Average Error: 22.6 → 7.7
Time: 4.1s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.03224386563627215 \cdot 10^{35} \lor \neg \left(y \le 1572231659809556\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.03224386563627215 \cdot 10^{35} \lor \neg \left(y \le 1572231659809556\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r757038 = 1.0;
        double r757039 = x;
        double r757040 = r757038 - r757039;
        double r757041 = y;
        double r757042 = r757040 * r757041;
        double r757043 = r757041 + r757038;
        double r757044 = r757042 / r757043;
        double r757045 = r757038 - r757044;
        return r757045;
}
↓
double f(double x, double y) {
        double r757046 = y;
        double r757047 = -1.0322438656362721e+35;
        bool r757048 = r757046 <= r757047;
        double r757049 = 1572231659809556.0;
        bool r757050 = r757046 <= r757049;
        double r757051 = !r757050;
        bool r757052 = r757048 || r757051;
        double r757053 = x;
        double r757054 = r757053 / r757046;
        double r757055 = 1.0;
        double r757056 = r757055 / r757046;
        double r757057 = r757056 - r757055;
        double r757058 = fma(r757054, r757057, r757053);
        double r757059 = r757046 * r757046;
        double r757060 = r757055 * r757055;
        double r757061 = r757059 - r757060;
        double r757062 = r757046 / r757061;
        double r757063 = r757046 - r757055;
        double r757064 = r757062 * r757063;
        double r757065 = r757053 - r757055;
        double r757066 = fma(r757064, r757065, r757055);
        double r757067 = r757052 ? r757058 : r757066;
        return r757067;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.6
Target0.2
Herbie7.7
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.0322438656362721e+35 or 1572231659809556.0 < y

    1. Initial program 46.9

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

    2. Simplified29.7

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

    3. Taylor expanded around inf 14.9

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{x}{y}}\]

    4. Simplified14.9

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

    if -1.0322438656362721e+35 < y < 1572231659809556.0

    1. Initial program 1.5

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

    2. Simplified1.4

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

    4. Applied flip-+1.4

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

    5. Applied associate-/r/1.4

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.03224386563627215 \cdot 10^{35} \lor \neg \left(y \le 1572231659809556\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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))))