Average Error: 7.3 → 0.1
Time: 13.0s
Precision: 64
\[\frac{x \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.7521480753717866 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y} - 1.0, x\right)\\ \mathbf{elif}\;y \le 257574.27222550486:\\ \;\;\;\;\frac{x \cdot y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y} - 1.0, x\right)\\ \end{array}\]
\frac{x \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.7521480753717866 \cdot 10^{+46}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y} - 1.0, x\right)\\

\mathbf{elif}\;y \le 257574.27222550486:\\
\;\;\;\;\frac{x \cdot y}{1.0 + y}\\

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

\end{array}
double f(double x, double y) {
        double r29013727 = x;
        double r29013728 = y;
        double r29013729 = r29013727 * r29013728;
        double r29013730 = 1.0;
        double r29013731 = r29013728 + r29013730;
        double r29013732 = r29013729 / r29013731;
        return r29013732;
}
↓
double f(double x, double y) {
        double r29013733 = y;
        double r29013734 = -3.7521480753717866e+46;
        bool r29013735 = r29013733 <= r29013734;
        double r29013736 = x;
        double r29013737 = r29013736 / r29013733;
        double r29013738 = 1.0;
        double r29013739 = r29013738 / r29013733;
        double r29013740 = r29013739 - r29013738;
        double r29013741 = fma(r29013737, r29013740, r29013736);
        double r29013742 = 257574.27222550486;
        bool r29013743 = r29013733 <= r29013742;
        double r29013744 = r29013736 * r29013733;
        double r29013745 = r29013738 + r29013733;
        double r29013746 = r29013744 / r29013745;
        double r29013747 = r29013743 ? r29013746 : r29013741;
        double r29013748 = r29013735 ? r29013741 : r29013747;
        return r29013748;
}

Error

Bits error versus x

Bits error versus y

Target

Original7.3
Target0.0
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;\frac{x \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.7521480753717866e+46 or 257574.27222550486 < y

    1. Initial program 15.7

      \[\frac{x \cdot y}{y + 1.0}\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(1.0 \cdot \frac{x}{{y}^{2}} + x\right) - 1.0 \cdot \frac{x}{y}}\]

    3. Simplified0

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

    if -3.7521480753717866e+46 < y < 257574.27222550486

    1. Initial program 0.1

      \[\frac{x \cdot y}{y + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.7521480753717866 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y} - 1.0, x\right)\\ \mathbf{elif}\;y \le 257574.27222550486:\\ \;\;\;\;\frac{x \cdot y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y} - 1.0, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) (- (/ x (* y y)) (- (/ x y) x))))

  (/ (* x y) (+ y 1.0)))