Average Error: 29.4 → 0.1
Time: 6.5s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -14610.34704770997814193833619356155395508 \lor \neg \left(x \le 11258.21718550441983097698539495468139648\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -14610.34704770997814193833619356155395508 \lor \neg \left(x \le 11258.21718550441983097698539495468139648\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r112141 = x;
        double r112142 = 1.0;
        double r112143 = r112141 + r112142;
        double r112144 = r112141 / r112143;
        double r112145 = r112141 - r112142;
        double r112146 = r112143 / r112145;
        double r112147 = r112144 - r112146;
        return r112147;
}
↓
double f(double x) {
        double r112148 = x;
        double r112149 = -14610.347047709978;
        bool r112150 = r112148 <= r112149;
        double r112151 = 11258.21718550442;
        bool r112152 = r112148 <= r112151;
        double r112153 = !r112152;
        bool r112154 = r112150 || r112153;
        double r112155 = 1.0;
        double r112156 = -r112155;
        double r112157 = 2.0;
        double r112158 = pow(r112148, r112157);
        double r112159 = r112156 / r112158;
        double r112160 = 3.0;
        double r112161 = r112160 / r112148;
        double r112162 = r112159 - r112161;
        double r112163 = 3.0;
        double r112164 = pow(r112148, r112163);
        double r112165 = r112160 / r112164;
        double r112166 = r112162 - r112165;
        double r112167 = r112148 - r112155;
        double r112168 = r112148 * r112167;
        double r112169 = r112148 + r112155;
        double r112170 = r112169 * r112169;
        double r112171 = r112168 - r112170;
        double r112172 = r112148 * r112148;
        double r112173 = r112155 * r112155;
        double r112174 = r112172 - r112173;
        double r112175 = r112171 / r112174;
        double r112176 = exp(r112175);
        double r112177 = log(r112176);
        double r112178 = r112154 ? r112166 : r112177;
        return r112178;
}

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 < -14610.347047709978 or 11258.21718550442 < x

    1. Initial program 59.2

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

    3. Applied flip-+60.4

      \[\leadsto \frac{x}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}} - \frac{x + 1}{x - 1}\]

    4. Applied associate-/r/60.4

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

    5. Applied fma-neg60.2

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

    6. Taylor expanded around inf 0.4

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

    7. Simplified0.0

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

    if -14610.347047709978 < x < 11258.21718550442

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    6. Simplified0.1

      \[\leadsto \log \color{blue}{\left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)}\]
    7. Using strategy rm

    8. Applied frac-sub0.1

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

    9. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -14610.34704770997814193833619356155395508 \lor \neg \left(x \le 11258.21718550441983097698539495468139648\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))