Average Error: 21.9 → 0.1
Time: 52.7s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -150725849.0902552902698516845703125:\\ \;\;\;\;\left(\frac{1}{y} - \frac{x}{y} \cdot 1\right) + x\\ \mathbf{elif}\;y \le 89892178.74303682148456573486328125:\\ \;\;\;\;1 - \left(\frac{1 - x}{1 + y} \cdot \frac{y}{y - 1}\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{y} - \frac{x}{y} \cdot 1\right) + x\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -150725849.0902552902698516845703125:\\
\;\;\;\;\left(\frac{1}{y} - \frac{x}{y} \cdot 1\right) + x\\

\mathbf{elif}\;y \le 89892178.74303682148456573486328125:\\
\;\;\;\;1 - \left(\frac{1 - x}{1 + y} \cdot \frac{y}{y - 1}\right) \cdot \left(y - 1\right)\\

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

\end{array}
double f(double x, double y) {
        double r30100139 = 1.0;
        double r30100140 = x;
        double r30100141 = r30100139 - r30100140;
        double r30100142 = y;
        double r30100143 = r30100141 * r30100142;
        double r30100144 = r30100142 + r30100139;
        double r30100145 = r30100143 / r30100144;
        double r30100146 = r30100139 - r30100145;
        return r30100146;
}
↓
double f(double x, double y) {
        double r30100147 = y;
        double r30100148 = -150725849.0902553;
        bool r30100149 = r30100147 <= r30100148;
        double r30100150 = 1.0;
        double r30100151 = r30100150 / r30100147;
        double r30100152 = x;
        double r30100153 = r30100152 / r30100147;
        double r30100154 = r30100153 * r30100150;
        double r30100155 = r30100151 - r30100154;
        double r30100156 = r30100155 + r30100152;
        double r30100157 = 89892178.74303682;
        bool r30100158 = r30100147 <= r30100157;
        double r30100159 = r30100150 - r30100152;
        double r30100160 = r30100150 + r30100147;
        double r30100161 = r30100159 / r30100160;
        double r30100162 = r30100147 - r30100150;
        double r30100163 = r30100147 / r30100162;
        double r30100164 = r30100161 * r30100163;
        double r30100165 = r30100164 * r30100162;
        double r30100166 = r30100150 - r30100165;
        double r30100167 = r30100158 ? r30100166 : r30100156;
        double r30100168 = r30100149 ? r30100156 : r30100167;
        return r30100168;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.9
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 < -150725849.0902553 or 89892178.74303682 < y

    1. Initial program 45.6

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

    3. Applied flip-+50.4

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

    4. Applied associate-/r/50.4

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

    5. Simplified29.4

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

    6. Taylor expanded around inf 0.1

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

    7. Simplified0.1

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

    if -150725849.0902553 < y < 89892178.74303682

    1. Initial program 0.2

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

    3. Applied flip-+0.2

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

    4. Applied associate-/r/0.2

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

    5. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -150725849.0902552902698516845703125:\\ \;\;\;\;\left(\frac{1}{y} - \frac{x}{y} \cdot 1\right) + x\\ \mathbf{elif}\;y \le 89892178.74303682148456573486328125:\\ \;\;\;\;1 - \left(\frac{1 - x}{1 + y} \cdot \frac{y}{y - 1}\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{y} - \frac{x}{y} \cdot 1\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))