Average Error: 14.3 → 0.4
Time: 37.0s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{1}{x \cdot \left(-1 - x\right)}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{1}{x \cdot \left(-1 - x\right)}
double f(double x) {
        double r1006783 = 1.0;
        double r1006784 = x;
        double r1006785 = r1006784 + r1006783;
        double r1006786 = r1006783 / r1006785;
        double r1006787 = r1006783 / r1006784;
        double r1006788 = r1006786 - r1006787;
        return r1006788;
}
↓
double f(double x) {
        double r1006789 = 1.0;
        double r1006790 = x;
        double r1006791 = -1.0;
        double r1006792 = r1006791 - r1006790;
        double r1006793 = r1006790 * r1006792;
        double r1006794 = r1006789 / r1006793;
        return r1006794;
}

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

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

  5. Simplified13.7

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

  7. Applied frac-2neg13.7

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))