Average Error: 28.7 → 0.1
Time: 15.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13946.2737556041702 \lor \neg \left(x \le 12811.273113389769\right):\\ \;\;\;\;\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\left(\frac{x}{x + 1} \cdot x\right) \cdot \left(x - 1\right) - \frac{{\left(x + 1\right)}^{3}}{x - 1}}{x \cdot x - 1 \cdot 1}\right)}^{3}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13946.2737556041702 \lor \neg \left(x \le 12811.273113389769\right):\\
\;\;\;\;\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\

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

\end{array}
double f(double x) {
        double r103994 = x;
        double r103995 = 1.0;
        double r103996 = r103994 + r103995;
        double r103997 = r103994 / r103996;
        double r103998 = r103994 - r103995;
        double r103999 = r103996 / r103998;
        double r104000 = r103997 - r103999;
        return r104000;
}
↓
double f(double x) {
        double r104001 = x;
        double r104002 = -13946.27375560417;
        bool r104003 = r104001 <= r104002;
        double r104004 = 12811.273113389769;
        bool r104005 = r104001 <= r104004;
        double r104006 = !r104005;
        bool r104007 = r104003 || r104006;
        double r104008 = 3.0;
        double r104009 = 3.0;
        double r104010 = pow(r104001, r104009);
        double r104011 = r104008 / r104010;
        double r104012 = -r104011;
        double r104013 = r104008 / r104001;
        double r104014 = 1.0;
        double r104015 = r104001 * r104001;
        double r104016 = r104014 / r104015;
        double r104017 = r104013 + r104016;
        double r104018 = r104012 - r104017;
        double r104019 = r104001 + r104014;
        double r104020 = r104001 / r104019;
        double r104021 = r104020 * r104001;
        double r104022 = r104001 - r104014;
        double r104023 = r104021 * r104022;
        double r104024 = pow(r104019, r104009);
        double r104025 = r104024 / r104022;
        double r104026 = r104023 - r104025;
        double r104027 = r104014 * r104014;
        double r104028 = r104015 - r104027;
        double r104029 = r104026 / r104028;
        double r104030 = pow(r104029, r104009);
        double r104031 = cbrt(r104030);
        double r104032 = r104019 / r104022;
        double r104033 = r104020 + r104032;
        double r104034 = r104031 / r104033;
        double r104035 = r104007 ? r104018 : r104034;
        return r104035;
}

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 < -13946.27375560417 or 12811.273113389769 < x

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -13946.27375560417 < x < 12811.273113389769

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    5. Applied add-cbrt-cube0.1

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

    6. Simplified0.1

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]
    7. Using strategy rm

    8. Applied associate-*r/0.1

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

    9. Applied associate-*r/0.1

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

    10. Applied frac-sub0.1

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

    11. Simplified0.1

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

    12. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13946.2737556041702 \lor \neg \left(x \le 12811.273113389769\right):\\ \;\;\;\;\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\left(\frac{x}{x + 1} \cdot x\right) \cdot \left(x - 1\right) - \frac{{\left(x + 1\right)}^{3}}{x - 1}}{x \cdot x - 1 \cdot 1}\right)}^{3}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))