Average Error: 12.1 → 1.8
Time: 9.2s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.198736726510580317627942676532283275075 \cdot 10^{73}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;x \le 4.322929192599831483537001033408001044383 \cdot 10^{-59}:\\ \;\;\;\;x + \frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y + z}{z}}{\frac{1}{x}}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.198736726510580317627942676532283275075 \cdot 10^{73}:\\
\;\;\;\;x \cdot \frac{y + z}{z}\\

\mathbf{elif}\;x \le 4.322929192599831483537001033408001044383 \cdot 10^{-59}:\\
\;\;\;\;x + \frac{x}{z} \cdot y\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y + z}{z}}{\frac{1}{x}}\\

\end{array}
double f(double x, double y, double z) {
        double r303228 = x;
        double r303229 = y;
        double r303230 = z;
        double r303231 = r303229 + r303230;
        double r303232 = r303228 * r303231;
        double r303233 = r303232 / r303230;
        return r303233;
}
↓
double f(double x, double y, double z) {
        double r303234 = x;
        double r303235 = -4.1987367265105803e+73;
        bool r303236 = r303234 <= r303235;
        double r303237 = y;
        double r303238 = z;
        double r303239 = r303237 + r303238;
        double r303240 = r303239 / r303238;
        double r303241 = r303234 * r303240;
        double r303242 = 4.3229291925998315e-59;
        bool r303243 = r303234 <= r303242;
        double r303244 = r303234 / r303238;
        double r303245 = r303244 * r303237;
        double r303246 = r303234 + r303245;
        double r303247 = 1.0;
        double r303248 = r303247 / r303234;
        double r303249 = r303240 / r303248;
        double r303250 = r303243 ? r303246 : r303249;
        double r303251 = r303236 ? r303241 : r303250;
        return r303251;
}

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

Original12.1
Target3.1
Herbie1.8
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.1987367265105803e+73

    1. Initial program 28.4

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

    2. Simplified8.6

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

    4. Applied div-inv8.8

      \[\leadsto \frac{y + z}{\color{blue}{z \cdot \frac{1}{x}}}\]

    5. Applied associate-/r*0.2

      \[\leadsto \color{blue}{\frac{\frac{y + z}{z}}{\frac{1}{x}}}\]
    6. Using strategy rm

    7. Applied div-inv0.2

      \[\leadsto \color{blue}{\frac{y + z}{z} \cdot \frac{1}{\frac{1}{x}}}\]

    8. Simplified0.1

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

    if -4.1987367265105803e+73 < x < 4.3229291925998315e-59

    1. Initial program 5.0

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

    2. Simplified15.4

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

    3. Taylor expanded around 0 2.6

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

    4. Simplified2.8

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

    if 4.3229291925998315e-59 < x

    1. Initial program 18.7

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

    2. Simplified6.7

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

    4. Applied div-inv6.8

      \[\leadsto \frac{y + z}{\color{blue}{z \cdot \frac{1}{x}}}\]

    5. Applied associate-/r*0.6

      \[\leadsto \color{blue}{\frac{\frac{y + z}{z}}{\frac{1}{x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.198736726510580317627942676532283275075 \cdot 10^{73}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;x \le 4.322929192599831483537001033408001044383 \cdot 10^{-59}:\\ \;\;\;\;x + \frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y + z}{z}}{\frac{1}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))