Average Error: 9.8 → 0.1
Time: 2.8s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x \cdot x - 1}}{x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x \cdot x - 1}}{x}
(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ (/ 2.0 (- (* x x) 1.0)) x))
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 * x) - 1.0)) / x;
}

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-sub_binary64_143426.0

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

  4. Applied frac-add_binary64_143325.5

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

  5. Simplified25.5

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

  6. Simplified25.5

    \[\leadsto \frac{\left(x - 1\right) \cdot \left(x - \left(1 + x\right) \cdot 2\right) + x \cdot \left(1 + x\right)}{\color{blue}{{x}^{3} - x}}\]

  7. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{{x}^{3} - x}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity_binary64_14250.3

    \[\leadsto \frac{2}{{x}^{3} - \color{blue}{1 \cdot x}}\]

  10. Applied unpow3_binary64_14880.3

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

  11. Applied distribute-rgt-out--_binary64_13810.3

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

  12. Applied *-un-lft-identity_binary64_14250.3

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

  13. Applied times-frac_binary64_14310.1

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

  15. Applied associate-*l/_binary64_13700.1

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

  16. Simplified0.1

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

  17. Final simplification0.1

    \[\leadsto \frac{\frac{2}{x \cdot x - 1}}{x}\]

Reproduce

herbie shell --seed 2020281 
(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))))