Average Error: 22.0 → 7.5
Time: 23.5s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5186263861.88195:\\ \;\;\;\;x + \left(\frac{1.0}{y} - 1.0\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;y \le 5.169922070464494 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \frac{x - 1.0}{1.0 + y} + 1.0\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1.0}{y} - 1.0\right) \cdot \frac{x}{y}\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -5186263861.88195:\\
\;\;\;\;x + \left(\frac{1.0}{y} - 1.0\right) \cdot \frac{x}{y}\\

\mathbf{elif}\;y \le 5.169922070464494 \cdot 10^{+30}:\\
\;\;\;\;y \cdot \frac{x - 1.0}{1.0 + y} + 1.0\\

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

\end{array}
double f(double x, double y) {
        double r28199721 = 1.0;
        double r28199722 = x;
        double r28199723 = r28199721 - r28199722;
        double r28199724 = y;
        double r28199725 = r28199723 * r28199724;
        double r28199726 = r28199724 + r28199721;
        double r28199727 = r28199725 / r28199726;
        double r28199728 = r28199721 - r28199727;
        return r28199728;
}
↓
double f(double x, double y) {
        double r28199729 = y;
        double r28199730 = -5186263861.88195;
        bool r28199731 = r28199729 <= r28199730;
        double r28199732 = x;
        double r28199733 = 1.0;
        double r28199734 = r28199733 / r28199729;
        double r28199735 = r28199734 - r28199733;
        double r28199736 = r28199732 / r28199729;
        double r28199737 = r28199735 * r28199736;
        double r28199738 = r28199732 + r28199737;
        double r28199739 = 5.169922070464494e+30;
        bool r28199740 = r28199729 <= r28199739;
        double r28199741 = r28199732 - r28199733;
        double r28199742 = r28199733 + r28199729;
        double r28199743 = r28199741 / r28199742;
        double r28199744 = r28199729 * r28199743;
        double r28199745 = r28199744 + r28199733;
        double r28199746 = r28199740 ? r28199745 : r28199738;
        double r28199747 = r28199731 ? r28199738 : r28199746;
        return r28199747;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.0
Target0.2
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -5186263861.88195 or 5.169922070464494e+30 < y

    1. Initial program 45.3

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

    2. Simplified29.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x - 1.0}{1.0 + y}, 1.0\right)}\]
    3. Using strategy rm

    4. Applied fma-udef29.4

      \[\leadsto \color{blue}{y \cdot \frac{x - 1.0}{1.0 + y} + 1.0}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt30.1

      \[\leadsto y \cdot \frac{x - 1.0}{\color{blue}{\left(\sqrt[3]{1.0 + y} \cdot \sqrt[3]{1.0 + y}\right) \cdot \sqrt[3]{1.0 + y}}} + 1.0\]

    7. Applied *-un-lft-identity30.1

      \[\leadsto y \cdot \frac{\color{blue}{1 \cdot \left(x - 1.0\right)}}{\left(\sqrt[3]{1.0 + y} \cdot \sqrt[3]{1.0 + y}\right) \cdot \sqrt[3]{1.0 + y}} + 1.0\]

    8. Applied times-frac30.1

      \[\leadsto y \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{1.0 + y} \cdot \sqrt[3]{1.0 + y}} \cdot \frac{x - 1.0}{\sqrt[3]{1.0 + y}}\right)} + 1.0\]

    9. Applied associate-*r*30.1

      \[\leadsto \color{blue}{\left(y \cdot \frac{1}{\sqrt[3]{1.0 + y} \cdot \sqrt[3]{1.0 + y}}\right) \cdot \frac{x - 1.0}{\sqrt[3]{1.0 + y}}} + 1.0\]

    10. Taylor expanded around inf 14.7

      \[\leadsto \color{blue}{\left(1.0 \cdot \frac{x}{{y}^{2}} + x\right) - 1.0 \cdot \frac{x}{y}}\]

    11. Simplified14.7

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

    if -5186263861.88195 < y < 5.169922070464494e+30

    1. Initial program 1.1

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

    2. Simplified1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x - 1.0}{1.0 + y}, 1.0\right)}\]
    3. Using strategy rm

    4. Applied fma-udef1.0

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5186263861.88195:\\ \;\;\;\;x + \left(\frac{1.0}{y} - 1.0\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;y \le 5.169922070464494 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \frac{x - 1.0}{1.0 + y} + 1.0\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1.0}{y} - 1.0\right) \cdot \frac{x}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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))))