Average Error: 9.5 → 0.0
Time: 3.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -115.218709183760566 \lor \neg \left(x \le 113.211583501684103\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right) - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -115.218709183760566 \lor \neg \left(x \le 113.211583501684103\right):\\
\;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\

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

\end{array}
double code(double x) {
	return ((double) (((double) (((double) (1.0 / ((double) (x + 1.0)))) - ((double) (2.0 / x)))) + ((double) (1.0 / ((double) (x - 1.0))))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -115.21870918376057) || !(x <= 113.2115835016841))) {
		VAR = ((double) (2.0 * ((double) (((double) (1.0 / ((double) pow(x, 7.0)))) + ((double) (((double) (1.0 / ((double) pow(x, 5.0)))) + ((double) pow(x, ((double) -(3.0))))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (1.0 / ((double) (((double) (x * x)) - ((double) (1.0 * 1.0)))))) * ((double) (x - 1.0)))) - ((double) (2.0 / x)))) + ((double) (1.0 / ((double) (x - 1.0))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.5
Target0.3
Herbie0.0
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -115.21870918376057 or 113.2115835016841 < x

    1. Initial program 19.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

    2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.5

      \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)}\]
    4. Using strategy rm

    5. Applied pow-flip0.0

      \[\leadsto 2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \color{blue}{{x}^{\left(-3\right)}}\right)\right)\]

    if -115.21870918376057 < x < 113.2115835016841

    1. Initial program 0.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
    2. Using strategy rm

    3. Applied flip-+0.0

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

    4. Applied associate-/r/0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -115.218709183760566 \lor \neg \left(x \le 113.211583501684103\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right) - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))