Average Error: 22.2 → 0.2
Time: 7.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -104628043.80156818 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\frac{1}{y} + \left(x - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -104628043.80156818 \lor \neg \left(y \le 212488777.898407\right):\\
\;\;\;\;\frac{1}{y} + \left(x - 1 \cdot \frac{x}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r1428088 = 1.0;
        double r1428089 = x;
        double r1428090 = r1428088 - r1428089;
        double r1428091 = y;
        double r1428092 = r1428090 * r1428091;
        double r1428093 = r1428091 + r1428088;
        double r1428094 = r1428092 / r1428093;
        double r1428095 = r1428088 - r1428094;
        return r1428095;
}
↓
double f(double x, double y) {
        double r1428096 = y;
        double r1428097 = -104628043.80156818;
        bool r1428098 = r1428096 <= r1428097;
        double r1428099 = 212488777.89840698;
        bool r1428100 = r1428096 <= r1428099;
        double r1428101 = !r1428100;
        bool r1428102 = r1428098 || r1428101;
        double r1428103 = 1.0;
        double r1428104 = r1428103 / r1428096;
        double r1428105 = x;
        double r1428106 = r1428105 / r1428096;
        double r1428107 = r1428103 * r1428106;
        double r1428108 = r1428105 - r1428107;
        double r1428109 = r1428104 + r1428108;
        double r1428110 = r1428103 - r1428105;
        double r1428111 = r1428096 + r1428103;
        double r1428112 = r1428096 / r1428111;
        double r1428113 = r1428110 * r1428112;
        double r1428114 = r1428103 - r1428113;
        double r1428115 = r1428102 ? r1428109 : r1428114;
        return r1428115;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.2
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -104628043.80156818 or 212488777.89840698 < y

    1. Initial program 45.6

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -104628043.80156818 < y < 212488777.89840698

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -104628043.80156818 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\frac{1}{y} + \left(x - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))