Average Error: 21.8 → 0.2
Time: 4.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -111259902.92763877 \lor \neg \left(y \le 179306177.53728765\right):\\ \;\;\;\;\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -111259902.92763877 \lor \neg \left(y \le 179306177.53728765\right):\\
\;\;\;\;\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\

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

\end{array}
double f(double x, double y) {
        double r704773 = 1.0;
        double r704774 = x;
        double r704775 = r704773 - r704774;
        double r704776 = y;
        double r704777 = r704775 * r704776;
        double r704778 = r704776 + r704773;
        double r704779 = r704777 / r704778;
        double r704780 = r704773 - r704779;
        return r704780;
}
↓
double f(double x, double y) {
        double r704781 = y;
        double r704782 = -111259902.92763877;
        bool r704783 = r704781 <= r704782;
        double r704784 = 179306177.53728765;
        bool r704785 = r704781 <= r704784;
        double r704786 = !r704785;
        bool r704787 = r704783 || r704786;
        double r704788 = x;
        double r704789 = 1.0;
        double r704790 = 1.0;
        double r704791 = r704790 / r704781;
        double r704792 = r704789 * r704791;
        double r704793 = r704788 + r704792;
        double r704794 = r704788 / r704781;
        double r704795 = r704789 * r704794;
        double r704796 = r704793 - r704795;
        double r704797 = r704789 - r704788;
        double r704798 = r704797 * r704781;
        double r704799 = r704781 * r704781;
        double r704800 = r704789 * r704789;
        double r704801 = r704799 - r704800;
        double r704802 = r704798 / r704801;
        double r704803 = r704781 - r704789;
        double r704804 = r704802 * r704803;
        double r704805 = r704789 - r704804;
        double r704806 = r704787 ? r704796 : r704805;
        return r704806;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.8
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 < -111259902.92763877 or 179306177.53728765 < y

    1. Initial program 45.6

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

    3. Applied *-un-lft-identity45.6

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

    4. Applied times-frac29.1

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

    5. Simplified29.1

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

    7. Applied flip3--50.6

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

    8. Simplified50.6

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

    9. Taylor expanded around inf 0.1

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

    if -111259902.92763877 < y < 179306177.53728765

    1. Initial program 0.2

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

    3. Applied flip-+0.2

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

    4. Applied associate-/r/0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -111259902.92763877 \lor \neg \left(y \le 179306177.53728765\right):\\ \;\;\;\;\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))))