Average Error: 9.9 → 0.2
Time: 24.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(\left(x - 1\right) \cdot \left(x + 1\right)\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(\left(x - 1\right) \cdot \left(x + 1\right)\right) \cdot x}
double f(double x) {
        double r4175424 = 1.0;
        double r4175425 = x;
        double r4175426 = r4175425 + r4175424;
        double r4175427 = r4175424 / r4175426;
        double r4175428 = 2.0;
        double r4175429 = r4175428 / r4175425;
        double r4175430 = r4175427 - r4175429;
        double r4175431 = r4175425 - r4175424;
        double r4175432 = r4175424 / r4175431;
        double r4175433 = r4175430 + r4175432;
        return r4175433;
}
↓
double f(double x) {
        double r4175434 = 2.0;
        double r4175435 = x;
        double r4175436 = 1.0;
        double r4175437 = r4175435 - r4175436;
        double r4175438 = r4175435 + r4175436;
        double r4175439 = r4175437 * r4175438;
        double r4175440 = r4175439 * r4175435;
        double r4175441 = r4175434 / r4175440;
        return r4175441;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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 +-commutative9.9

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

  5. Applied associate-+r-10.0

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

  7. Applied frac-add26.3

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

  8. Applied frac-sub26.0

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

  9. Taylor expanded around 0 0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 +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))))