Average Error: 21.1 → 0.2
Time: 10.7s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -178146608.6426316:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x \cdot 1.0}{y}\\ \mathbf{elif}\;y \le 170810479.021181:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1.0}{y + 1.0}, y, 1.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x \cdot 1.0}{y}\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -178146608.6426316:\\
\;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x \cdot 1.0}{y}\\

\mathbf{elif}\;y \le 170810479.021181:\\
\;\;\;\;\mathsf{fma}\left(\frac{x - 1.0}{y + 1.0}, y, 1.0\right)\\

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

\end{array}
double f(double x, double y) {
        double r30892823 = 1.0;
        double r30892824 = x;
        double r30892825 = r30892823 - r30892824;
        double r30892826 = y;
        double r30892827 = r30892825 * r30892826;
        double r30892828 = r30892826 + r30892823;
        double r30892829 = r30892827 / r30892828;
        double r30892830 = r30892823 - r30892829;
        return r30892830;
}
↓
double f(double x, double y) {
        double r30892831 = y;
        double r30892832 = -178146608.6426316;
        bool r30892833 = r30892831 <= r30892832;
        double r30892834 = x;
        double r30892835 = 1.0;
        double r30892836 = r30892835 / r30892831;
        double r30892837 = r30892834 + r30892836;
        double r30892838 = r30892834 * r30892835;
        double r30892839 = r30892838 / r30892831;
        double r30892840 = r30892837 - r30892839;
        double r30892841 = 170810479.021181;
        bool r30892842 = r30892831 <= r30892841;
        double r30892843 = r30892834 - r30892835;
        double r30892844 = r30892831 + r30892835;
        double r30892845 = r30892843 / r30892844;
        double r30892846 = fma(r30892845, r30892831, r30892835);
        double r30892847 = r30892842 ? r30892846 : r30892840;
        double r30892848 = r30892833 ? r30892840 : r30892847;
        return r30892848;
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Split input into 2 regimes

  2. if y < -178146608.6426316 or 170810479.021181 < y

    1. Initial program 44.5

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

    2. Simplified29.6

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

    if -178146608.6426316 < y < 170810479.021181

    1. Initial program 0.2

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

    2. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - 1.0}{1.0 + y}, y, 1.0\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -178146608.6426316:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x \cdot 1.0}{y}\\ \mathbf{elif}\;y \le 170810479.021181:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1.0}{y + 1.0}, y, 1.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x \cdot 1.0}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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))))