Average Error: 29.7 → 0.1
Time: 9.1s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13568.460963923884 \lor \neg \left(x \le 10580.4331898339569\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({x}^{3} + \left(1 \cdot \left(1 - x\right)\right) \cdot x\right) - \left(x + 1\right) \cdot \frac{{x}^{3} + {1}^{3}}{x - 1}}{\left(1 \cdot \left(1 - x\right) + x \cdot x\right) \cdot \left(x + 1\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13568.460963923884 \lor \neg \left(x \le 10580.4331898339569\right):\\
\;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\

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

\end{array}
double f(double x) {
        double r94504 = x;
        double r94505 = 1.0;
        double r94506 = r94504 + r94505;
        double r94507 = r94504 / r94506;
        double r94508 = r94504 - r94505;
        double r94509 = r94506 / r94508;
        double r94510 = r94507 - r94509;
        return r94510;
}
↓
double f(double x) {
        double r94511 = x;
        double r94512 = -13568.460963923884;
        bool r94513 = r94511 <= r94512;
        double r94514 = 10580.433189833957;
        bool r94515 = r94511 <= r94514;
        double r94516 = !r94515;
        bool r94517 = r94513 || r94516;
        double r94518 = 1.0;
        double r94519 = r94511 * r94511;
        double r94520 = r94518 / r94519;
        double r94521 = -r94520;
        double r94522 = 3.0;
        double r94523 = 3.0;
        double r94524 = pow(r94511, r94523);
        double r94525 = r94522 / r94524;
        double r94526 = r94522 / r94511;
        double r94527 = r94525 + r94526;
        double r94528 = r94521 - r94527;
        double r94529 = r94518 - r94511;
        double r94530 = r94518 * r94529;
        double r94531 = r94530 * r94511;
        double r94532 = r94524 + r94531;
        double r94533 = r94511 + r94518;
        double r94534 = pow(r94518, r94523);
        double r94535 = r94524 + r94534;
        double r94536 = r94511 - r94518;
        double r94537 = r94535 / r94536;
        double r94538 = r94533 * r94537;
        double r94539 = r94532 - r94538;
        double r94540 = r94530 + r94519;
        double r94541 = r94540 * r94533;
        double r94542 = r94539 / r94541;
        double r94543 = r94517 ? r94528 : r94542;
        return r94543;
}

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 < -13568.460963923884 or 10580.433189833957 < x

    1. Initial program 59.2

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

    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

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

    if -13568.460963923884 < x < 10580.433189833957

    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

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

    9. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13568.460963923884 \lor \neg \left(x \le 10580.4331898339569\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({x}^{3} + \left(1 \cdot \left(1 - x\right)\right) \cdot x\right) - \left(x + 1\right) \cdot \frac{{x}^{3} + {1}^{3}}{x - 1}}{\left(1 \cdot \left(1 - x\right) + x \cdot x\right) \cdot \left(x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))