Average Error: 9.7 → 0.1
Time: 4.3s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x}}{x \cdot x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x}}{x \cdot x - 1}
(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 x) 1.0)))
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 * x) - 1.0);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

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

  3. Applied frac-sub_binary6425.7

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

  4. Applied frac-add_binary6425.3

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

  5. Simplified25.4

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

  6. Simplified25.4

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

  8. Applied add-cbrt-cube_binary6425.3

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

  9. Simplified25.3

    \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(x + x \cdot x\right) + \left(-2 - x\right) \cdot \left(x + -1\right)\right)}^{3}}}}{{x}^{3} - x}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity_binary6425.3

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

  12. Applied unpow3_binary6425.3

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

  13. Applied distribute-rgt-out--_binary6425.3

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

  14. Applied associate-/r*_binary6425.3

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

  15. Simplified25.4

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

  16. Taylor expanded around 0 0.1

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

  17. Final simplification0.1

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

Alternatives

Reproduce

herbie shell --seed 2021110 
(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))))