Average Error: 23.6 → 0.1
Time: 45.7s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -124597878.5240411:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 144059933.3700765:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -124597878.5240411:\\
\;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\

\mathbf{elif}\;y \le 144059933.3700765:\\
\;\;\;\;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 r29003913 = 1.0;
        double r29003914 = x;
        double r29003915 = r29003913 - r29003914;
        double r29003916 = y;
        double r29003917 = r29003915 * r29003916;
        double r29003918 = r29003916 + r29003913;
        double r29003919 = r29003917 / r29003918;
        double r29003920 = r29003913 - r29003919;
        return r29003920;
}
↓
double f(double x, double y) {
        double r29003921 = y;
        double r29003922 = -124597878.5240411;
        bool r29003923 = r29003921 <= r29003922;
        double r29003924 = 1.0;
        double r29003925 = 1.0;
        double r29003926 = r29003925 / r29003921;
        double r29003927 = x;
        double r29003928 = r29003927 / r29003921;
        double r29003929 = r29003926 - r29003928;
        double r29003930 = fma(r29003924, r29003929, r29003927);
        double r29003931 = 144059933.3700765;
        bool r29003932 = r29003921 <= r29003931;
        double r29003933 = r29003924 - r29003927;
        double r29003934 = r29003933 * r29003921;
        double r29003935 = r29003924 + r29003921;
        double r29003936 = r29003934 / r29003935;
        double r29003937 = r29003924 - r29003936;
        double r29003938 = r29003932 ? r29003937 : r29003930;
        double r29003939 = r29003923 ? r29003930 : r29003938;
        return r29003939;
}

Error

Bits error versus x

Bits error versus y

Target

Original23.6
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 < -124597878.5240411 or 144059933.3700765 < y

    1. Initial program 46.8

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

    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 -124597878.5240411 < y < 144059933.3700765

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -124597878.5240411:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 144059933.3700765:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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))))