Average Error: 14.8 → 0.1
Time: 11.7s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[-\frac{\frac{1}{1 + x}}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
-\frac{\frac{1}{1 + x}}{x}
double f(double x) {
        double r65121 = 1.0;
        double r65122 = x;
        double r65123 = r65122 + r65121;
        double r65124 = r65121 / r65123;
        double r65125 = r65121 / r65122;
        double r65126 = r65124 - r65125;
        return r65126;
}
↓
double f(double x) {
        double r65127 = 1.0;
        double r65128 = x;
        double r65129 = r65127 + r65128;
        double r65130 = r65127 / r65129;
        double r65131 = r65130 / r65128;
        double r65132 = -r65131;
        return r65132;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.8

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

  3. Applied frac-sub14.2

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

  4. Simplified14.2

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

  5. Taylor expanded around 0 0.4

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

  7. Applied neg-sub00.4

    \[\leadsto \frac{\color{blue}{0 - 1}}{\left(x + 1\right) \cdot x}\]

  8. Applied div-sub0.4

    \[\leadsto \color{blue}{\frac{0}{\left(x + 1\right) \cdot x} - \frac{1}{\left(x + 1\right) \cdot x}}\]

  9. Simplified0.4

    \[\leadsto \color{blue}{0} - \frac{1}{\left(x + 1\right) \cdot x}\]

  10. Simplified0.1

    \[\leadsto 0 - \color{blue}{\frac{\frac{1}{1 + x}}{x}}\]

  11. Final simplification0.1

    \[\leadsto -\frac{\frac{1}{1 + x}}{x}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))