Average Error: 22.1 → 7.4
Time: 4.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -18067858855719976 \lor \neg \left(y \le 7.03747293035493089 \cdot 10^{31}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{y + 1}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -18067858855719976 \lor \neg \left(y \le 7.03747293035493089 \cdot 10^{31}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r622856 = 1.0;
        double r622857 = x;
        double r622858 = r622856 - r622857;
        double r622859 = y;
        double r622860 = r622858 * r622859;
        double r622861 = r622859 + r622856;
        double r622862 = r622860 / r622861;
        double r622863 = r622856 - r622862;
        return r622863;
}
↓
double f(double x, double y) {
        double r622864 = y;
        double r622865 = -18067858855719976.0;
        bool r622866 = r622864 <= r622865;
        double r622867 = 7.037472930354931e+31;
        bool r622868 = r622864 <= r622867;
        double r622869 = !r622868;
        bool r622870 = r622866 || r622869;
        double r622871 = x;
        double r622872 = r622871 / r622864;
        double r622873 = 1.0;
        double r622874 = r622873 / r622864;
        double r622875 = r622874 - r622873;
        double r622876 = fma(r622872, r622875, r622871);
        double r622877 = 1.0;
        double r622878 = r622864 + r622873;
        double r622879 = r622877 / r622878;
        double r622880 = r622864 * r622879;
        double r622881 = r622871 - r622873;
        double r622882 = fma(r622880, r622881, r622873);
        double r622883 = r622870 ? r622876 : r622882;
        return r622883;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.1
Target0.2
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -18067858855719976.0 or 7.037472930354931e+31 < y

    1. Initial program 46.8

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

    2. Simplified28.9

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

    3. Taylor expanded around inf 14.8

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

    4. Simplified14.8

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

    if -18067858855719976.0 < y < 7.037472930354931e+31

    1. Initial program 1.3

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

    2. Simplified1.2

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

    4. Applied div-inv1.2

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -18067858855719976 \lor \neg \left(y \le 7.03747293035493089 \cdot 10^{31}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{y + 1}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))))