Average Error: 29.1 → 0.2
Time: 8.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13596.2266202856026 \lor \neg \left(x \le 14233.51059306751\right):\\ \;\;\;\;-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13596.2266202856026 \lor \neg \left(x \le 14233.51059306751\right):\\
\;\;\;\;-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\\

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

\end{array}
double f(double x) {
        double r118294 = x;
        double r118295 = 1.0;
        double r118296 = r118294 + r118295;
        double r118297 = r118294 / r118296;
        double r118298 = r118294 - r118295;
        double r118299 = r118296 / r118298;
        double r118300 = r118297 - r118299;
        return r118300;
}
↓
double f(double x) {
        double r118301 = x;
        double r118302 = -13596.226620285603;
        bool r118303 = r118301 <= r118302;
        double r118304 = 14233.510593067505;
        bool r118305 = r118301 <= r118304;
        double r118306 = !r118305;
        bool r118307 = r118303 || r118306;
        double r118308 = 1.0;
        double r118309 = 1.0;
        double r118310 = 2.0;
        double r118311 = pow(r118301, r118310);
        double r118312 = r118309 / r118311;
        double r118313 = r118308 * r118312;
        double r118314 = 3.0;
        double r118315 = r118309 / r118301;
        double r118316 = r118314 * r118315;
        double r118317 = 3.0;
        double r118318 = pow(r118301, r118317);
        double r118319 = r118309 / r118318;
        double r118320 = r118314 * r118319;
        double r118321 = r118316 + r118320;
        double r118322 = r118313 + r118321;
        double r118323 = -r118322;
        double r118324 = r118301 + r118308;
        double r118325 = r118309 / r118324;
        double r118326 = r118301 - r118308;
        double r118327 = r118324 / r118326;
        double r118328 = -r118327;
        double r118329 = fma(r118301, r118325, r118328);
        double r118330 = r118307 ? r118323 : r118329;
        return r118330;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -13596.226620285603 or 14233.510593067505 < x

    1. Initial program 59.2

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

    3. Applied add-cbrt-cube60.4

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

    4. Applied add-cbrt-cube62.0

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

    5. Applied cbrt-undiv62.0

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

    6. Simplified59.2

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

    7. Taylor expanded around inf 0.3

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

    if -13596.226620285603 < x < 14233.510593067505

    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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13596.2266202856026 \lor \neg \left(x \le 14233.51059306751\right):\\ \;\;\;\;-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))