Average Error: 14.8 → 0.1
Time: 11.1s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-\frac{1}{x + 1}}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{-\frac{1}{x + 1}}{x}
double f(double x) {
        double r34846 = 1.0;
        double r34847 = x;
        double r34848 = r34847 + r34846;
        double r34849 = r34846 / r34848;
        double r34850 = r34846 / r34847;
        double r34851 = r34849 - r34850;
        return r34851;
}
↓
double f(double x) {
        double r34852 = 1.0;
        double r34853 = x;
        double r34854 = r34853 + r34852;
        double r34855 = r34852 / r34854;
        double r34856 = -r34855;
        double r34857 = r34856 / r34853;
        return r34857;
}

Error

Bits error versus x

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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 0.3

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

  7. Applied associate-/r*0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))