Average Error: 9.8 → 0.1
Time: 5.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}
double f(double x) {
        double r144836 = 1.0;
        double r144837 = x;
        double r144838 = r144837 + r144836;
        double r144839 = r144836 / r144838;
        double r144840 = 2.0;
        double r144841 = r144840 / r144837;
        double r144842 = r144839 - r144841;
        double r144843 = r144837 - r144836;
        double r144844 = r144836 / r144843;
        double r144845 = r144842 + r144844;
        return r144845;
}
↓
double f(double x) {
        double r144846 = 1.0;
        double r144847 = x;
        double r144848 = 1.0;
        double r144849 = r144847 + r144848;
        double r144850 = r144849 * r144847;
        double r144851 = r144846 / r144850;
        double r144852 = 2.0;
        double r144853 = r144847 - r144848;
        double r144854 = r144852 / r144853;
        double r144855 = r144851 * r144854;
        return r144855;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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-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-add25.6

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

  5. Simplified26.0

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

  6. Taylor expanded around 0 0.2

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

  8. Applied *-un-lft-identity0.2

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

  9. Applied times-frac0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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))))