Average Error: 9.6 → 0.1
Time: 20.7s
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 r101521 = 1.0;
        double r101522 = x;
        double r101523 = r101522 + r101521;
        double r101524 = r101521 / r101523;
        double r101525 = 2.0;
        double r101526 = r101525 / r101522;
        double r101527 = r101524 - r101526;
        double r101528 = r101522 - r101521;
        double r101529 = r101521 / r101528;
        double r101530 = r101527 + r101529;
        return r101530;
}
↓
double f(double x) {
        double r101531 = 2.0;
        double r101532 = x;
        double r101533 = r101531 / r101532;
        double r101534 = r101532 * r101532;
        double r101535 = 1.0;
        double r101536 = r101534 - r101535;
        double r101537 = r101533 / r101536;
        return r101537;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.6

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

  3. Applied frac-sub25.5

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

  4. Applied frac-add25.1

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

  5. Simplified25.5

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

  6. Taylor expanded around 0 0.3

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

  7. Taylor expanded around 0 0.2

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

  9. Applied unpow30.2

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

  10. Applied distribute-rgt-out--0.2

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

  11. Applied associate-/r*0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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))))