Average Error: 14.8 → 0.1
Time: 21.0s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{x + 1}\right)\right)}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{x + 1}\right)\right)}{x}
double f(double x) {
        double r3893651 = 1.0;
        double r3893652 = x;
        double r3893653 = r3893652 + r3893651;
        double r3893654 = r3893651 / r3893653;
        double r3893655 = r3893651 / r3893652;
        double r3893656 = r3893654 - r3893655;
        return r3893656;
}
↓
double f(double x) {
        double r3893657 = 1.0;
        double r3893658 = -r3893657;
        double r3893659 = x;
        double r3893660 = r3893659 + r3893657;
        double r3893661 = r3893658 / r3893660;
        double r3893662 = expm1(r3893661);
        double r3893663 = log1p(r3893662);
        double r3893664 = r3893663 / r3893659;
        return r3893664;
}

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-sub14.1

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

  4. Taylor expanded around 0 0.4

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

  6. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{-1}{x + 1}}{x}}\]
  7. Using strategy rm

  8. Applied log1p-expm1-u0.1

    \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{x + 1}\right)\right)}}{x}\]

  9. Final simplification0.1

    \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{x + 1}\right)\right)}{x}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))