Average Error: 10.2 → 0.3
Time: 9.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}
double f(double x) {
        double r140784 = 1.0;
        double r140785 = x;
        double r140786 = r140785 + r140784;
        double r140787 = r140784 / r140786;
        double r140788 = 2.0;
        double r140789 = r140788 / r140785;
        double r140790 = r140787 - r140789;
        double r140791 = r140785 - r140784;
        double r140792 = r140784 / r140791;
        double r140793 = r140790 + r140792;
        return r140793;
}
↓
double f(double x) {
        double r140794 = 2.0;
        double r140795 = x;
        double r140796 = 1.0;
        double r140797 = r140795 + r140796;
        double r140798 = r140797 * r140795;
        double r140799 = r140795 - r140796;
        double r140800 = r140798 * r140799;
        double r140801 = r140794 / r140800;
        return r140801;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.2
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.2

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

  3. Applied frac-sub26.1

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

  4. Applied frac-add25.7

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 197574269 
(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))))