Average Error: 22.4 → 7.4
Time: 17.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2404433391549375:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{elif}\;y \le 3.069168683971201003607075545609233850904 \cdot 10^{49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}} \cdot \frac{\sqrt[3]{\sqrt[3]{\left(1 + y\right) \cdot \left(y - 1\right)}}}{\sqrt[3]{\sqrt[3]{y - 1}}}}, x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2404433391549375:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\mathbf{elif}\;y \le 3.069168683971201003607075545609233850904 \cdot 10^{49}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}} \cdot \frac{\sqrt[3]{\sqrt[3]{\left(1 + y\right) \cdot \left(y - 1\right)}}}{\sqrt[3]{\sqrt[3]{y - 1}}}}, x - 1, 1\right)\\

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

\end{array}
double f(double x, double y) {
        double r27627666 = 1.0;
        double r27627667 = x;
        double r27627668 = r27627666 - r27627667;
        double r27627669 = y;
        double r27627670 = r27627668 * r27627669;
        double r27627671 = r27627669 + r27627666;
        double r27627672 = r27627670 / r27627671;
        double r27627673 = r27627666 - r27627672;
        return r27627673;
}
↓
double f(double x, double y) {
        double r27627674 = y;
        double r27627675 = -2404433391549375.0;
        bool r27627676 = r27627674 <= r27627675;
        double r27627677 = x;
        double r27627678 = r27627677 / r27627674;
        double r27627679 = 1.0;
        double r27627680 = r27627679 / r27627674;
        double r27627681 = r27627680 - r27627679;
        double r27627682 = fma(r27627678, r27627681, r27627677);
        double r27627683 = 3.069168683971201e+49;
        bool r27627684 = r27627674 <= r27627683;
        double r27627685 = 1.0;
        double r27627686 = r27627679 + r27627674;
        double r27627687 = cbrt(r27627686);
        double r27627688 = r27627687 * r27627687;
        double r27627689 = r27627685 / r27627688;
        double r27627690 = cbrt(r27627688);
        double r27627691 = r27627674 - r27627679;
        double r27627692 = r27627686 * r27627691;
        double r27627693 = cbrt(r27627692);
        double r27627694 = cbrt(r27627693);
        double r27627695 = cbrt(r27627691);
        double r27627696 = cbrt(r27627695);
        double r27627697 = r27627694 / r27627696;
        double r27627698 = r27627690 * r27627697;
        double r27627699 = r27627674 / r27627698;
        double r27627700 = r27627689 * r27627699;
        double r27627701 = r27627677 - r27627679;
        double r27627702 = fma(r27627700, r27627701, r27627679);
        double r27627703 = r27627684 ? r27627702 : r27627682;
        double r27627704 = r27627676 ? r27627682 : r27627703;
        return r27627704;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.4
Target0.2
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2404433391549375.0 or 3.069168683971201e+49 < y

    1. Initial program 47.0

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

    2. Simplified29.3

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

    4. Applied *-un-lft-identity29.3

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

    5. Applied add-cube-cbrt30.1

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

    6. Applied times-frac30.1

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

    7. Simplified30.1

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

    8. Taylor expanded around inf 14.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{x}{{y}^{2}} + x\right) - 1 \cdot \frac{x}{y}}\]

    9. Simplified14.0

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

    if -2404433391549375.0 < y < 3.069168683971201e+49

    1. Initial program 2.2

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

    2. Simplified1.9

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

    4. Applied add-cube-cbrt2.0

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

    5. Applied *-un-lft-identity2.0

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

    6. Applied times-frac2.0

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

    8. Applied add-cube-cbrt2.0

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

    9. Applied cbrt-prod2.0

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

    11. Applied flip-+2.0

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

    12. Applied cbrt-div2.0

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

    13. Applied cbrt-div2.0

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

    14. Simplified2.0

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2404433391549375:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{elif}\;y \le 3.069168683971201003607075545609233850904 \cdot 10^{49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}} \cdot \frac{\sqrt[3]{\sqrt[3]{\left(1 + y\right) \cdot \left(y - 1\right)}}}{\sqrt[3]{\sqrt[3]{y - 1}}}}, x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))))