Average Error: 21.9 → 0.2
Time: 3.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -131869364.42794824 \lor \neg \left(y \le 45126711.214655101\right):\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -131869364.42794824 \lor \neg \left(y \le 45126711.214655101\right):\\
\;\;\;\;1 \cdot \left(1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\right)\\

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

\end{array}
double f(double x, double y) {
        double r729515 = 1.0;
        double r729516 = x;
        double r729517 = r729515 - r729516;
        double r729518 = y;
        double r729519 = r729517 * r729518;
        double r729520 = r729518 + r729515;
        double r729521 = r729519 / r729520;
        double r729522 = r729515 - r729521;
        return r729522;
}
↓
double f(double x, double y) {
        double r729523 = y;
        double r729524 = -131869364.42794824;
        bool r729525 = r729523 <= r729524;
        double r729526 = 45126711.2146551;
        bool r729527 = r729523 <= r729526;
        double r729528 = !r729527;
        bool r729529 = r729525 || r729528;
        double r729530 = 1.0;
        double r729531 = 1.0;
        double r729532 = r729530 / r729523;
        double r729533 = x;
        double r729534 = r729533 / r729523;
        double r729535 = r729532 - r729534;
        double r729536 = r729531 * r729535;
        double r729537 = r729536 + r729533;
        double r729538 = r729530 * r729537;
        double r729539 = r729531 - r729533;
        double r729540 = r729539 * r729523;
        double r729541 = r729523 + r729531;
        double r729542 = r729540 / r729541;
        double r729543 = r729531 - r729542;
        double r729544 = r729530 * r729543;
        double r729545 = r729529 ? r729538 : r729544;
        return r729545;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.9
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 < -131869364.42794824 or 45126711.2146551 < y

    1. Initial program 45.2

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

    3. Applied flip--48.9

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

    5. Applied flip-+50.2

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

    6. Applied associate-/r/50.2

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

    7. Simplified45.2

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

    8. Taylor expanded around inf 0.2

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

    9. Simplified0.2

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

    if -131869364.42794824 < y < 45126711.2146551

    1. Initial program 0.1

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

    3. Applied flip--4.3

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

    5. Applied flip-+4.3

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

    6. Applied associate-/r/4.3

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

    7. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -131869364.42794824 \lor \neg \left(y \le 45126711.214655101\right):\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))