Average Error: 63.0 → 0.0
Time: 4.0s
Precision: binary64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 + \left(0.5 \cdot \frac{1}{n} + 0.29166666666666663 \cdot \frac{1}{{n}^{3}}\right)\right) - 1 \cdot \log \left(\frac{1}{n}\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 + \left(0.5 \cdot \frac{1}{n} + 0.29166666666666663 \cdot \frac{1}{{n}^{3}}\right)\right) - 1 \cdot \log \left(\frac{1}{n}\right)\right) - 1
double code(double n) {
	return ((double) (((double) (((double) (((double) (n + 1.0)) * ((double) log(((double) (n + 1.0)))))) - ((double) (n * ((double) log(n)))))) - 1.0));
}
↓
double code(double n) {
	return ((double) (((double) (((double) (1.0 + ((double) (((double) (0.5 * ((double) (1.0 / n)))) + ((double) (0.29166666666666663 * ((double) (1.0 / ((double) pow(n, 3.0)))))))))) - ((double) (1.0 * ((double) log(((double) (1.0 / n)))))))) - 1.0));
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 63.0

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

  2. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\left(\left(1 + \left(0.5 \cdot \frac{1}{n} + 0.29166666666666663 \cdot \frac{1}{{n}^{3}}\right)\right) - 1 \cdot \log \left(\frac{1}{n}\right)\right)} - 1\]

  3. Final simplification0.0

    \[\leadsto \left(\left(1 + \left(0.5 \cdot \frac{1}{n} + 0.29166666666666663 \cdot \frac{1}{{n}^{3}}\right)\right) - 1 \cdot \log \left(\frac{1}{n}\right)\right) - 1\]

Reproduce

herbie shell --seed 2020147 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))