Average Error: 10.1 → 0.1
Time: 19.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[2 \cdot \frac{\frac{1}{x + 1}}{x \cdot \left(x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
2 \cdot \frac{\frac{1}{x + 1}}{x \cdot \left(x - 1\right)}
double f(double x) {
        double r112456 = 1.0;
        double r112457 = x;
        double r112458 = r112457 + r112456;
        double r112459 = r112456 / r112458;
        double r112460 = 2.0;
        double r112461 = r112460 / r112457;
        double r112462 = r112459 - r112461;
        double r112463 = r112457 - r112456;
        double r112464 = r112456 / r112463;
        double r112465 = r112462 + r112464;
        return r112465;
}
↓
double f(double x) {
        double r112466 = 2.0;
        double r112467 = 1.0;
        double r112468 = x;
        double r112469 = 1.0;
        double r112470 = r112468 + r112469;
        double r112471 = r112467 / r112470;
        double r112472 = r112468 - r112469;
        double r112473 = r112468 * r112472;
        double r112474 = r112471 / r112473;
        double r112475 = r112466 * r112474;
        return r112475;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 10.1

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

  2. Simplified10.1

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

  4. Applied frac-sub25.9

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

  5. Applied frac-add25.4

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

  6. Simplified25.4

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

  7. Simplified25.4

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

  8. Taylor expanded around 0 0.3

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

  10. Applied div-inv0.3

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 
(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))))