Average Error: 63.0 → 0
Time: 12.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\log n + \left(\frac{\frac{-1}{6}}{n \cdot n} + \frac{\frac{1}{2}}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\log n + \left(\frac{\frac{-1}{6}}{n \cdot n} + \frac{\frac{1}{2}}{n}\right)
double f(double n) {
        double r694233 = n;
        double r694234 = 1.0;
        double r694235 = r694233 + r694234;
        double r694236 = log(r694235);
        double r694237 = r694235 * r694236;
        double r694238 = log(r694233);
        double r694239 = r694233 * r694238;
        double r694240 = r694237 - r694239;
        double r694241 = r694240 - r694234;
        return r694241;
}
↓
double f(double n) {
        double r694242 = n;
        double r694243 = log(r694242);
        double r694244 = -0.16666666666666666;
        double r694245 = r694242 * r694242;
        double r694246 = r694244 / r694245;
        double r694247 = 0.5;
        double r694248 = r694247 / r694242;
        double r694249 = r694246 + r694248;
        double r694250 = r694243 + r694249;
        return r694250;
}

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

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

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019130 +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))