Average Error: 63.0 → 0
Time: 19.1s
Precision: 64
Internal Precision: 128
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{-1}{6}}{n \cdot n} + \left(\frac{\frac{1}{2}}{n} + \log n\right)\]

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. Initial simplification62.0

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

  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

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

  5. Final simplification0

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

Runtime

Time bar (total: 19.1s)Debug logProfile

start304.0ms

Algorithm
intervals

setup265.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 62.0b

localize30.0ms

Local error

Found 3 expressions with local error:

4.6b
(- (* (+ 1 n) (log (+ 1 n))) (- (* (log n) n) -1))
0.2b
(* (log n) n)
0.2b
(* (+ 1 n) (log (+ 1 n)))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
3 → 62
Calls

3 calls. Slowest were:

18.0ms
(- (* (+ 1 n) (log (+ 1 n))) (- (* (log n) n) -1))
4.0ms
(* (+ 1 n) (log (+ 1 n)))
2.0ms
(* (log n) n)

series388.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

242.0ms
(* (log n) n)
97.0ms
(- (* (+ 1 n) (log (+ 1 n))) (- (* (log n) n) -1))
50.0ms
(* (+ 1 n) (log (+ 1 n)))

simplify6.1s

Counts
45 → 71
Calls

45 calls. Slowest were:

762.0ms
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* n (log (/ 1 n))) (log (/ 1 n))))
645.0ms
(- (+ (* n (log -1)) (+ 1 (+ (log -1) (* 1/2 (/ 1 n))))) (+ (* n (log (/ -1 n))) (log (/ -1 n))))
532.0ms
(* (- 1 n) (+ (* (log n) n) -1))

prune901.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize31.0ms

Local error

Found 1 expressions with local error:

1.5b
(/ -1/6 (* n n))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 11
Calls

1 calls. Slowest were:

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

series13.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

13.0ms
(/ -1/6 (* n n))

simplify10.0ms

Counts
3 → 14
Calls

3 calls. Slowest were:

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

prune211.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018357 
(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))