Average Error: 9.9 → 0.3
Time: 4.3s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{{x}^{3} - x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{{x}^{3} - x}
(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ 2.0 (- (pow x 3.0) 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 / (pow(x, 3.0) - x);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

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-sub_binary64_120726.4

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

  4. Applied frac-add_binary64_120625.9

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

  5. Simplified25.9

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

  6. Simplified25.9

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

  7. Taylor expanded around 0 0.3

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

  8. Final simplification0.3

    \[\leadsto \frac{2}{{x}^{3} - x}\]

Reproduce

herbie shell --seed 2020231 
(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))))