Average Error: 10.0 → 0.1
Time: 1.1m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
\[\frac{\frac{2}{1 + x}}{\left(x - 1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
\frac{\frac{2}{1 + x}}{\left(x - 1\right) \cdot x}
double f(double x) {
        double r5366645 = 1.0;
        double r5366646 = x;
        double r5366647 = r5366646 + r5366645;
        double r5366648 = r5366645 / r5366647;
        double r5366649 = 2.0;
        double r5366650 = r5366649 / r5366646;
        double r5366651 = r5366648 - r5366650;
        double r5366652 = r5366646 - r5366645;
        double r5366653 = r5366645 / r5366652;
        double r5366654 = r5366651 + r5366653;
        return r5366654;
}

double f(double x) {
        double r5366655 = 2.0;
        double r5366656 = 1.0;
        double r5366657 = x;
        double r5366658 = r5366656 + r5366657;
        double r5366659 = r5366655 / r5366658;
        double r5366660 = r5366657 - r5366656;
        double r5366661 = r5366660 * r5366657;
        double r5366662 = r5366659 / r5366661;
        return r5366662;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.0

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  2. Using strategy rm
  3. Applied associate-+l-10.0

    \[\leadsto \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)}\]
  4. Using strategy rm
  5. Applied frac-sub26.2

    \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{2 \cdot \left(x - 1\right) - x \cdot 1}{x \cdot \left(x - 1\right)}}\]
  6. Applied frac-sub25.8

    \[\leadsto \color{blue}{\frac{1 \cdot \left(x \cdot \left(x - 1\right)\right) - \left(x + 1\right) \cdot \left(2 \cdot \left(x - 1\right) - x \cdot 1\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x - 1\right)\right)}}\]
  7. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(x + 1\right) \cdot \left(x \cdot \left(x - 1\right)\right)}\]
  8. Using strategy rm
  9. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{2}{x + 1}}{x \cdot \left(x - 1\right)}}\]
  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019200 +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))))