Average Error: 9.5 → 0.1
Time: 3.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}
double f(double x) {
        double r120660 = 1.0;
        double r120661 = x;
        double r120662 = r120661 + r120660;
        double r120663 = r120660 / r120662;
        double r120664 = 2.0;
        double r120665 = r120664 / r120661;
        double r120666 = r120663 - r120665;
        double r120667 = r120661 - r120660;
        double r120668 = r120660 / r120667;
        double r120669 = r120666 + r120668;
        return r120669;
}
↓
double f(double x) {
        double r120670 = 2.0;
        double r120671 = x;
        double r120672 = 1.0;
        double r120673 = r120671 + r120672;
        double r120674 = r120670 / r120673;
        double r120675 = r120674 / r120671;
        double r120676 = r120671 - r120672;
        double r120677 = r120675 / r120676;
        return r120677;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.5
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.5

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

  3. Applied frac-sub26.0

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

  4. Applied frac-add25.5

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

  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. Using strategy rm

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 
(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))))