Average Error: 10.1 → 0.1
Time: 1.2m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{\frac{x}{2} \cdot \left(x + 1\right)}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{1}{\frac{x}{2} \cdot \left(x + 1\right)}}{x - 1}
double f(double x) {
        double r6903713 = 1.0;
        double r6903714 = x;
        double r6903715 = r6903714 + r6903713;
        double r6903716 = r6903713 / r6903715;
        double r6903717 = 2.0;
        double r6903718 = r6903717 / r6903714;
        double r6903719 = r6903716 - r6903718;
        double r6903720 = r6903714 - r6903713;
        double r6903721 = r6903713 / r6903720;
        double r6903722 = r6903719 + r6903721;
        return r6903722;
}
↓
double f(double x) {
        double r6903723 = 1.0;
        double r6903724 = x;
        double r6903725 = 2.0;
        double r6903726 = r6903724 / r6903725;
        double r6903727 = r6903724 + r6903723;
        double r6903728 = r6903726 * r6903727;
        double r6903729 = r6903723 / r6903728;
        double r6903730 = r6903724 - r6903723;
        double r6903731 = r6903729 / r6903730;
        return r6903731;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 10.1

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

  3. Applied clear-num10.1

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

  4. Applied frac-sub25.5

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

  5. Applied frac-add24.9

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

  6. Simplified24.9

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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