Average Error: 22.3 → 0.2
Time: 10.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 0.9519809141393587026414024876430630683899 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{y + 1} \le 1.000000023499299839357945529627613723278\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 0.9519809141393587026414024876430630683899 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{y + 1} \le 1.000000023499299839357945529627613723278\right):\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\

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

\end{array}
double f(double x, double y) {
        double r493010 = 1.0;
        double r493011 = x;
        double r493012 = r493010 - r493011;
        double r493013 = y;
        double r493014 = r493012 * r493013;
        double r493015 = r493013 + r493010;
        double r493016 = r493014 / r493015;
        double r493017 = r493010 - r493016;
        return r493017;
}
↓
double f(double x, double y) {
        double r493018 = 1.0;
        double r493019 = x;
        double r493020 = r493018 - r493019;
        double r493021 = y;
        double r493022 = r493020 * r493021;
        double r493023 = r493021 + r493018;
        double r493024 = r493022 / r493023;
        double r493025 = 0.9519809141393587;
        bool r493026 = r493024 <= r493025;
        double r493027 = 1.0000000234992998;
        bool r493028 = r493024 <= r493027;
        double r493029 = !r493028;
        bool r493030 = r493026 || r493029;
        double r493031 = r493021 / r493023;
        double r493032 = r493020 * r493031;
        double r493033 = r493018 - r493032;
        double r493034 = 1.0;
        double r493035 = r493034 / r493021;
        double r493036 = r493019 / r493021;
        double r493037 = r493035 - r493036;
        double r493038 = r493018 * r493037;
        double r493039 = r493038 + r493019;
        double r493040 = r493030 ? r493033 : r493039;
        return r493040;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (- 1.0 x) y) (+ y 1.0)) < 0.9519809141393587 or 1.0000000234992998 < (/ (* (- 1.0 x) y) (+ y 1.0))

    1. Initial program 10.9

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

    3. Applied *-un-lft-identity10.9

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

    4. Applied times-frac0.1

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{1} \cdot \frac{y}{y + 1}}\]

    5. Simplified0.1

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

    if 0.9519809141393587 < (/ (* (- 1.0 x) y) (+ y 1.0)) < 1.0000000234992998

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.7

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 0.9519809141393587026414024876430630683899 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{y + 1} \le 1.000000023499299839357945529627613723278\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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))))