Average Error: 15.0 → 0.1
Time: 6.2s
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 r39094 = 1.0;
        double r39095 = x;
        double r39096 = r39095 + r39094;
        double r39097 = r39094 / r39096;
        double r39098 = r39094 / r39095;
        double r39099 = r39097 - r39098;
        return r39099;
}
↓
double f(double x) {
        double r39100 = 1.0;
        double r39101 = x;
        double r39102 = r39101 + r39100;
        double r39103 = r39100 / r39102;
        double r39104 = -r39100;
        double r39105 = r39103 * r39104;
        double r39106 = r39105 / r39101;
        return r39106;
}

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)))