Average Error: 14.3 → 0.1
Time: 16.4s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{-1}{1 + x}}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\frac{-1}{1 + x}}{x}
double f(double x) {
        double r1680165 = 1.0;
        double r1680166 = x;
        double r1680167 = r1680166 + r1680165;
        double r1680168 = r1680165 / r1680167;
        double r1680169 = r1680165 / r1680166;
        double r1680170 = r1680168 - r1680169;
        return r1680170;
}
↓
double f(double x) {
        double r1680171 = -1.0;
        double r1680172 = 1.0;
        double r1680173 = x;
        double r1680174 = r1680172 + r1680173;
        double r1680175 = r1680171 / r1680174;
        double r1680176 = r1680175 / r1680173;
        return r1680176;
}

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

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

  6. Applied associate-/r*13.8

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))