Average Error: 9.7 → 0.1
Time: 4.3s
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 r140979 = 1.0;
        double r140980 = x;
        double r140981 = r140980 + r140979;
        double r140982 = r140979 / r140981;
        double r140983 = 2.0;
        double r140984 = r140983 / r140980;
        double r140985 = r140982 - r140984;
        double r140986 = r140980 - r140979;
        double r140987 = r140979 / r140986;
        double r140988 = r140985 + r140987;
        return r140988;
}
↓
double f(double x) {
        double r140989 = 2.0;
        double r140990 = x;
        double r140991 = 1.0;
        double r140992 = r140990 - r140991;
        double r140993 = r140989 / r140992;
        double r140994 = r140990 + r140991;
        double r140995 = r140994 * r140990;
        double r140996 = r140993 / r140995;
        return r140996;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

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

  3. Applied frac-sub25.5

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

  4. Applied frac-add25.0

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

  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. Using strategy rm

  7. Applied *-un-lft-identity0.3

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

  8. Applied times-frac0.1

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

  10. Applied pow10.1

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

  11. Applied pow10.1

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

  12. Applied pow-prod-down0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020035 
(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))))