Average Error: 9.9 → 0.1
Time: 18.0s
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 r6853645 = 1.0;
        double r6853646 = x;
        double r6853647 = r6853646 + r6853645;
        double r6853648 = r6853645 / r6853647;
        double r6853649 = 2.0;
        double r6853650 = r6853649 / r6853646;
        double r6853651 = r6853648 - r6853650;
        double r6853652 = r6853646 - r6853645;
        double r6853653 = r6853645 / r6853652;
        double r6853654 = r6853651 + r6853653;
        return r6853654;
}
↓
double f(double x) {
        double r6853655 = 2.0;
        double r6853656 = x;
        double r6853657 = 1.0;
        double r6853658 = r6853656 + r6853657;
        double r6853659 = r6853658 * r6853656;
        double r6853660 = r6853655 / r6853659;
        double r6853661 = r6853656 - r6853657;
        double r6853662 = r6853660 / r6853661;
        return r6853662;
}

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. Using strategy rm

  3. Applied frac-sub25.7

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

  4. Applied frac-add25.3

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

  5. Taylor expanded around 0 0.3

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 
(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))))