Average Error: 9.9 → 0.1
Time: 1.1m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x + 1}}{\left(x - 1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x + 1}}{\left(x - 1\right) \cdot x}
double f(double x) {
        double r10997838 = 1.0;
        double r10997839 = x;
        double r10997840 = r10997839 + r10997838;
        double r10997841 = r10997838 / r10997840;
        double r10997842 = 2.0;
        double r10997843 = r10997842 / r10997839;
        double r10997844 = r10997841 - r10997843;
        double r10997845 = r10997839 - r10997838;
        double r10997846 = r10997838 / r10997845;
        double r10997847 = r10997844 + r10997846;
        return r10997847;
}
↓
double f(double x) {
        double r10997848 = 2.0;
        double r10997849 = x;
        double r10997850 = 1.0;
        double r10997851 = r10997849 + r10997850;
        double r10997852 = r10997848 / r10997851;
        double r10997853 = r10997849 - r10997850;
        double r10997854 = r10997853 * r10997849;
        double r10997855 = r10997852 / r10997854;
        return r10997855;
}

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 associate-+l-9.9

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

  5. Applied frac-sub26.0

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

  6. Applied frac-sub25.1

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

  7. Simplified8.9

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

  8. Taylor expanded around -inf 0.3

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

  10. Applied associate-/r*0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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