Average Error: 63.0 → 0
Time: 17.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{\frac{-1}{6}}{n \cdot n} + \log n\right) + \frac{\frac{1}{2}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{\frac{-1}{6}}{n \cdot n} + \log n\right) + \frac{\frac{1}{2}}{n}
double f(double n) {
        double r1675752 = n;
        double r1675753 = 1.0;
        double r1675754 = r1675752 + r1675753;
        double r1675755 = log(r1675754);
        double r1675756 = r1675754 * r1675755;
        double r1675757 = log(r1675752);
        double r1675758 = r1675752 * r1675757;
        double r1675759 = r1675756 - r1675758;
        double r1675760 = r1675759 - r1675753;
        return r1675760;
}
↓
double f(double n) {
        double r1675761 = -0.16666666666666666;
        double r1675762 = n;
        double r1675763 = r1675762 * r1675762;
        double r1675764 = r1675761 / r1675763;
        double r1675765 = log(r1675762);
        double r1675766 = r1675764 + r1675765;
        double r1675767 = 0.5;
        double r1675768 = r1675767 / r1675762;
        double r1675769 = r1675766 + r1675768;
        return r1675769;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

Derivation

  1. Initial program 63.0

    \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

  2. Simplified44.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(n\right) + \mathsf{fma}\left(n, \mathsf{log1p}\left(n\right) - \log n, -1\right)}\]

  3. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)}\]

  4. Simplified0

    \[\leadsto \color{blue}{\left(\frac{\frac{-1}{6}}{n \cdot n} + \log n\right) + \frac{\frac{1}{2}}{n}}\]

  5. Final simplification0

    \[\leadsto \left(\frac{\frac{-1}{6}}{n \cdot n} + \log n\right) + \frac{\frac{1}{2}}{n}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1)) (- (/ 1 (* 2 n)) (- (/ 1 (* 3 (* n n))) (/ 4 (pow n 3)))))

  (- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1))