Average Error: 14.8 → 0.1
Time: 10.2s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[1 \cdot \frac{\frac{-1}{x + 1}}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
1 \cdot \frac{\frac{-1}{x + 1}}{x}
double f(double x) {
        double r60786 = 1.0;
        double r60787 = x;
        double r60788 = r60787 + r60786;
        double r60789 = r60786 / r60788;
        double r60790 = r60786 / r60787;
        double r60791 = r60789 - r60790;
        return r60791;
}
↓
double f(double x) {
        double r60792 = 1.0;
        double r60793 = -1.0;
        double r60794 = x;
        double r60795 = r60794 + r60792;
        double r60796 = r60793 / r60795;
        double r60797 = r60796 / r60794;
        double r60798 = r60792 * r60797;
        return r60798;
}

Error

Bits error versus x

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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 0.3

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

  7. Applied div-inv0.3

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

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