Average Error: 15.0 → 0.1
Time: 7.5s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{1}{x + 1} \cdot \left(-1\right)}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\frac{1}{x + 1} \cdot \left(-1\right)}{x}
double f(double x) {
        double r50619 = 1.0;
        double r50620 = x;
        double r50621 = r50620 + r50619;
        double r50622 = r50619 / r50621;
        double r50623 = r50619 / r50620;
        double r50624 = r50622 - r50623;
        return r50624;
}
↓
double f(double x) {
        double r50625 = 1.0;
        double r50626 = x;
        double r50627 = r50626 + r50625;
        double r50628 = r50625 / r50627;
        double r50629 = -r50625;
        double r50630 = r50628 * r50629;
        double r50631 = r50630 / r50626;
        return r50631;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.0

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

  3. Applied frac-sub14.3

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

  4. Simplified14.3

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

  6. Applied associate-/r*14.3

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))