Average Error: 63.0 → 0
Time: 2.8min
Precision: binary64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[1 \cdot \log n + \frac{0.5}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
1 \cdot \log n + \frac{0.5}{n}
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) (1.0 * ((double) log(n)))) + (0.5 / n)));
}

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. Taylor expanded around -inf 64.0

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

  3. Simplified0.0

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

  4. Taylor expanded around 0 0

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

  5. Simplified0

    \[\leadsto \color{blue}{1 \cdot \log n + \frac{0.5}{n}}\]

  6. Final simplification0

    \[\leadsto 1 \cdot \log n + \frac{0.5}{n}\]

Reproduce

herbie shell --seed 2020181 
(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))