Average Error: 21.7 → 0.2
Time: 19.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -171525914.4309198558330535888671875 \lor \neg \left(y \le 283365302.9953787326812744140625\right):\\ \;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -171525914.4309198558330535888671875 \lor \neg \left(y \le 283365302.9953787326812744140625\right):\\
\;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r463704 = 1.0;
        double r463705 = x;
        double r463706 = r463704 - r463705;
        double r463707 = y;
        double r463708 = r463706 * r463707;
        double r463709 = r463707 + r463704;
        double r463710 = r463708 / r463709;
        double r463711 = r463704 - r463710;
        return r463711;
}
↓
double f(double x, double y) {
        double r463712 = y;
        double r463713 = -171525914.43091986;
        bool r463714 = r463712 <= r463713;
        double r463715 = 283365302.99537873;
        bool r463716 = r463712 <= r463715;
        double r463717 = !r463716;
        bool r463718 = r463714 || r463717;
        double r463719 = 1.0;
        double r463720 = r463719 / r463712;
        double r463721 = x;
        double r463722 = r463721 / r463712;
        double r463723 = r463719 * r463722;
        double r463724 = r463720 - r463723;
        double r463725 = r463724 + r463721;
        double r463726 = r463719 - r463721;
        double r463727 = r463712 + r463719;
        double r463728 = r463712 / r463727;
        double r463729 = r463726 * r463728;
        double r463730 = r463719 - r463729;
        double r463731 = r463718 ? r463725 : r463730;
        return r463731;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.7
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 < -171525914.43091986 or 283365302.99537873 < y

    1. Initial program 44.7

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    if -171525914.43091986 < y < 283365302.99537873

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -171525914.4309198558330535888671875 \lor \neg \left(y \le 283365302.9953787326812744140625\right):\\ \;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

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