Average Error: 22.6 → 0.2
Time: 12.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -124109135.66881044209003448486328125 \lor \neg \left(y \le 228723108.999044954776763916015625\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -124109135.66881044209003448486328125 \lor \neg \left(y \le 228723108.999044954776763916015625\right):\\
\;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\

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

\end{array}
double f(double x, double y) {
        double r511728 = 1.0;
        double r511729 = x;
        double r511730 = r511728 - r511729;
        double r511731 = y;
        double r511732 = r511730 * r511731;
        double r511733 = r511731 + r511728;
        double r511734 = r511732 / r511733;
        double r511735 = r511728 - r511734;
        return r511735;
}
↓
double f(double x, double y) {
        double r511736 = y;
        double r511737 = -124109135.66881044;
        bool r511738 = r511736 <= r511737;
        double r511739 = 228723108.99904495;
        bool r511740 = r511736 <= r511739;
        double r511741 = !r511740;
        bool r511742 = r511738 || r511741;
        double r511743 = x;
        double r511744 = 1.0;
        double r511745 = r511744 / r511736;
        double r511746 = r511743 + r511745;
        double r511747 = r511743 / r511736;
        double r511748 = r511744 * r511747;
        double r511749 = r511746 - r511748;
        double r511750 = r511744 - r511743;
        double r511751 = r511736 + r511744;
        double r511752 = r511736 / r511751;
        double r511753 = r511750 * r511752;
        double r511754 = r511744 - r511753;
        double r511755 = r511742 ? r511749 : r511754;
        return r511755;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.6
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -124109135.66881044 or 228723108.99904495 < y

    1. Initial program 45.6

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

    3. Applied *-un-lft-identity45.6

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

    4. Applied times-frac28.7

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

    5. Simplified28.7

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

    6. Taylor expanded around inf 0.2

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

    7. Simplified0.2

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

    if -124109135.66881044 < y < 228723108.99904495

    1. Initial program 0.2

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

    3. Applied *-un-lft-identity0.2

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

    4. Applied times-frac0.2

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

    5. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -124109135.66881044209003448486328125 \lor \neg \left(y \le 228723108.999044954776763916015625\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.84827882972468) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891003) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))