Average Error: 29.7 → 0.1
Time: 5.5s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9657.02298143569533 \lor \neg \left(x \le 12590.6930419104156\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \cdot \left(x - 1\right) - \left({x}^{3} + {1}^{3}\right) \cdot \left(x + 1\right)}{\left({x}^{3} + {1}^{3}\right) \cdot \left(x - 1\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9657.02298143569533 \lor \neg \left(x \le 12590.6930419104156\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r141114 = x;
        double r141115 = 1.0;
        double r141116 = r141114 + r141115;
        double r141117 = r141114 / r141116;
        double r141118 = r141114 - r141115;
        double r141119 = r141116 / r141118;
        double r141120 = r141117 - r141119;
        return r141120;
}
↓
double f(double x) {
        double r141121 = x;
        double r141122 = -9657.022981435695;
        bool r141123 = r141121 <= r141122;
        double r141124 = 12590.693041910416;
        bool r141125 = r141121 <= r141124;
        double r141126 = !r141125;
        bool r141127 = r141123 || r141126;
        double r141128 = 1.0;
        double r141129 = -r141128;
        double r141130 = 2.0;
        double r141131 = pow(r141121, r141130);
        double r141132 = r141129 / r141131;
        double r141133 = 3.0;
        double r141134 = r141133 / r141121;
        double r141135 = r141132 - r141134;
        double r141136 = 3.0;
        double r141137 = pow(r141121, r141136);
        double r141138 = r141133 / r141137;
        double r141139 = r141135 - r141138;
        double r141140 = r141121 * r141121;
        double r141141 = r141128 * r141128;
        double r141142 = r141121 * r141128;
        double r141143 = r141141 - r141142;
        double r141144 = r141140 + r141143;
        double r141145 = r141121 * r141144;
        double r141146 = r141121 - r141128;
        double r141147 = r141145 * r141146;
        double r141148 = pow(r141128, r141136);
        double r141149 = r141137 + r141148;
        double r141150 = r141121 + r141128;
        double r141151 = r141149 * r141150;
        double r141152 = r141147 - r141151;
        double r141153 = r141149 * r141146;
        double r141154 = r141152 / r141153;
        double r141155 = r141127 ? r141139 : r141154;
        return r141155;
}

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 < -9657.022981435695 or 12590.693041910416 < x

    1. Initial program 59.3

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

    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. Simplified0.0

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

    if -9657.022981435695 < x < 12590.693041910416

    1. Initial program 0.1

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

    3. Applied flip3-+0.1

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

    4. Applied associate-/r/0.1

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

    6. Applied associate-*l/0.1

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

    7. Applied frac-sub0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9657.02298143569533 \lor \neg \left(x \le 12590.6930419104156\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \cdot \left(x - 1\right) - \left({x}^{3} + {1}^{3}\right) \cdot \left(x + 1\right)}{\left({x}^{3} + {1}^{3}\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))