Average Error: 22.4 → 0.4
Time: 21.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8762655710739029794226176:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 32042329.830661833286285400390625:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -8762655710739029794226176:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

\mathbf{elif}\;y \le 32042329.830661833286285400390625:\\
\;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\

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

\end{array}
double f(double x, double y) {
        double r29205663 = 1.0;
        double r29205664 = x;
        double r29205665 = r29205663 - r29205664;
        double r29205666 = y;
        double r29205667 = r29205665 * r29205666;
        double r29205668 = r29205666 + r29205663;
        double r29205669 = r29205667 / r29205668;
        double r29205670 = r29205663 - r29205669;
        return r29205670;
}
↓
double f(double x, double y) {
        double r29205671 = y;
        double r29205672 = -8.76265571073903e+24;
        bool r29205673 = r29205671 <= r29205672;
        double r29205674 = 1.0;
        double r29205675 = 1.0;
        double r29205676 = r29205675 / r29205671;
        double r29205677 = x;
        double r29205678 = r29205677 / r29205671;
        double r29205679 = r29205676 - r29205678;
        double r29205680 = fma(r29205674, r29205679, r29205677);
        double r29205681 = 32042329.830661833;
        bool r29205682 = r29205671 <= r29205681;
        double r29205683 = r29205674 - r29205677;
        double r29205684 = r29205683 * r29205671;
        double r29205685 = r29205674 + r29205671;
        double r29205686 = r29205684 / r29205685;
        double r29205687 = r29205674 - r29205686;
        double r29205688 = r29205682 ? r29205687 : r29205680;
        double r29205689 = r29205673 ? r29205680 : r29205688;
        return r29205689;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.4
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -8.76265571073903e+24 or 32042329.830661833 < y

    1. Initial program 46.3

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

    2. Simplified29.4

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

    if -8.76265571073903e+24 < y < 32042329.830661833

    1. Initial program 0.8

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8762655710739029794226176:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 32042329.830661833286285400390625:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))))