Average Error: 29.5 → 0.1
Time: 4.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12526.87883169779706804547458887100219727 \lor \neg \left(x \le 12009.41967600402495008893311023712158203\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{1}{x}}{x}, \frac{-3}{x}\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12526.87883169779706804547458887100219727 \lor \neg \left(x \le 12009.41967600402495008893311023712158203\right):\\
\;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{1}{x}}{x}, \frac{-3}{x}\right) - 3 \cdot \frac{1}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r124518 = x;
        double r124519 = 1.0;
        double r124520 = r124518 + r124519;
        double r124521 = r124518 / r124520;
        double r124522 = r124518 - r124519;
        double r124523 = r124520 / r124522;
        double r124524 = r124521 - r124523;
        return r124524;
}
↓
double f(double x) {
        double r124525 = x;
        double r124526 = -12526.878831697797;
        bool r124527 = r124525 <= r124526;
        double r124528 = 12009.419676004025;
        bool r124529 = r124525 <= r124528;
        double r124530 = !r124529;
        bool r124531 = r124527 || r124530;
        double r124532 = -1.0;
        double r124533 = 1.0;
        double r124534 = r124533 / r124525;
        double r124535 = r124534 / r124525;
        double r124536 = 3.0;
        double r124537 = -r124536;
        double r124538 = r124537 / r124525;
        double r124539 = fma(r124532, r124535, r124538);
        double r124540 = 1.0;
        double r124541 = 3.0;
        double r124542 = pow(r124525, r124541);
        double r124543 = r124540 / r124542;
        double r124544 = r124536 * r124543;
        double r124545 = r124539 - r124544;
        double r124546 = r124525 + r124533;
        double r124547 = r124540 / r124546;
        double r124548 = r124525 - r124533;
        double r124549 = r124546 / r124548;
        double r124550 = -r124549;
        double r124551 = fma(r124525, r124547, r124550);
        double r124552 = r124531 ? r124545 : r124551;
        return r124552;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -12526.878831697797 or 12009.419676004025 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)}\]
    4. Using strategy rm

    5. Applied fma-udef0.3

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

    6. Applied associate--r+0.3

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

    7. Simplified0.0

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

    if -12526.878831697797 < x < 12009.419676004025

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    4. Applied fma-neg0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12526.87883169779706804547458887100219727 \lor \neg \left(x \le 12009.41967600402495008893311023712158203\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{1}{x}}{x}, \frac{-3}{x}\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))