Average Error: 29.4 → 0.1
Time: 6.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10429.8349440664733 \lor \neg \left(x \le 11484.316894500327\right):\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + 1}\right)\right) - \frac{x + 1}{x - 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10429.8349440664733 \lor \neg \left(x \le 11484.316894500327\right):\\
\;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + 1}\right)\right) - \frac{x + 1}{x - 1}\\

\end{array}
double f(double x) {
        double r128209 = x;
        double r128210 = 1.0;
        double r128211 = r128209 + r128210;
        double r128212 = r128209 / r128211;
        double r128213 = r128209 - r128210;
        double r128214 = r128211 / r128213;
        double r128215 = r128212 - r128214;
        return r128215;
}
↓
double f(double x) {
        double r128216 = x;
        double r128217 = -10429.834944066473;
        bool r128218 = r128216 <= r128217;
        double r128219 = 11484.316894500327;
        bool r128220 = r128216 <= r128219;
        double r128221 = !r128220;
        bool r128222 = r128218 || r128221;
        double r128223 = 1.0;
        double r128224 = 2.0;
        double r128225 = pow(r128216, r128224);
        double r128226 = r128223 / r128225;
        double r128227 = 3.0;
        double r128228 = r128227 / r128216;
        double r128229 = r128226 + r128228;
        double r128230 = -r128229;
        double r128231 = 1.0;
        double r128232 = 3.0;
        double r128233 = pow(r128216, r128232);
        double r128234 = r128231 / r128233;
        double r128235 = r128227 * r128234;
        double r128236 = r128230 - r128235;
        double r128237 = r128216 + r128223;
        double r128238 = r128216 / r128237;
        double r128239 = log1p(r128238);
        double r128240 = expm1(r128239);
        double r128241 = r128216 - r128223;
        double r128242 = r128237 / r128241;
        double r128243 = r128240 - r128242;
        double r128244 = r128222 ? r128236 : r128243;
        return r128244;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -10429.834944066473 or 11484.316894500327 < x

    1. Initial program 59.2

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

    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

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

    if -10429.834944066473 < x < 11484.316894500327

    1. Initial program 0.1

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

    3. Applied expm1-log1p-u0.2

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + 1}\right)\right)} - \frac{x + 1}{x - 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10429.8349440664733 \lor \neg \left(x \le 11484.316894500327\right):\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + 1}\right)\right) - \frac{x + 1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))