Average Error: 22.3 → 0.2
Time: 16.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -126285643.80532795 \lor \neg \left(y \le 184531412.18592519\right):\\ \;\;\;\;\frac{1}{y} \cdot \left(1 - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-\left(1 - x\right)\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -126285643.80532795 \lor \neg \left(y \le 184531412.18592519\right):\\
\;\;\;\;\frac{1}{y} \cdot \left(1 - x\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r842090 = 1.0;
        double r842091 = x;
        double r842092 = r842090 - r842091;
        double r842093 = y;
        double r842094 = r842092 * r842093;
        double r842095 = r842093 + r842090;
        double r842096 = r842094 / r842095;
        double r842097 = r842090 - r842096;
        return r842097;
}
↓
double f(double x, double y) {
        double r842098 = y;
        double r842099 = -126285643.80532795;
        bool r842100 = r842098 <= r842099;
        double r842101 = 184531412.1859252;
        bool r842102 = r842098 <= r842101;
        double r842103 = !r842102;
        bool r842104 = r842100 || r842103;
        double r842105 = 1.0;
        double r842106 = r842105 / r842098;
        double r842107 = 1.0;
        double r842108 = x;
        double r842109 = r842107 - r842108;
        double r842110 = r842106 * r842109;
        double r842111 = r842110 + r842108;
        double r842112 = r842105 - r842108;
        double r842113 = -r842112;
        double r842114 = r842098 + r842105;
        double r842115 = r842098 / r842114;
        double r842116 = r842113 * r842115;
        double r842117 = r842105 + r842116;
        double r842118 = r842104 ? r842111 : r842117;
        return r842118;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.3
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -126285643.80532795 or 184531412.1859252 < y

    1. Initial program 45.2

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

    2. Simplified29.6

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

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

    if -126285643.80532795 < y < 184531412.1859252

    1. Initial program 0.2

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
    2. Using strategy rm

    3. Applied sub-neg0.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -126285643.80532795 \lor \neg \left(y \le 184531412.18592519\right):\\ \;\;\;\;\frac{1}{y} \cdot \left(1 - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-\left(1 - x\right)\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

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