Average Error: 29.7 → 0.1
Time: 6.1m
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6263.833804380613:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 7862.0498067640765:\\ \;\;\;\;\mathsf{fma}\left(-\left(x \cdot x + \left(x + 1\right)\right), \frac{1 + x}{{x}^{3} - 1}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(x \cdot x + \left(x + 1\right)\right)\right) + \mathsf{fma}\left(\sqrt[3]{\frac{x}{1 + x}} \cdot \sqrt[3]{\frac{x}{1 + x}}, \sqrt[3]{\frac{x}{1 + x}}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(-\left(x \cdot x + \left(x + 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6263.833804380613:\\
\;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\

\mathbf{elif}\;x \le 7862.0498067640765:\\
\;\;\;\;\mathsf{fma}\left(-\left(x \cdot x + \left(x + 1\right)\right), \frac{1 + x}{{x}^{3} - 1}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(x \cdot x + \left(x + 1\right)\right)\right) + \mathsf{fma}\left(\sqrt[3]{\frac{x}{1 + x}} \cdot \sqrt[3]{\frac{x}{1 + x}}, \sqrt[3]{\frac{x}{1 + x}}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(-\left(x \cdot x + \left(x + 1\right)\right)\right)\right)\\

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

\end{array}
double f(double x) {
        double r2330345 = x;
        double r2330346 = 1.0;
        double r2330347 = r2330345 + r2330346;
        double r2330348 = r2330345 / r2330347;
        double r2330349 = r2330345 - r2330346;
        double r2330350 = r2330347 / r2330349;
        double r2330351 = r2330348 - r2330350;
        return r2330351;
}
↓
double f(double x) {
        double r2330352 = x;
        double r2330353 = -6263.833804380613;
        bool r2330354 = r2330352 <= r2330353;
        double r2330355 = -3.0;
        double r2330356 = r2330355 / r2330352;
        double r2330357 = -1.0;
        double r2330358 = r2330352 * r2330352;
        double r2330359 = r2330357 / r2330358;
        double r2330360 = r2330356 + r2330359;
        double r2330361 = r2330358 * r2330352;
        double r2330362 = r2330355 / r2330361;
        double r2330363 = r2330360 + r2330362;
        double r2330364 = 7862.0498067640765;
        bool r2330365 = r2330352 <= r2330364;
        double r2330366 = 1.0;
        double r2330367 = r2330352 + r2330366;
        double r2330368 = r2330358 + r2330367;
        double r2330369 = -r2330368;
        double r2330370 = r2330366 + r2330352;
        double r2330371 = 3.0;
        double r2330372 = pow(r2330352, r2330371);
        double r2330373 = r2330372 - r2330366;
        double r2330374 = r2330370 / r2330373;
        double r2330375 = r2330374 * r2330368;
        double r2330376 = fma(r2330369, r2330374, r2330375);
        double r2330377 = r2330352 / r2330370;
        double r2330378 = cbrt(r2330377);
        double r2330379 = r2330378 * r2330378;
        double r2330380 = r2330374 * r2330369;
        double r2330381 = fma(r2330379, r2330378, r2330380);
        double r2330382 = r2330376 + r2330381;
        double r2330383 = r2330365 ? r2330382 : r2330363;
        double r2330384 = r2330354 ? r2330363 : r2330383;
        return r2330384;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -6263.833804380613 or 7862.0498067640765 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -6263.833804380613 < x < 7862.0498067640765

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied add-cube-cbrt0.1

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

    6. Applied prod-diff0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6263.833804380613:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 7862.0498067640765:\\ \;\;\;\;\mathsf{fma}\left(-\left(x \cdot x + \left(x + 1\right)\right), \frac{1 + x}{{x}^{3} - 1}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(x \cdot x + \left(x + 1\right)\right)\right) + \mathsf{fma}\left(\sqrt[3]{\frac{x}{1 + x}} \cdot \sqrt[3]{\frac{x}{1 + x}}, \sqrt[3]{\frac{x}{1 + x}}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(-\left(x \cdot x + \left(x + 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))