Average Error: 29.2 → 0.1
Time: 14.4s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12597.72366167998370656277984380722045898 \lor \neg \left(x \le 10072.80263363166523049585521221160888672\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \frac{\left(x \cdot x - 1 \cdot 1\right) \cdot \left({x}^{3} + {1}^{3}\right)}{\left(x - 1\right) \cdot \left({x}^{2} + 1 \cdot \left(1 - x\right)\right)}}{x \cdot x - 1 \cdot 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12597.72366167998370656277984380722045898 \lor \neg \left(x \le 10072.80263363166523049585521221160888672\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot \left(x - 1\right) - \frac{\left(x \cdot x - 1 \cdot 1\right) \cdot \left({x}^{3} + {1}^{3}\right)}{\left(x - 1\right) \cdot \left({x}^{2} + 1 \cdot \left(1 - x\right)\right)}}{x \cdot x - 1 \cdot 1}\\

\end{array}
double f(double x) {
        double r105913 = x;
        double r105914 = 1.0;
        double r105915 = r105913 + r105914;
        double r105916 = r105913 / r105915;
        double r105917 = r105913 - r105914;
        double r105918 = r105915 / r105917;
        double r105919 = r105916 - r105918;
        return r105919;
}
↓
double f(double x) {
        double r105920 = x;
        double r105921 = -12597.723661679984;
        bool r105922 = r105920 <= r105921;
        double r105923 = 10072.802633631665;
        bool r105924 = r105920 <= r105923;
        double r105925 = !r105924;
        bool r105926 = r105922 || r105925;
        double r105927 = 1.0;
        double r105928 = -r105927;
        double r105929 = 2.0;
        double r105930 = pow(r105920, r105929);
        double r105931 = r105928 / r105930;
        double r105932 = 3.0;
        double r105933 = r105932 / r105920;
        double r105934 = r105931 - r105933;
        double r105935 = 3.0;
        double r105936 = pow(r105920, r105935);
        double r105937 = r105932 / r105936;
        double r105938 = r105934 - r105937;
        double r105939 = r105920 - r105927;
        double r105940 = r105920 * r105939;
        double r105941 = r105920 * r105920;
        double r105942 = r105927 * r105927;
        double r105943 = r105941 - r105942;
        double r105944 = pow(r105927, r105935);
        double r105945 = r105936 + r105944;
        double r105946 = r105943 * r105945;
        double r105947 = r105927 - r105920;
        double r105948 = r105927 * r105947;
        double r105949 = r105930 + r105948;
        double r105950 = r105939 * r105949;
        double r105951 = r105946 / r105950;
        double r105952 = r105940 - r105951;
        double r105953 = r105952 / r105943;
        double r105954 = r105926 ? r105938 : r105953;
        return r105954;
}

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 < -12597.723661679984 or 10072.802633631665 < 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 -12597.723661679984 < x < 10072.802633631665

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

    4. Simplified0.1

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

    6. Applied flip-+0.1

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

    7. Applied flip3-+0.1

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

    8. Applied frac-times0.1

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

    9. Simplified0.1

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

    10. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12597.72366167998370656277984380722045898 \lor \neg \left(x \le 10072.80263363166523049585521221160888672\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \frac{\left(x \cdot x - 1 \cdot 1\right) \cdot \left({x}^{3} + {1}^{3}\right)}{\left(x - 1\right) \cdot \left({x}^{2} + 1 \cdot \left(1 - x\right)\right)}}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))