Average Error: 22.4 → 0.4
Time: 16.8s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8762655710739029794226176:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{\frac{y}{1}}\\ \mathbf{elif}\;y \le 361767606.440016448497772216796875:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{\frac{y}{1}}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -8762655710739029794226176:\\
\;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{\frac{y}{1}}\\

\mathbf{elif}\;y \le 361767606.440016448497772216796875:\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\

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

\end{array}
double f(double x, double y) {
        double r30452414 = 1.0;
        double r30452415 = x;
        double r30452416 = r30452414 - r30452415;
        double r30452417 = y;
        double r30452418 = r30452416 * r30452417;
        double r30452419 = r30452417 + r30452414;
        double r30452420 = r30452418 / r30452419;
        double r30452421 = r30452414 - r30452420;
        return r30452421;
}
↓
double f(double x, double y) {
        double r30452422 = y;
        double r30452423 = -8.76265571073903e+24;
        bool r30452424 = r30452422 <= r30452423;
        double r30452425 = x;
        double r30452426 = 1.0;
        double r30452427 = r30452426 / r30452422;
        double r30452428 = r30452425 + r30452427;
        double r30452429 = r30452422 / r30452426;
        double r30452430 = r30452425 / r30452429;
        double r30452431 = r30452428 - r30452430;
        double r30452432 = 361767606.44001645;
        bool r30452433 = r30452422 <= r30452432;
        double r30452434 = r30452426 - r30452425;
        double r30452435 = r30452426 + r30452422;
        double r30452436 = r30452422 / r30452435;
        double r30452437 = r30452434 * r30452436;
        double r30452438 = r30452426 - r30452437;
        double r30452439 = r30452433 ? r30452438 : r30452431;
        double r30452440 = r30452424 ? r30452431 : r30452439;
        return r30452440;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.4
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -8.76265571073903e+24 or 361767606.44001645 < y

    1. Initial program 46.3

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

    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

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

    if -8.76265571073903e+24 < y < 361767606.44001645

    1. Initial program 0.8

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

    3. Applied *-un-lft-identity0.8

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

    4. Applied times-frac0.7

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

    5. Simplified0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8762655710739029794226176:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{\frac{y}{1}}\\ \mathbf{elif}\;y \le 361767606.440016448497772216796875:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{\frac{y}{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))