Average Error: 63.0 → 0
Time: 13.0s
Precision: 64
Internal Precision: 128
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{\frac{-1}{6}}{n}}{n} + \left(\frac{\frac{1}{2}}{n} + \log n\right)\]

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. Simplified62.0

    \[\leadsto \color{blue}{(n \cdot \left(\log_* (1 + n)\right) + \left(\log_* (1 + n)\right))_* - (n \cdot \left(\log n\right) + 1)_*}\]

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019007 +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))

Details

Time bar (total: 12.7s)Debug log

sample166.0ms

Algorithm
intervals

simplify456.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

456.0ms
(- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1)

prune24.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 62.0b

localize24.0ms

Local error

Found 3 expressions with local error:

4.6b
(- (fma n (log1p n) (log1p n)) (fma n (log n) 1))
0.2b
(fma n (log1p n) (log1p n))
0.2b
(fma n (log n) 1)

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
3 → 50
Calls

3 calls. Slowest were:

4.0ms
(- (fma n (log1p n) (log1p n)) (fma n (log n) 1))
0.0ms
(fma n (log1p n) (log1p n))
0.0ms
(fma n (log n) 1)

series435.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

291.0ms
(fma n (log n) 1)
99.0ms
(- (fma n (log1p n) (log1p n)) (fma n (log n) 1))
44.0ms
(fma n (log1p n) (log1p n))

simplify5.0s

Counts
34 → 59
Calls

34 calls. Slowest were:

600.0ms
(- (+ n (* 1/2 (pow n 2))) (+ (* n (log n)) 1))
496.0ms
(- (+ (log -1) (* 1/2 (/ 1 n))) (+ (* 1/6 (/ 1 (pow n 2))) (log (/ -1 n))))
459.0ms
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* n (log (/ 1 n))) (log (/ 1 n))))

prune260.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize15.0ms

Local error

Found 2 expressions with local error:

0.4b
(/ -1/6 n)
0.1b
(/ (/ -1/6 n) n)

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 29
Calls

2 calls. Slowest were:

4.0ms
(/ (/ -1/6 n) n)
0.0ms
(/ -1/6 n)

series22.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

11.0ms
(/ -1/6 n)
11.0ms
(/ (/ -1/6 n) n)

simplify19.0ms

Counts
8 → 35
Calls

8 calls. Slowest were:

6.0ms
(/ -1/6 (pow n 2))
5.0ms
(/ -1/6 (pow n 2))
3.0ms
(/ -1/6 (pow n 2))

prune254.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample6.0s

Algorithm
intervals