Average Error: 9.7 → 0.3
Time: 13.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(x - 1\right) \cdot \left(x \cdot \left(x + 1\right)\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(x - 1\right) \cdot \left(x \cdot \left(x + 1\right)\right)}
double f(double x) {
        double r4819741 = 1.0;
        double r4819742 = x;
        double r4819743 = r4819742 + r4819741;
        double r4819744 = r4819741 / r4819743;
        double r4819745 = 2.0;
        double r4819746 = r4819745 / r4819742;
        double r4819747 = r4819744 - r4819746;
        double r4819748 = r4819742 - r4819741;
        double r4819749 = r4819741 / r4819748;
        double r4819750 = r4819747 + r4819749;
        return r4819750;
}
↓
double f(double x) {
        double r4819751 = 2.0;
        double r4819752 = x;
        double r4819753 = 1.0;
        double r4819754 = r4819752 - r4819753;
        double r4819755 = r4819752 + r4819753;
        double r4819756 = r4819752 * r4819755;
        double r4819757 = r4819754 * r4819756;
        double r4819758 = r4819751 / r4819757;
        return r4819758;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 9.7

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

  3. Applied +-commutative9.7

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

  5. Applied frac-sub26.3

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

  6. Applied frac-add25.6

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

  7. Simplified25.6

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

  8. Taylor expanded around 0 0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))