Average Error: 29.3 → 0.0
Time: 6.0m
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.991816941079669 \cdot 10^{+45}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 137847.80161016053:\\ \;\;\;\;\mathsf{fma}\left(-3, x, -1\right) \cdot \frac{1}{\left(1 + x\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.991816941079669 \cdot 10^{+45}:\\
\;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\

\mathbf{elif}\;x \le 137847.80161016053:\\
\;\;\;\;\mathsf{fma}\left(-3, x, -1\right) \cdot \frac{1}{\left(1 + x\right) \cdot \left(x - 1\right)}\\

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

\end{array}
double f(double x) {
        double r23482252 = x;
        double r23482253 = 1.0;
        double r23482254 = r23482252 + r23482253;
        double r23482255 = r23482252 / r23482254;
        double r23482256 = r23482252 - r23482253;
        double r23482257 = r23482254 / r23482256;
        double r23482258 = r23482255 - r23482257;
        return r23482258;
}
↓
double f(double x) {
        double r23482259 = x;
        double r23482260 = -4.991816941079669e+45;
        bool r23482261 = r23482259 <= r23482260;
        double r23482262 = -3.0;
        double r23482263 = r23482262 / r23482259;
        double r23482264 = 1.0;
        double r23482265 = r23482259 * r23482259;
        double r23482266 = r23482264 / r23482265;
        double r23482267 = r23482263 - r23482266;
        double r23482268 = r23482263 / r23482265;
        double r23482269 = r23482267 + r23482268;
        double r23482270 = 137847.80161016053;
        bool r23482271 = r23482259 <= r23482270;
        double r23482272 = -1.0;
        double r23482273 = fma(r23482262, r23482259, r23482272);
        double r23482274 = r23482264 + r23482259;
        double r23482275 = r23482259 - r23482264;
        double r23482276 = r23482274 * r23482275;
        double r23482277 = r23482264 / r23482276;
        double r23482278 = r23482273 * r23482277;
        double r23482279 = r23482271 ? r23482278 : r23482269;
        double r23482280 = r23482261 ? r23482269 : r23482279;
        return r23482280;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -4.991816941079669e+45 or 137847.80161016053 < x

    1. Initial program 59.9

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

    3. Applied frac-sub61.7

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

    4. Taylor expanded around 0 32.1

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

    5. Simplified32.1

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

    6. Taylor expanded around -inf 0.3

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

    7. Simplified0.0

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

    if -4.991816941079669e+45 < x < 137847.80161016053

    1. Initial program 3.4

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

    3. Applied frac-sub3.5

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

    4. Taylor expanded around 0 0.1

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

    5. Simplified0.1

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

    7. Applied div-inv0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.991816941079669 \cdot 10^{+45}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 137847.80161016053:\\ \;\;\;\;\mathsf{fma}\left(-3, x, -1\right) \cdot \frac{1}{\left(1 + x\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))