Average Error: 22.2 → 7.7
Time: 4.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.4231983371330456 \cdot 10^{86} \lor \neg \left(y \le 1.2211508164976468 \cdot 10^{60}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.4231983371330456 \cdot 10^{86} \lor \neg \left(y \le 1.2211508164976468 \cdot 10^{60}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\

\end{array}
double f(double x, double y) {
        double r664795 = 1.0;
        double r664796 = x;
        double r664797 = r664795 - r664796;
        double r664798 = y;
        double r664799 = r664797 * r664798;
        double r664800 = r664798 + r664795;
        double r664801 = r664799 / r664800;
        double r664802 = r664795 - r664801;
        return r664802;
}
↓
double f(double x, double y) {
        double r664803 = y;
        double r664804 = -2.4231983371330456e+86;
        bool r664805 = r664803 <= r664804;
        double r664806 = 1.2211508164976468e+60;
        bool r664807 = r664803 <= r664806;
        double r664808 = !r664807;
        bool r664809 = r664805 || r664808;
        double r664810 = x;
        double r664811 = r664810 / r664803;
        double r664812 = 1.0;
        double r664813 = r664812 / r664803;
        double r664814 = r664813 - r664812;
        double r664815 = fma(r664811, r664814, r664810);
        double r664816 = 3.0;
        double r664817 = pow(r664803, r664816);
        double r664818 = pow(r664812, r664816);
        double r664819 = r664817 + r664818;
        double r664820 = r664803 / r664819;
        double r664821 = r664803 * r664803;
        double r664822 = r664812 * r664812;
        double r664823 = r664803 * r664812;
        double r664824 = r664822 - r664823;
        double r664825 = r664821 + r664824;
        double r664826 = r664820 * r664825;
        double r664827 = r664810 - r664812;
        double r664828 = fma(r664826, r664827, r664812);
        double r664829 = r664809 ? r664815 : r664828;
        return r664829;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.2
Target0.3
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 < -2.4231983371330456e+86 or 1.2211508164976468e+60 < y

    1. Initial program 49.0

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

    2. Simplified29.4

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

    3. Taylor expanded around inf 11.7

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

    4. Simplified11.7

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

    if -2.4231983371330456e+86 < y < 1.2211508164976468e+60

    1. Initial program 6.0

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

    2. Simplified5.2

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

    4. Applied flip3-+5.2

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}}, x - 1, 1\right)\]

    5. Applied associate-/r/5.2

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.4231983371330456 \cdot 10^{86} \lor \neg \left(y \le 1.2211508164976468 \cdot 10^{60}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))))