Average Error: 22.6 → 0.3
Time: 3.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -0.99061126335873073 \lor \neg \left(y \le 77921780.708448023\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\left(1 - x\right) \cdot y}{\sqrt{y + 1}}}{\sqrt{y + 1}}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -0.99061126335873073 \lor \neg \left(y \le 77921780.708448023\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r790348 = 1.0;
        double r790349 = x;
        double r790350 = r790348 - r790349;
        double r790351 = y;
        double r790352 = r790350 * r790351;
        double r790353 = r790351 + r790348;
        double r790354 = r790352 / r790353;
        double r790355 = r790348 - r790354;
        return r790355;
}
↓
double f(double x, double y) {
        double r790356 = y;
        double r790357 = -0.9906112633587307;
        bool r790358 = r790356 <= r790357;
        double r790359 = 77921780.70844802;
        bool r790360 = r790356 <= r790359;
        double r790361 = !r790360;
        bool r790362 = r790358 || r790361;
        double r790363 = 1.0;
        double r790364 = 1.0;
        double r790365 = r790364 / r790356;
        double r790366 = x;
        double r790367 = r790366 / r790356;
        double r790368 = r790365 - r790367;
        double r790369 = r790363 * r790368;
        double r790370 = r790369 + r790366;
        double r790371 = r790363 - r790366;
        double r790372 = r790371 * r790356;
        double r790373 = r790356 + r790363;
        double r790374 = sqrt(r790373);
        double r790375 = r790372 / r790374;
        double r790376 = r790375 / r790374;
        double r790377 = r790363 - r790376;
        double r790378 = r790362 ? r790370 : r790377;
        return r790378;
}

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
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -0.9906112633587307 or 77921780.70844802 < y

    1. Initial program 45.4

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

    if -0.9906112633587307 < y < 77921780.70844802

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied associate-/r*0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -0.99061126335873073 \lor \neg \left(y \le 77921780.708448023\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\left(1 - x\right) \cdot y}{\sqrt{y + 1}}}{\sqrt{y + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))))