Average Error: 9.7 → 0.1
Time: 12.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x \cdot x - 1 \cdot 1}}{x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x \cdot x - 1 \cdot 1}}{x}
double f(double x) {
        double r95623 = 1.0;
        double r95624 = x;
        double r95625 = r95624 + r95623;
        double r95626 = r95623 / r95625;
        double r95627 = 2.0;
        double r95628 = r95627 / r95624;
        double r95629 = r95626 - r95628;
        double r95630 = r95624 - r95623;
        double r95631 = r95623 / r95630;
        double r95632 = r95629 + r95631;
        return r95632;
}
↓
double f(double x) {
        double r95633 = 2.0;
        double r95634 = x;
        double r95635 = r95634 * r95634;
        double r95636 = 1.0;
        double r95637 = r95636 * r95636;
        double r95638 = r95635 - r95637;
        double r95639 = r95633 / r95638;
        double r95640 = r95639 / r95634;
        return r95640;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

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

  2. Simplified9.7

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

  4. Applied frac-sub26.0

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

  5. Applied frac-sub25.6

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

  6. Taylor expanded around 0 0.3

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

  8. Applied associate-/r*0.1

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

  10. Applied flip--0.1

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

  11. Applied associate-/r/0.1

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

  12. Applied times-frac0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019305 
(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))))