Average Error: 63.0 → 0
Time: 13.9s
Precision: 64
Internal Precision: 128
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{-1}{6}}{n \cdot n} + \left(\log n + \frac{\frac{1}{2}}{n}\right)\]

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0.0
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. Initial simplification62.0

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

  3. Taylor expanded around -inf 62.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2018360 +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: 6.9s)Debug log

start147.0ms

Algorithm
intervals

setup178.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 62.0b

localize17.0ms

Local error

Found 3 expressions with local error:

6.1b
(- (fma n (log1p n) (log1p n)) (fma n (log n) 1))
0.3b
(fma n (log1p n) (log1p n))
0.3b
(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)

series461.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

267.0ms
(fma n (log n) 1)
128.0ms
(- (fma n (log1p n) (log1p n)) (fma n (log n) 1))
67.0ms
(fma n (log1p n) (log1p n))

simplify5.5s

Counts
34 → 59
Calls

34 calls. Slowest were:

990.0ms
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* n (log (/ 1 n))) (log (/ 1 n))))
737.0ms
(- (+ n (* 1/2 (pow n 2))) (+ (* n (log n)) 1))
533.0ms
(- (+ (* n (log -1)) (+ 1 (+ (log -1) (* 1/2 (/ 1 n))))) (+ (* n (log (/ -1 n))) (log (/ -1 n))))

prune278.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize27.0ms

Local error

Found 1 expressions with local error:

0.7b
(/ -1/6 (* n n))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 13
Calls

1 calls. Slowest were:

1.0ms
(/ -1/6 (* n n))

series20.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

20.0ms
(/ -1/6 (* n n))

simplify20.0ms

Counts
3 → 16
Calls

3 calls. Slowest were:

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

prune171.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b