Average Error: 10.0 → 0.1
Time: 3.5s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x + 1}}{\left(x - 1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x + 1}}{\left(x - 1\right) \cdot x}
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) {
	return ((double) (((double) (2.0 / ((double) (x + 1.0)))) / ((double) (((double) (x - 1.0)) * x))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.0

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

  3. Applied sub-neg10.0

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

  4. Applied associate-+l+10.0

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

  5. Simplified10.0

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

  7. Applied frac-sub26.0

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

  8. Applied frac-add25.3

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

  9. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(x + 1\right) \cdot \left(\left(x - 1\right) \cdot x\right)}\]
  10. Using strategy rm

  11. Applied associate-/r*0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020130 
(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))))