Average Error: 9.9 → 0.1
Time: 3.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{x + 1}}{\frac{x}{2} \cdot \left(x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{1}{x + 1}}{\frac{x}{2} \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 ((1.0 / (x + 1.0)) / ((x / 2.0) * (x - 1.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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 clear-num9.9

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

  5. Applied frac-2neg9.9

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

  6. Applied frac-sub26.3

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

  7. Applied frac-add25.7

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

  8. Simplified26.2

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

  9. Simplified26.2

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

  10. Taylor expanded around 0 0.2

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

  12. Applied associate-/r*0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020078 +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))))