Average Error: 21.7 → 0.1
Time: 18.9s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -148588481.45705655:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 221673187.8082131:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -148588481.45705655:\\
\;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\

\mathbf{elif}\;y \le 221673187.8082131:\\
\;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{1.0 + y}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\

\end{array}
double f(double x, double y) {
        double r21233794 = 1.0;
        double r21233795 = x;
        double r21233796 = r21233794 - r21233795;
        double r21233797 = y;
        double r21233798 = r21233796 * r21233797;
        double r21233799 = r21233797 + r21233794;
        double r21233800 = r21233798 / r21233799;
        double r21233801 = r21233794 - r21233800;
        return r21233801;
}
↓
double f(double x, double y) {
        double r21233802 = y;
        double r21233803 = -148588481.45705655;
        bool r21233804 = r21233802 <= r21233803;
        double r21233805 = 1.0;
        double r21233806 = 1.0;
        double r21233807 = r21233806 / r21233802;
        double r21233808 = x;
        double r21233809 = r21233808 / r21233802;
        double r21233810 = r21233807 - r21233809;
        double r21233811 = fma(r21233805, r21233810, r21233808);
        double r21233812 = 221673187.8082131;
        bool r21233813 = r21233802 <= r21233812;
        double r21233814 = r21233805 - r21233808;
        double r21233815 = r21233814 * r21233802;
        double r21233816 = r21233805 + r21233802;
        double r21233817 = r21233815 / r21233816;
        double r21233818 = r21233805 - r21233817;
        double r21233819 = r21233813 ? r21233818 : r21233811;
        double r21233820 = r21233804 ? r21233811 : r21233819;
        return r21233820;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.7
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -148588481.45705655 or 221673187.8082131 < y

    1. Initial program 45.3

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)}\]

    if -148588481.45705655 < y < 221673187.8082131

    1. Initial program 0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -148588481.45705655:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 221673187.8082131:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))))