Average Error: 9.4 → 9.6
Time: 21.5s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.356774142317772 \cdot 10^{+154} \lor \neg \left(x \leq 1.865958468837717 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{{x}^{3} - x}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.356774142317772 \cdot 10^{+154} \lor \neg \left(x \leq 1.865958468837717 \cdot 10^{+117}\right):\\
\;\;\;\;\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\\

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

\end{array}
(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -1.356774142317772e+154) (not (<= x 1.865958468837717e+117)))
   (/
    (-
     (* (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)))
     (/ (/ 1.0 (- x 1.0)) (- x 1.0)))
    (- (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
   (/
    (+ (* (- x 1.0) (- x (* (+ x 1.0) 2.0))) (* x (+ x 1.0)))
    (- (pow x 3.0) x))))
double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -1.356774142317772e+154) || !(x <= 1.865958468837717e+117)) {
		tmp = ((((1.0 / (x + 1.0)) - (2.0 / x)) * ((1.0 / (x + 1.0)) - (2.0 / x))) - ((1.0 / (x - 1.0)) / (x - 1.0))) / (((1.0 / (x + 1.0)) - (2.0 / x)) - (1.0 / (x - 1.0)));
	} else {
		tmp = (((x - 1.0) * (x - ((x + 1.0) * 2.0))) + (x * (x + 1.0))) / (pow(x, 3.0) - x);
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.4
Target0.3
Herbie9.6
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.35677414231777208e154 or 1.8659584688377169e117 < x

    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-+_binary640

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

    4. Simplified1.6

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

    5. Simplified1.6

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

    if -1.35677414231777208e154 < x < 1.8659584688377169e117

    1. Initial program 13.0

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

    3. Applied frac-sub_binary6414.2

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

    4. Applied frac-add_binary6412.6

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

    5. Simplified12.6

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

    6. Simplified12.6

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.356774142317772 \cdot 10^{+154} \lor \neg \left(x \leq 1.865958468837717 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{{x}^{3} - x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))