Average Error: 22.4 → 0.4
Time: 17.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8762655710739029794226176:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \mathbf{elif}\;y \le 361767606.440016448497772216796875:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -8762655710739029794226176:\\
\;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\

\mathbf{elif}\;y \le 361767606.440016448497772216796875:\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\

\end{array}
double f(double x, double y) {
        double r33548021 = 1.0;
        double r33548022 = x;
        double r33548023 = r33548021 - r33548022;
        double r33548024 = y;
        double r33548025 = r33548023 * r33548024;
        double r33548026 = r33548024 + r33548021;
        double r33548027 = r33548025 / r33548026;
        double r33548028 = r33548021 - r33548027;
        return r33548028;
}
↓
double f(double x, double y) {
        double r33548029 = y;
        double r33548030 = -8.76265571073903e+24;
        bool r33548031 = r33548029 <= r33548030;
        double r33548032 = 1.0;
        double r33548033 = r33548032 / r33548029;
        double r33548034 = x;
        double r33548035 = r33548029 / r33548034;
        double r33548036 = r33548032 / r33548035;
        double r33548037 = r33548036 - r33548034;
        double r33548038 = r33548033 - r33548037;
        double r33548039 = 361767606.44001645;
        bool r33548040 = r33548029 <= r33548039;
        double r33548041 = r33548032 - r33548034;
        double r33548042 = r33548032 + r33548029;
        double r33548043 = r33548029 / r33548042;
        double r33548044 = r33548041 * r33548043;
        double r33548045 = r33548032 - r33548044;
        double r33548046 = r33548040 ? r33548045 : r33548038;
        double r33548047 = r33548031 ? r33548038 : r33548046;
        return r33548047;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.4
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -8.76265571073903e+24 or 361767606.44001645 < y

    1. Initial program 46.3

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity46.3

      \[\leadsto 1 - \frac{\left(1 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1\right)}}\]

    4. Applied times-frac29.3

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

    5. Simplified29.3

      \[\leadsto 1 - \color{blue}{\left(1 - x\right)} \cdot \frac{y}{y + 1}\]

    6. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}}\]

    7. Simplified0.1

      \[\leadsto \color{blue}{\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)}\]

    if -8.76265571073903e+24 < y < 361767606.44001645

    1. Initial program 0.8

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.8

      \[\leadsto 1 - \frac{\left(1 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1\right)}}\]

    4. Applied times-frac0.7

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

    5. Simplified0.7

      \[\leadsto 1 - \color{blue}{\left(1 - x\right)} \cdot \frac{y}{y + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8762655710739029794226176:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \mathbf{elif}\;y \le 361767606.440016448497772216796875:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))