Average Error: 9.9 → 0.1
Time: 7.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x}}{x \cdot x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x}}{x \cdot x - 1}
double f(double x) {
        double r88614 = 1.0;
        double r88615 = x;
        double r88616 = r88615 + r88614;
        double r88617 = r88614 / r88616;
        double r88618 = 2.0;
        double r88619 = r88618 / r88615;
        double r88620 = r88617 - r88619;
        double r88621 = r88615 - r88614;
        double r88622 = r88614 / r88621;
        double r88623 = r88620 + r88622;
        return r88623;
}
↓
double f(double x) {
        double r88624 = 2.0;
        double r88625 = x;
        double r88626 = r88624 / r88625;
        double r88627 = r88625 * r88625;
        double r88628 = 1.0;
        double r88629 = r88627 - r88628;
        double r88630 = r88626 / r88629;
        return r88630;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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-sub25.6

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

  4. Applied frac-add25.0

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

  5. Taylor expanded around 0 0.3

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

  6. Taylor expanded around 0 0.3

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

  8. Applied unpow30.3

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

  9. Applied distribute-rgt-out--0.3

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

  10. Applied associate-/r*0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019350 
(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))))