Average Error: 14.3 → 0.1
Time: 3.2s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{1}{\frac{x + 1}{0 - 1}}}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\frac{1}{\frac{x + 1}{0 - 1}}}{x}
double f(double x) {
        double r24981 = 1.0;
        double r24982 = x;
        double r24983 = r24982 + r24981;
        double r24984 = r24981 / r24983;
        double r24985 = r24981 / r24982;
        double r24986 = r24984 - r24985;
        return r24986;
}
↓
double f(double x) {
        double r24987 = 1.0;
        double r24988 = x;
        double r24989 = r24988 + r24987;
        double r24990 = 0.0;
        double r24991 = r24990 - r24987;
        double r24992 = r24989 / r24991;
        double r24993 = r24987 / r24992;
        double r24994 = r24993 / r24988;
        return r24994;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.3

    \[\frac{1}{x + 1} - \frac{1}{x}\]
  2. Using strategy rm

  3. Applied frac-sub13.8

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

  4. Simplified13.8

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

  6. Applied associate-/r*13.8

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

  7. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x + 1}{0 - 1}}}}{x}\]

  8. Final simplification0.1

    \[\leadsto \frac{\frac{1}{\frac{x + 1}{0 - 1}}}{x}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))