Average Error: 29.2 → 0.1
Time: 4.3s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -19531.3574826783042 \lor \neg \left(x \le 10797.6607588330535\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right) - \frac{x + 1}{x - 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -19531.3574826783042 \lor \neg \left(x \le 10797.6607588330535\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
double f(double x) {
        double r142200 = x;
        double r142201 = 1.0;
        double r142202 = r142200 + r142201;
        double r142203 = r142200 / r142202;
        double r142204 = r142200 - r142201;
        double r142205 = r142202 / r142204;
        double r142206 = r142203 - r142205;
        return r142206;
}
↓
double f(double x) {
        double r142207 = x;
        double r142208 = -19531.357482678304;
        bool r142209 = r142207 <= r142208;
        double r142210 = 10797.660758833053;
        bool r142211 = r142207 <= r142210;
        double r142212 = !r142211;
        bool r142213 = r142209 || r142212;
        double r142214 = 1.0;
        double r142215 = -r142214;
        double r142216 = 2.0;
        double r142217 = pow(r142207, r142216);
        double r142218 = r142215 / r142217;
        double r142219 = 3.0;
        double r142220 = r142219 / r142207;
        double r142221 = r142218 - r142220;
        double r142222 = 3.0;
        double r142223 = pow(r142207, r142222);
        double r142224 = r142219 / r142223;
        double r142225 = r142221 - r142224;
        double r142226 = pow(r142214, r142222);
        double r142227 = r142223 + r142226;
        double r142228 = r142207 / r142227;
        double r142229 = r142207 * r142207;
        double r142230 = r142214 * r142214;
        double r142231 = r142207 * r142214;
        double r142232 = r142230 - r142231;
        double r142233 = r142229 + r142232;
        double r142234 = r142228 * r142233;
        double r142235 = r142207 + r142214;
        double r142236 = r142207 - r142214;
        double r142237 = r142235 / r142236;
        double r142238 = r142234 - r142237;
        double r142239 = r142213 ? r142225 : r142238;
        return r142239;
}

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 < -19531.357482678304 or 10797.660758833053 < 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 -19531.357482678304 < x < 10797.660758833053

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -19531.3574826783042 \lor \neg \left(x \le 10797.6607588330535\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right) - \frac{x + 1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))