Average Error: 9.8 → 0.2
Time: 13.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{{x}^{3} - 1 \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{{x}^{3} - 1 \cdot x}
double f(double x) {
        double r66906 = 1.0;
        double r66907 = x;
        double r66908 = r66907 + r66906;
        double r66909 = r66906 / r66908;
        double r66910 = 2.0;
        double r66911 = r66910 / r66907;
        double r66912 = r66909 - r66911;
        double r66913 = r66907 - r66906;
        double r66914 = r66906 / r66913;
        double r66915 = r66912 + r66914;
        return r66915;
}
↓
double f(double x) {
        double r66916 = 2.0;
        double r66917 = x;
        double r66918 = 3.0;
        double r66919 = pow(r66917, r66918);
        double r66920 = 1.0;
        double r66921 = r66920 * r66917;
        double r66922 = r66919 - r66921;
        double r66923 = r66916 / r66922;
        return r66923;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.2
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.8

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

  3. Applied frac-sub25.9

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

  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. Taylor expanded around 0 0.2

    \[\leadsto \frac{2}{\color{blue}{{x}^{3} - 1 \cdot x}}\]

  7. Final simplification0.2

    \[\leadsto \frac{2}{{x}^{3} - 1 \cdot x}\]

Reproduce

herbie shell --seed 2019303 
(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))))