Average Error: 9.8 → 0.1
Time: 20.1min
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{1}{x} \cdot \frac{2}{\left(x - 1\right) \cdot \left(x + 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{1}{x} \cdot \frac{2}{\left(x - 1\right) \cdot \left(x + 1\right)}
double code(double x) {
	return ((double) (((double) ((1.0 / ((double) (x + 1.0))) - (2.0 / x))) + (1.0 / ((double) (x - 1.0)))));
}
↓
double code(double x) {
	return ((double) ((1.0 / x) * (2.0 / ((double) (((double) (x - 1.0)) * ((double) (x + 1.0)))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.8

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

  3. Applied frac-sub26.0

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

  4. Applied frac-add25.6

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

  5. Simplified25.6

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

  6. Taylor expanded around 0 0.3

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

  8. Applied *-un-lft-identity0.3

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

  9. Applied times-frac0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020181 
(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))))