Average Error: 14.7 → 0.1
Time: 13.1s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{1 \cdot \left(-1\right)}{x + 1}}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\frac{1 \cdot \left(-1\right)}{x + 1}}{x}
double f(double x) {
        double r2653556 = 1.0;
        double r2653557 = x;
        double r2653558 = r2653557 + r2653556;
        double r2653559 = r2653556 / r2653558;
        double r2653560 = r2653556 / r2653557;
        double r2653561 = r2653559 - r2653560;
        return r2653561;
}
↓
double f(double x) {
        double r2653562 = 1.0;
        double r2653563 = -r2653562;
        double r2653564 = r2653562 * r2653563;
        double r2653565 = x;
        double r2653566 = r2653565 + r2653562;
        double r2653567 = r2653564 / r2653566;
        double r2653568 = r2653567 / r2653565;
        return r2653568;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.7

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

  3. Applied frac-sub14.1

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

  4. Simplified14.1

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

  5. Taylor expanded around 0 0.3

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))