Average Error: 29.4 → 0.2
Time: 20.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9906799218317323685312203451758250594139:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 9433.186745037075525033287703990936279297:\\ \;\;\;\;\frac{x + 1}{x - 1} \cdot 0 + \mathsf{fma}\left(x, \frac{1}{x + 1}, -\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{x + 1} \cdot \left(\frac{1}{x - 1} \cdot \sqrt{x + 1}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9906799218317323685312203451758250594139:\\
\;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\

\mathbf{elif}\;x \le 9433.186745037075525033287703990936279297:\\
\;\;\;\;\frac{x + 1}{x - 1} \cdot 0 + \mathsf{fma}\left(x, \frac{1}{x + 1}, -\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{x + 1} \cdot \left(\frac{1}{x - 1} \cdot \sqrt{x + 1}\right)\right)\right)\right)\\

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

\end{array}
double f(double x) {
        double r5449201 = x;
        double r5449202 = 1.0;
        double r5449203 = r5449201 + r5449202;
        double r5449204 = r5449201 / r5449203;
        double r5449205 = r5449201 - r5449202;
        double r5449206 = r5449203 / r5449205;
        double r5449207 = r5449204 - r5449206;
        return r5449207;
}
↓
double f(double x) {
        double r5449208 = x;
        double r5449209 = -0.9906799218317324;
        bool r5449210 = r5449208 <= r5449209;
        double r5449211 = 3.0;
        double r5449212 = r5449208 * r5449208;
        double r5449213 = r5449212 * r5449208;
        double r5449214 = r5449211 / r5449213;
        double r5449215 = -r5449214;
        double r5449216 = 1.0;
        double r5449217 = r5449216 / r5449212;
        double r5449218 = r5449211 / r5449208;
        double r5449219 = r5449217 + r5449218;
        double r5449220 = r5449215 - r5449219;
        double r5449221 = 9433.186745037076;
        bool r5449222 = r5449208 <= r5449221;
        double r5449223 = r5449208 + r5449216;
        double r5449224 = r5449208 - r5449216;
        double r5449225 = r5449223 / r5449224;
        double r5449226 = 0.0;
        double r5449227 = r5449225 * r5449226;
        double r5449228 = 1.0;
        double r5449229 = r5449228 / r5449223;
        double r5449230 = sqrt(r5449223);
        double r5449231 = r5449228 / r5449224;
        double r5449232 = r5449231 * r5449230;
        double r5449233 = r5449230 * r5449232;
        double r5449234 = expm1(r5449233);
        double r5449235 = log1p(r5449234);
        double r5449236 = -r5449235;
        double r5449237 = fma(r5449208, r5449229, r5449236);
        double r5449238 = r5449227 + r5449237;
        double r5449239 = r5449222 ? r5449238 : r5449220;
        double r5449240 = r5449210 ? r5449220 : r5449239;
        return r5449240;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.9906799218317324 or 9433.186745037076 < x

    1. Initial program 58.9

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.2

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

    if -0.9906799218317324 < x < 9433.186745037076

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    4. Applied div-inv0.1

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

    5. Applied prod-diff0.1

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

    6. Simplified0.1

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{1}{x - 1} \cdot \left(x + 1\right)\right) + \color{blue}{\frac{x + 1}{x - 1} \cdot 0}\]
    7. Using strategy rm

    8. Applied log1p-expm1-u0.1

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{x + 1}, -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x - 1} \cdot \left(x + 1\right)\right)\right)}\right) + \frac{x + 1}{x - 1} \cdot 0\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt0.1

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{x + 1}, -\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x - 1} \cdot \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}\right)\right)\right) + \frac{x + 1}{x - 1} \cdot 0\]

    11. Applied associate-*r*0.1

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{x + 1}, -\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\frac{1}{x - 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}}\right)\right)\right) + \frac{x + 1}{x - 1} \cdot 0\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9906799218317323685312203451758250594139:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 9433.186745037075525033287703990936279297:\\ \;\;\;\;\frac{x + 1}{x - 1} \cdot 0 + \mathsf{fma}\left(x, \frac{1}{x + 1}, -\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{x + 1} \cdot \left(\frac{1}{x - 1} \cdot \sqrt{x + 1}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))