Average Error: 9.9 → 0.2
Time: 2.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{-2}{1 \cdot x - {x}^{3}}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{-2}{1 \cdot x - {x}^{3}}
double code(double x) {
	return (((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0)));
}
↓
double code(double x) {
	return (-2.0 / ((1.0 * x) - pow(x, 3.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.2
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.9

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

  3. Applied frac-2neg9.9

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

  4. Applied frac-sub26.3

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

  5. Applied frac-add25.7

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

  6. Simplified25.7

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

  7. Simplified25.7

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

  8. Taylor expanded around 0 0.2

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

  9. Taylor expanded around 0 0.2

    \[\leadsto \frac{-2}{\color{blue}{1 \cdot x - {x}^{3}}}\]

  10. Final simplification0.2

    \[\leadsto \frac{-2}{1 \cdot x - {x}^{3}}\]

Reproduce

herbie shell --seed 2020078 
(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))))