Average Error: 14.3 → 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 r42017 = 1.0;
        double r42018 = x;
        double r42019 = r42018 + r42017;
        double r42020 = r42017 / r42019;
        double r42021 = r42017 / r42018;
        double r42022 = r42020 - r42021;
        return r42022;
}
↓
double f(double x) {
        double r42023 = 1.0;
        double r42024 = x;
        double r42025 = r42024 + r42023;
        double r42026 = r42023 / r42025;
        double r42027 = -r42023;
        double r42028 = r42026 * r42027;
        double r42029 = r42028 / r42024;
        return r42029;
}

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-sub13.7

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

  4. Simplified13.7

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

  6. Applied associate-/r*13.7

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

  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 2020046 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))