346465194992080086 \cdot 10^{-306}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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))