Average Error: 22.8 → 0.3
Time: 14.1s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3086863162769103360 \lor \neg \left(y \le 166171510.4706774652004241943359375\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3086863162769103360 \lor \neg \left(y \le 166171510.4706774652004241943359375\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r411081 = 1.0;
        double r411082 = x;
        double r411083 = r411081 - r411082;
        double r411084 = y;
        double r411085 = r411083 * r411084;
        double r411086 = r411084 + r411081;
        double r411087 = r411085 / r411086;
        double r411088 = r411081 - r411087;
        return r411088;
}
↓
double f(double x, double y) {
        double r411089 = y;
        double r411090 = -3.0868631627691034e+18;
        bool r411091 = r411089 <= r411090;
        double r411092 = 166171510.47067747;
        bool r411093 = r411089 <= r411092;
        double r411094 = !r411093;
        bool r411095 = r411091 || r411094;
        double r411096 = 1.0;
        double r411097 = 1.0;
        double r411098 = r411097 / r411089;
        double r411099 = x;
        double r411100 = r411099 / r411089;
        double r411101 = r411098 - r411100;
        double r411102 = fma(r411096, r411101, r411099);
        double r411103 = r411099 - r411096;
        double r411104 = r411089 + r411096;
        double r411105 = r411103 / r411104;
        double r411106 = fma(r411105, r411089, r411096);
        double r411107 = r411095 ? r411102 : r411106;
        return r411107;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.8
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.0868631627691034e+18 or 166171510.47067747 < y

    1. Initial program 46.8

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

    2. Simplified30.4

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

    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 -3.0868631627691034e+18 < y < 166171510.47067747

    1. Initial program 0.6

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

    2. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3086863162769103360 \lor \neg \left(y \le 166171510.4706774652004241943359375\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.84827882972468) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891003) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))