Average Error: 9.9 → 0.1
Time: 25.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x - 1\right) \cdot x}}{x + 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\left(x - 1\right) \cdot x}}{x + 1}
double f(double x) {
        double r150481 = 1.0;
        double r150482 = x;
        double r150483 = r150482 + r150481;
        double r150484 = r150481 / r150483;
        double r150485 = 2.0;
        double r150486 = r150485 / r150482;
        double r150487 = r150484 - r150486;
        double r150488 = r150482 - r150481;
        double r150489 = r150481 / r150488;
        double r150490 = r150487 + r150489;
        return r150490;
}
↓
double f(double x) {
        double r150491 = 2.0;
        double r150492 = x;
        double r150493 = 1.0;
        double r150494 = r150492 - r150493;
        double r150495 = r150494 * r150492;
        double r150496 = r150491 / r150495;
        double r150497 = r150492 + r150493;
        double r150498 = r150496 / r150497;
        return r150498;
}

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. Simplified9.9

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

  4. Applied frac-sub25.7

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

  5. Applied frac-add25.3

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

  6. Simplified25.7

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

  7. Simplified25.7

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

  8. Taylor expanded around 0 0.3

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

  10. Applied associate-/r*0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +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))))