Average Error: 29.5 → 0.2
Time: 5.7m
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10673.842437006402:\\ \;\;\;\;(3 \cdot \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) + \left(\frac{-1}{x \cdot x}\right))_*\\ \mathbf{elif}\;x \le 13348.20229892933:\\ \;\;\;\;(\left(\frac{x}{(x \cdot \left(x \cdot x\right) + 1)_*}\right) \cdot \left(x \cdot x + \left(1 - x\right)\right) + \left(\frac{-1 + \left(-x\right)}{x - 1}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(3 \cdot \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) + \left(\frac{-1}{x \cdot x}\right))_*\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10673.842437006402:\\
\;\;\;\;(3 \cdot \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) + \left(\frac{-1}{x \cdot x}\right))_*\\

\mathbf{elif}\;x \le 13348.20229892933:\\
\;\;\;\;(\left(\frac{x}{(x \cdot \left(x \cdot x\right) + 1)_*}\right) \cdot \left(x \cdot x + \left(1 - x\right)\right) + \left(\frac{-1 + \left(-x\right)}{x - 1}\right))_*\\

\mathbf{else}:\\
\;\;\;\;(3 \cdot \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) + \left(\frac{-1}{x \cdot x}\right))_*\\

\end{array}
double f(double x) {
        double r19762690 = x;
        double r19762691 = 1.0;
        double r19762692 = r19762690 + r19762691;
        double r19762693 = r19762690 / r19762692;
        double r19762694 = r19762690 - r19762691;
        double r19762695 = r19762692 / r19762694;
        double r19762696 = r19762693 - r19762695;
        return r19762696;
}
↓
double f(double x) {
        double r19762697 = x;
        double r19762698 = -10673.842437006402;
        bool r19762699 = r19762697 <= r19762698;
        double r19762700 = 3.0;
        double r19762701 = -1.0;
        double r19762702 = r19762701 / r19762697;
        double r19762703 = r19762697 * r19762697;
        double r19762704 = r19762702 / r19762703;
        double r19762705 = r19762702 + r19762704;
        double r19762706 = r19762701 / r19762703;
        double r19762707 = fma(r19762700, r19762705, r19762706);
        double r19762708 = 13348.20229892933;
        bool r19762709 = r19762697 <= r19762708;
        double r19762710 = 1.0;
        double r19762711 = fma(r19762697, r19762703, r19762710);
        double r19762712 = r19762697 / r19762711;
        double r19762713 = r19762710 - r19762697;
        double r19762714 = r19762703 + r19762713;
        double r19762715 = -r19762697;
        double r19762716 = r19762701 + r19762715;
        double r19762717 = r19762697 - r19762710;
        double r19762718 = r19762716 / r19762717;
        double r19762719 = fma(r19762712, r19762714, r19762718);
        double r19762720 = r19762709 ? r19762719 : r19762707;
        double r19762721 = r19762699 ? r19762707 : r19762720;
        return r19762721;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -10673.842437006402 or 13348.20229892933 < x

    1. Initial program 59.4

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.3

      \[\leadsto \color{blue}{(3 \cdot \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) + \left(\frac{-1}{x \cdot x}\right))_*}\]

    if -10673.842437006402 < x < 13348.20229892933

    1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied flip3-+0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied fma-neg0.1

      \[\leadsto \color{blue}{(\left(\frac{x}{{x}^{3} + {1}^{3}}\right) \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right) + \left(-\frac{x + 1}{x - 1}\right))_*}\]

    6. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10673.842437006402:\\ \;\;\;\;(3 \cdot \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) + \left(\frac{-1}{x \cdot x}\right))_*\\ \mathbf{elif}\;x \le 13348.20229892933:\\ \;\;\;\;(\left(\frac{x}{(x \cdot \left(x \cdot x\right) + 1)_*}\right) \cdot \left(x \cdot x + \left(1 - x\right)\right) + \left(\frac{-1 + \left(-x\right)}{x - 1}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(3 \cdot \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) + \left(\frac{-1}{x \cdot x}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))