Average Error: 22.7 → 0.2
Time: 15.9s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -234223145.056681096553802490234375 \lor \neg \left(y \le 186709514.105293214321136474609375\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x \cdot 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(\frac{1}{y + 1} \cdot y\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -234223145.056681096553802490234375 \lor \neg \left(y \le 186709514.105293214321136474609375\right):\\
\;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x \cdot 1}{y}\\

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

\end{array}
double f(double x, double y) {
        double r596808 = 1.0;
        double r596809 = x;
        double r596810 = r596808 - r596809;
        double r596811 = y;
        double r596812 = r596810 * r596811;
        double r596813 = r596811 + r596808;
        double r596814 = r596812 / r596813;
        double r596815 = r596808 - r596814;
        return r596815;
}
↓
double f(double x, double y) {
        double r596816 = y;
        double r596817 = -234223145.0566811;
        bool r596818 = r596816 <= r596817;
        double r596819 = 186709514.1052932;
        bool r596820 = r596816 <= r596819;
        double r596821 = !r596820;
        bool r596822 = r596818 || r596821;
        double r596823 = x;
        double r596824 = 1.0;
        double r596825 = r596824 / r596816;
        double r596826 = r596823 + r596825;
        double r596827 = r596823 * r596824;
        double r596828 = r596827 / r596816;
        double r596829 = r596826 - r596828;
        double r596830 = r596824 - r596823;
        double r596831 = 1.0;
        double r596832 = r596816 + r596824;
        double r596833 = r596831 / r596832;
        double r596834 = r596833 * r596816;
        double r596835 = r596830 * r596834;
        double r596836 = r596824 - r596835;
        double r596837 = r596822 ? r596829 : r596836;
        return r596837;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.7
Target0.2
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 < -234223145.0566811 or 186709514.1052932 < y

    1. Initial program 46.3

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

    2. Simplified29.5

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

    4. Applied flip--45.8

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

    5. Simplified45.8

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

    6. Simplified45.8

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

    7. Taylor expanded around inf 0.1

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

    8. Simplified0.1

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

    if -234223145.0566811 < y < 186709514.1052932

    1. Initial program 0.2

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

    2. Simplified0.2

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

    4. Applied div-inv0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -234223145.056681096553802490234375 \lor \neg \left(y \le 186709514.105293214321136474609375\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x \cdot 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(\frac{1}{y + 1} \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))