Average Error: 9.6 → 0.1
Time: 3.9m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{-\frac{1}{-\frac{x + 1}{1}}}{\left(x - 1\right) \cdot \frac{x}{2}}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{-\frac{1}{-\frac{x + 1}{1}}}{\left(x - 1\right) \cdot \frac{x}{2}}
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) / 1.0)) / ((x - 1.0) * (x / 2.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.6

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

  3. Applied frac-2neg9.6

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

  4. Applied clear-num9.6

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

  5. Applied clear-num9.6

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

  6. Applied frac-sub25.4

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

  7. Applied frac-add25.0

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

  8. Taylor expanded around 0 0.2

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

  10. Applied *-un-lft-identity0.2

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

  11. Applied distribute-lft-neg-in0.2

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

  12. Applied associate-*r*0.2

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

  13. Applied associate-/r*0.1

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

  14. Simplified0.1

    \[\leadsto \frac{\color{blue}{-\frac{1}{-\frac{x + 1}{1} \cdot \frac{x}{2}}}}{x - 1}\]
  15. Using strategy rm

  16. Applied distribute-lft-neg-in0.1

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

  17. Applied associate-/r*0.1

    \[\leadsto \frac{-\color{blue}{\frac{\frac{1}{-\frac{x + 1}{1}}}{\frac{x}{2}}}}{x - 1}\]

  18. Applied distribute-neg-frac0.1

    \[\leadsto \frac{\color{blue}{\frac{-\frac{1}{-\frac{x + 1}{1}}}{\frac{x}{2}}}}{x - 1}\]

  19. Applied associate-/l/0.1

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

  20. Final simplification0.1

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

Reproduce

herbie shell --seed 2020066 
(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))))