Average Error: 29.2 → 0.0
Time: 13.2s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6041717527243.163:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 180005.41283506868:\\ \;\;\;\;\frac{-3 \cdot x + -1}{\left(x - 1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6041717527243.163:\\
\;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\

\mathbf{elif}\;x \le 180005.41283506868:\\
\;\;\;\;\frac{-3 \cdot x + -1}{\left(x - 1\right) \cdot \left(x + 1\right)}\\

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

\end{array}
double f(double x) {
        double r2165165 = x;
        double r2165166 = 1.0;
        double r2165167 = r2165165 + r2165166;
        double r2165168 = r2165165 / r2165167;
        double r2165169 = r2165165 - r2165166;
        double r2165170 = r2165167 / r2165169;
        double r2165171 = r2165168 - r2165170;
        return r2165171;
}
↓
double f(double x) {
        double r2165172 = x;
        double r2165173 = -6041717527243.163;
        bool r2165174 = r2165172 <= r2165173;
        double r2165175 = -3.0;
        double r2165176 = r2165175 / r2165172;
        double r2165177 = r2165172 * r2165172;
        double r2165178 = r2165176 / r2165177;
        double r2165179 = 3.0;
        double r2165180 = r2165179 / r2165172;
        double r2165181 = 1.0;
        double r2165182 = r2165181 / r2165177;
        double r2165183 = r2165180 + r2165182;
        double r2165184 = r2165178 - r2165183;
        double r2165185 = 180005.41283506868;
        bool r2165186 = r2165172 <= r2165185;
        double r2165187 = r2165175 * r2165172;
        double r2165188 = -1.0;
        double r2165189 = r2165187 + r2165188;
        double r2165190 = r2165172 - r2165181;
        double r2165191 = r2165172 + r2165181;
        double r2165192 = r2165190 * r2165191;
        double r2165193 = r2165189 / r2165192;
        double r2165194 = r2165186 ? r2165193 : r2165184;
        double r2165195 = r2165174 ? r2165184 : r2165194;
        return r2165195;
}

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 < -6041717527243.163 or 180005.41283506868 < x

    1. Initial program 59.9

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

    3. Applied add-cube-cbrt60.8

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

    4. Applied associate-/l*60.8

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}{\frac{x - 1}{\sqrt[3]{x + 1}}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt60.9

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

    7. Applied associate-*l*60.9

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

    8. Taylor expanded around inf 0.3

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

    9. Simplified0.0

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

    if -6041717527243.163 < x < 180005.41283506868

    1. Initial program 0.5

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

    3. Applied frac-sub0.5

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

    4. Taylor expanded around -inf 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6041717527243.163:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 180005.41283506868:\\ \;\;\;\;\frac{-3 \cdot x + -1}{\left(x - 1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))