Average Error: 21.7 → 0.2
Time: 18.7s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -177179944.066863715648651123046875 \lor \neg \left(y \le 283365302.9953787326812744140625\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -177179944.066863715648651123046875 \lor \neg \left(y \le 283365302.9953787326812744140625\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 r470590 = 1.0;
        double r470591 = x;
        double r470592 = r470590 - r470591;
        double r470593 = y;
        double r470594 = r470592 * r470593;
        double r470595 = r470593 + r470590;
        double r470596 = r470594 / r470595;
        double r470597 = r470590 - r470596;
        return r470597;
}
↓
double f(double x, double y) {
        double r470598 = y;
        double r470599 = -177179944.06686372;
        bool r470600 = r470598 <= r470599;
        double r470601 = 283365302.99537873;
        bool r470602 = r470598 <= r470601;
        double r470603 = !r470602;
        bool r470604 = r470600 || r470603;
        double r470605 = x;
        double r470606 = 1.0;
        double r470607 = r470606 / r470598;
        double r470608 = r470605 + r470607;
        double r470609 = r470605 / r470598;
        double r470610 = r470606 * r470609;
        double r470611 = r470608 - r470610;
        double r470612 = r470606 - r470605;
        double r470613 = r470598 + r470606;
        double r470614 = r470598 / r470613;
        double r470615 = r470612 * r470614;
        double r470616 = r470606 - r470615;
        double r470617 = r470604 ? r470611 : r470616;
        return r470617;
}

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 < -177179944.06686372 or 283365302.99537873 < y

    1. Initial program 44.7

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

    3. Applied *-un-lft-identity44.7

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

    4. Applied times-frac28.5

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

    5. Simplified28.5

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

    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 -177179944.06686372 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -177179944.066863715648651123046875 \lor \neg \left(y \le 283365302.9953787326812744140625\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 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))))