Average Error: 30.0 → 0.1
Time: 7.2s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9843.5346029313187 \lor \neg \left(x \le 9309.67452526344459\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{{x}^{3} + {1}^{3}}{\left(1 \cdot \left(1 - x\right) + {x}^{2}\right) \cdot \left(x - 1\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9843.5346029313187 \lor \neg \left(x \le 9309.67452526344459\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r146459 = x;
        double r146460 = 1.0;
        double r146461 = r146459 + r146460;
        double r146462 = r146459 / r146461;
        double r146463 = r146459 - r146460;
        double r146464 = r146461 / r146463;
        double r146465 = r146462 - r146464;
        return r146465;
}
↓
double f(double x) {
        double r146466 = x;
        double r146467 = -9843.534602931319;
        bool r146468 = r146466 <= r146467;
        double r146469 = 9309.674525263445;
        bool r146470 = r146466 <= r146469;
        double r146471 = !r146470;
        bool r146472 = r146468 || r146471;
        double r146473 = 1.0;
        double r146474 = -r146473;
        double r146475 = 2.0;
        double r146476 = pow(r146466, r146475);
        double r146477 = r146474 / r146476;
        double r146478 = 3.0;
        double r146479 = r146478 / r146466;
        double r146480 = r146477 - r146479;
        double r146481 = 3.0;
        double r146482 = pow(r146466, r146481);
        double r146483 = r146478 / r146482;
        double r146484 = r146480 - r146483;
        double r146485 = r146466 + r146473;
        double r146486 = r146466 / r146485;
        double r146487 = pow(r146473, r146481);
        double r146488 = r146482 + r146487;
        double r146489 = r146473 - r146466;
        double r146490 = r146473 * r146489;
        double r146491 = r146490 + r146476;
        double r146492 = r146466 - r146473;
        double r146493 = r146491 * r146492;
        double r146494 = r146488 / r146493;
        double r146495 = r146486 - r146494;
        double r146496 = r146472 ? r146484 : r146495;
        return r146496;
}

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 < -9843.534602931319 or 9309.674525263445 < 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

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

    if -9843.534602931319 < x < 9309.674525263445

    1. Initial program 0.1

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

    3. Applied flip3-+0.1

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

    4. Applied associate-/l/0.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9843.5346029313187 \lor \neg \left(x \le 9309.67452526344459\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{{x}^{3} + {1}^{3}}{\left(1 \cdot \left(1 - x\right) + {x}^{2}\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))