Average Error: 22.6 → 0.1
Time: 17.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -173188923.79640638828277587890625:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 127597529.49763254821300506591796875:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{1 + y}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -173188923.79640638828277587890625:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

\mathbf{elif}\;y \le 127597529.49763254821300506591796875:\\
\;\;\;\;\mathsf{fma}\left(\frac{x - 1}{1 + y}, y, 1\right)\\

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

\end{array}
double f(double x, double y) {
        double r33604019 = 1.0;
        double r33604020 = x;
        double r33604021 = r33604019 - r33604020;
        double r33604022 = y;
        double r33604023 = r33604021 * r33604022;
        double r33604024 = r33604022 + r33604019;
        double r33604025 = r33604023 / r33604024;
        double r33604026 = r33604019 - r33604025;
        return r33604026;
}
↓
double f(double x, double y) {
        double r33604027 = y;
        double r33604028 = -173188923.7964064;
        bool r33604029 = r33604027 <= r33604028;
        double r33604030 = 1.0;
        double r33604031 = 1.0;
        double r33604032 = r33604031 / r33604027;
        double r33604033 = x;
        double r33604034 = r33604033 / r33604027;
        double r33604035 = r33604032 - r33604034;
        double r33604036 = fma(r33604030, r33604035, r33604033);
        double r33604037 = 127597529.49763255;
        bool r33604038 = r33604027 <= r33604037;
        double r33604039 = r33604033 - r33604030;
        double r33604040 = r33604030 + r33604027;
        double r33604041 = r33604039 / r33604040;
        double r33604042 = fma(r33604041, r33604027, r33604030);
        double r33604043 = r33604038 ? r33604042 : r33604036;
        double r33604044 = r33604029 ? r33604036 : r33604043;
        return r33604044;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.6
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -173188923.7964064 or 127597529.49763255 < y

    1. Initial program 45.8

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

    2. Simplified29.5

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

    if -173188923.7964064 < y < 127597529.49763255

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - 1}{1 + y}, y, 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -173188923.79640638828277587890625:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 127597529.49763254821300506591796875:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{1 + y}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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))))