Average Error: 9.5 → 0.1
Time: 7.7s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -35323144.403240085 \lor \neg \left(x \leq 7216120.091792685\right):\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(\left(\frac{2}{x \cdot x} + \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{3}}\right)}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{{x}^{3} - x}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -35323144.403240085 \lor \neg \left(x \leq 7216120.091792685\right):\\
\;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(\left(\frac{2}{x \cdot x} + \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{3}}\right)}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\\

\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 -35323144.403240085) (not (<= x 7216120.091792685)))
   (/
    (*
     (+ (/ 1.0 (+ x 1.0)) (/ 2.0 x))
     (- (+ (/ 2.0 (* x x)) (/ 2.0 (pow x 4.0))) (/ 2.0 (pow x 3.0))))
    (* (+ (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (- 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 <= -35323144.403240085) || !(x <= 7216120.091792685)) {
		tmp = (((1.0 / (x + 1.0)) + (2.0 / x)) * (((2.0 / (x * x)) + (2.0 / pow(x, 4.0))) - (2.0 / pow(x, 3.0)))) / (((1.0 / (x + 1.0)) + (2.0 / x)) * (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

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

Derivation

  1. Split input into 2 regimes

  2. if x < -35323144.403240085 or 7216120.09179268498 < x

    1. Initial program 19.1

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

    3. Applied flip--_binary6453.3

      \[\leadsto \color{blue}{\frac{\frac{1}{x + 1} \cdot \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{2}{x}}{\frac{1}{x + 1} + \frac{2}{x}}} + \frac{1}{x - 1}\]

    4. Applied frac-add_binary6454.4

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

    5. Simplified25.2

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

    6. Simplified25.2

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

    7. Taylor expanded around inf 0.2

      \[\leadsto \frac{\left(\frac{1}{1 + x} + \frac{2}{x}\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{4}}\right) - 2 \cdot \frac{1}{{x}^{3}}\right)}}{\left(x - 1\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}\]

    8. Simplified0.2

      \[\leadsto \frac{\left(\frac{1}{1 + x} + \frac{2}{x}\right) \cdot \color{blue}{\left(\left(\frac{2}{x \cdot x} + \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{3}}\right)}}{\left(x - 1\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}\]

    if -35323144.403240085 < x < 7216120.09179268498

    1. Initial program 0.5

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

    3. Applied frac-sub_binary640.5

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

    4. Applied frac-add_binary640.0

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

    5. Simplified0.0

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

    6. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -35323144.403240085 \lor \neg \left(x \leq 7216120.091792685\right):\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(\left(\frac{2}{x \cdot x} + \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{3}}\right)}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{{x}^{3} - x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020275 
(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))))