Average Error: 10.0 → 0.3
Time: 57.4s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(\left(1 + x\right) \cdot \left(x - 1\right)\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(\left(1 + x\right) \cdot \left(x - 1\right)\right) \cdot x}
double f(double x) {
        double r4670514 = 1.0;
        double r4670515 = x;
        double r4670516 = r4670515 + r4670514;
        double r4670517 = r4670514 / r4670516;
        double r4670518 = 2.0;
        double r4670519 = r4670518 / r4670515;
        double r4670520 = r4670517 - r4670519;
        double r4670521 = r4670515 - r4670514;
        double r4670522 = r4670514 / r4670521;
        double r4670523 = r4670520 + r4670522;
        return r4670523;
}
↓
double f(double x) {
        double r4670524 = 2.0;
        double r4670525 = 1.0;
        double r4670526 = x;
        double r4670527 = r4670525 + r4670526;
        double r4670528 = r4670526 - r4670525;
        double r4670529 = r4670527 * r4670528;
        double r4670530 = r4670529 * r4670526;
        double r4670531 = r4670524 / r4670530;
        return r4670531;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.2
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.0

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

  3. Applied +-commutative10.0

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

  5. Applied associate-+r-10.1

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

  7. Applied frac-add26.2

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

  8. Applied frac-sub25.8

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

  9. Taylor expanded around 0 0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))