Average Error: 9.4 → 0.1
Time: 2.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{-2}{\left(x + 1\right) \cdot -1}}{x \cdot \left(x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{-2}{\left(x + 1\right) \cdot -1}}{x \cdot \left(x - 1\right)}
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 / ((x + 1.0) * -1.0)) / (x * (x - 1.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.4
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.4

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

  3. Applied frac-2neg9.4

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

  4. Applied frac-sub25.9

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

  5. Applied frac-add25.3

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

  6. Simplified25.7

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

  7. Simplified25.7

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

  8. Taylor expanded around 0 0.3

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

  10. Applied neg-mul-10.3

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

  11. Applied associate-*l*0.3

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

  12. Applied associate-*r*0.3

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

  13. Applied associate-/r*0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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))))