Average Error: 29.7 → 0.1
Time: 13.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10439.15205635957318008877336978912353516 \lor \neg \left(x \le 11917.97074271185010729823261499404907227\right):\\ \;\;\;\;\frac{-\left(\frac{1}{x} + 3\right)}{x} - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({x}^{3} + x \cdot \left(1 \cdot \left(1 - x\right)\right)\right) - \left(x + 1\right) \cdot \frac{{x}^{3} + {1}^{3}}{x - 1}}{\left(x \cdot x + 1 \cdot \left(1 - x\right)\right) \cdot \left(x + 1\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10439.15205635957318008877336978912353516 \lor \neg \left(x \le 11917.97074271185010729823261499404907227\right):\\
\;\;\;\;\frac{-\left(\frac{1}{x} + 3\right)}{x} - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r62984 = x;
        double r62985 = 1.0;
        double r62986 = r62984 + r62985;
        double r62987 = r62984 / r62986;
        double r62988 = r62984 - r62985;
        double r62989 = r62986 / r62988;
        double r62990 = r62987 - r62989;
        return r62990;
}
↓
double f(double x) {
        double r62991 = x;
        double r62992 = -10439.152056359573;
        bool r62993 = r62991 <= r62992;
        double r62994 = 11917.97074271185;
        bool r62995 = r62991 <= r62994;
        double r62996 = !r62995;
        bool r62997 = r62993 || r62996;
        double r62998 = 1.0;
        double r62999 = r62998 / r62991;
        double r63000 = 3.0;
        double r63001 = r62999 + r63000;
        double r63002 = -r63001;
        double r63003 = r63002 / r62991;
        double r63004 = 3.0;
        double r63005 = pow(r62991, r63004);
        double r63006 = r63000 / r63005;
        double r63007 = r63003 - r63006;
        double r63008 = r62998 - r62991;
        double r63009 = r62998 * r63008;
        double r63010 = r62991 * r63009;
        double r63011 = r63005 + r63010;
        double r63012 = r62991 + r62998;
        double r63013 = pow(r62998, r63004);
        double r63014 = r63005 + r63013;
        double r63015 = r62991 - r62998;
        double r63016 = r63014 / r63015;
        double r63017 = r63012 * r63016;
        double r63018 = r63011 - r63017;
        double r63019 = r62991 * r62991;
        double r63020 = r63019 + r63009;
        double r63021 = r63020 * r63012;
        double r63022 = r63018 / r63021;
        double r63023 = r62997 ? r63007 : r63022;
        return r63023;
}

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 < -10439.152056359573 or 11917.97074271185 < 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. Simplified0.3

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

    5. Applied pow10.3

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

    6. Applied pow10.3

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

    7. Applied pow-prod-down0.3

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

    8. Simplified0.0

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

    if -10439.152056359573 < x < 11917.97074271185

    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. Using strategy rm

    5. Applied flip3-+0.1

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

    6. Applied associate-*l/0.1

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

    7. Applied frac-sub0.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10439.15205635957318008877336978912353516 \lor \neg \left(x \le 11917.97074271185010729823261499404907227\right):\\ \;\;\;\;\frac{-\left(\frac{1}{x} + 3\right)}{x} - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({x}^{3} + x \cdot \left(1 \cdot \left(1 - x\right)\right)\right) - \left(x + 1\right) \cdot \frac{{x}^{3} + {1}^{3}}{x - 1}}{\left(x \cdot x + 1 \cdot \left(1 - x\right)\right) \cdot \left(x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))