Average Error: 22.8 → 7.9
Time: 4.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.2287957848170708 \cdot 10^{34} \lor \neg \left(y \le 92001499739.104462\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot y}{y + 1} + 1\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.2287957848170708 \cdot 10^{34} \lor \neg \left(y \le 92001499739.104462\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r799966 = 1.0;
        double r799967 = x;
        double r799968 = r799966 - r799967;
        double r799969 = y;
        double r799970 = r799968 * r799969;
        double r799971 = r799969 + r799966;
        double r799972 = r799970 / r799971;
        double r799973 = r799966 - r799972;
        return r799973;
}
↓
double f(double x, double y) {
        double r799974 = y;
        double r799975 = -2.228795784817071e+34;
        bool r799976 = r799974 <= r799975;
        double r799977 = 92001499739.10446;
        bool r799978 = r799974 <= r799977;
        double r799979 = !r799978;
        bool r799980 = r799976 || r799979;
        double r799981 = 1.0;
        double r799982 = x;
        double r799983 = 2.0;
        double r799984 = pow(r799974, r799983);
        double r799985 = r799982 / r799984;
        double r799986 = r799982 / r799974;
        double r799987 = r799985 - r799986;
        double r799988 = fma(r799981, r799987, r799982);
        double r799989 = r799982 - r799981;
        double r799990 = r799989 * r799974;
        double r799991 = r799974 + r799981;
        double r799992 = r799990 / r799991;
        double r799993 = r799992 + r799981;
        double r799994 = r799980 ? r799988 : r799993;
        return r799994;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.8
Target0.2
Herbie7.9
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.228795784817071e+34 or 92001499739.10446 < y

    1. Initial program 46.6

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

    2. Simplified29.5

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

    4. Applied add-cube-cbrt30.3

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

    5. Applied associate-/r*30.3

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

    6. Taylor expanded around inf 15.1

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

    7. Simplified15.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)}\]

    if -2.228795784817071e+34 < y < 92001499739.10446

    1. Initial program 1.5

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

    2. Simplified1.3

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

    4. Applied add-cube-cbrt1.4

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

    5. Applied associate-/r*1.4

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

    7. Applied fma-udef1.4

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

    8. Simplified1.5

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.2287957848170708 \cdot 10^{34} \lor \neg \left(y \le 92001499739.104462\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot y}{y + 1} + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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))))