Average Error: 21.7 → 0.3
Time: 15.5s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6.66464768403957 \cdot 10^{+16}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\ \mathbf{elif}\;y \le 205630141.98138198:\\ \;\;\;\;1.0 - \left(y - 1.0\right) \cdot \left(\frac{y}{y + 1.0} \cdot \frac{1.0 - x}{y - 1.0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -6.66464768403957 \cdot 10^{+16}:\\
\;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\

\mathbf{elif}\;y \le 205630141.98138198:\\
\;\;\;\;1.0 - \left(y - 1.0\right) \cdot \left(\frac{y}{y + 1.0} \cdot \frac{1.0 - x}{y - 1.0}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\

\end{array}
double f(double x, double y) {
        double r35504675 = 1.0;
        double r35504676 = x;
        double r35504677 = r35504675 - r35504676;
        double r35504678 = y;
        double r35504679 = r35504677 * r35504678;
        double r35504680 = r35504678 + r35504675;
        double r35504681 = r35504679 / r35504680;
        double r35504682 = r35504675 - r35504681;
        return r35504682;
}
↓
double f(double x, double y) {
        double r35504683 = y;
        double r35504684 = -6.66464768403957e+16;
        bool r35504685 = r35504683 <= r35504684;
        double r35504686 = x;
        double r35504687 = 1.0;
        double r35504688 = r35504687 / r35504683;
        double r35504689 = r35504686 + r35504688;
        double r35504690 = r35504683 / r35504687;
        double r35504691 = r35504686 / r35504690;
        double r35504692 = r35504689 - r35504691;
        double r35504693 = 205630141.98138198;
        bool r35504694 = r35504683 <= r35504693;
        double r35504695 = r35504683 - r35504687;
        double r35504696 = r35504683 + r35504687;
        double r35504697 = r35504683 / r35504696;
        double r35504698 = r35504687 - r35504686;
        double r35504699 = r35504698 / r35504695;
        double r35504700 = r35504697 * r35504699;
        double r35504701 = r35504695 * r35504700;
        double r35504702 = r35504687 - r35504701;
        double r35504703 = r35504694 ? r35504702 : r35504692;
        double r35504704 = r35504685 ? r35504692 : r35504703;
        return r35504704;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.7
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -6.66464768403957e+16 or 205630141.98138198 < y

    1. Initial program 45.7

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity45.7

      \[\leadsto 1.0 - \frac{\left(1.0 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1.0\right)}}\]

    4. Applied times-frac30.5

      \[\leadsto 1.0 - \color{blue}{\frac{1.0 - x}{1} \cdot \frac{y}{y + 1.0}}\]

    5. Simplified30.5

      \[\leadsto 1.0 - \color{blue}{\left(1.0 - x\right)} \cdot \frac{y}{y + 1.0}\]
    6. Using strategy rm

    7. Applied flip--45.2

      \[\leadsto \color{blue}{\frac{1.0 \cdot 1.0 - \left(\left(1.0 - x\right) \cdot \frac{y}{y + 1.0}\right) \cdot \left(\left(1.0 - x\right) \cdot \frac{y}{y + 1.0}\right)}{1.0 + \left(1.0 - x\right) \cdot \frac{y}{y + 1.0}}}\]

    8. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(x + 1.0 \cdot \frac{1}{y}\right) - 1.0 \cdot \frac{x}{y}}\]

    9. Simplified0.1

      \[\leadsto \color{blue}{\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}}\]

    if -6.66464768403957e+16 < y < 205630141.98138198

    1. Initial program 0.5

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.5

      \[\leadsto 1.0 - \frac{\left(1.0 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1.0\right)}}\]

    4. Applied times-frac0.4

      \[\leadsto 1.0 - \color{blue}{\frac{1.0 - x}{1} \cdot \frac{y}{y + 1.0}}\]

    5. Simplified0.4

      \[\leadsto 1.0 - \color{blue}{\left(1.0 - x\right)} \cdot \frac{y}{y + 1.0}\]
    6. Using strategy rm

    7. Applied flip-+0.4

      \[\leadsto 1.0 - \left(1.0 - x\right) \cdot \frac{y}{\color{blue}{\frac{y \cdot y - 1.0 \cdot 1.0}{y - 1.0}}}\]

    8. Applied associate-/r/0.4

      \[\leadsto 1.0 - \left(1.0 - x\right) \cdot \color{blue}{\left(\frac{y}{y \cdot y - 1.0 \cdot 1.0} \cdot \left(y - 1.0\right)\right)}\]

    9. Applied associate-*r*0.4

      \[\leadsto 1.0 - \color{blue}{\left(\left(1.0 - x\right) \cdot \frac{y}{y \cdot y - 1.0 \cdot 1.0}\right) \cdot \left(y - 1.0\right)}\]

    10. Simplified0.4

      \[\leadsto 1.0 - \color{blue}{\left(\frac{y}{1.0 + y} \cdot \frac{1.0 - x}{y - 1.0}\right)} \cdot \left(y - 1.0\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.66464768403957 \cdot 10^{+16}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\ \mathbf{elif}\;y \le 205630141.98138198:\\ \;\;\;\;1.0 - \left(y - 1.0\right) \cdot \left(\frac{y}{y + 1.0} \cdot \frac{1.0 - x}{y - 1.0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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))))