Average Error: 63.0 → 0
Time: 12.9s
Precision: 64
Internal Precision: 128
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{\frac{-1}{6}}{n}}{n} + \left(\frac{\frac{1}{2}}{n} + \log n\right)\]

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Taylor expanded around 0 0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019018 
(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: 12.6s)Debug log

sample147.0ms

Algorithm
intervals
Results
80.0ms118×body1280valid
34.0ms74×body640valid
16.0ms47×body320valid
6.0ms256×pre80true
4.0ms17×body160valid

simplify268.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
268.0ms
(- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1)

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 63.0b

localize27.0ms

Local error

Found 3 expressions with local error:

5.6b
(- (* (+ n 1) (log (+ n 1))) (* n (log n)))
0.3b
(* n (log n))
0.3b
(* (+ n 1) (log (+ n 1)))

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
12×log-prod
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
7×associate-*r*
6×associate--l+
6×distribute-lft-in
6×pow1
6×distribute-rgt-in
5×add-log-exp
5×add-exp-log
5×add-cbrt-cube
3×associate-*l*
2×*-commutative
2×associate-*l/
1×flip--
1×flip-+
1×cbrt-unprod
1×log-pow
1×prod-exp
1×pow-prod-down
1×diff-log
1×flip3--
1×flip3-+
1×sub-neg
Counts
3 → 55
Calls
3 calls:
Slowest
14.0ms
(- (* (+ n 1) (log (+ n 1))) (* n (log n)))
9.0ms
(* (+ n 1) (log (+ n 1)))
3.0ms
(* n (log n))

series412.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
254.0ms
(* n (log n))
127.0ms
(- (* (+ n 1) (log (+ n 1))) (* n (log n)))
32.0ms
(* (+ n 1) (log (+ n 1)))

simplify4.0s

Counts
36 → 64
Calls
36 calls:
Slowest
667.0ms
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* n (log (/ 1 n))) (log (/ 1 n))))
619.0ms
(- (+ n (* 1/2 (pow n 2))) (+ (* n (log n)) (* 1/6 (pow n 3))))
554.0ms
(- (+ (* n (log -1)) (+ 1 (+ (log -1) (* 1/2 (/ 1 n))))) (+ (* n (log (/ -1 n))) (log (/ -1 n))))
386.0ms
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* 1/6 (/ 1 (pow n 2))) (log (/ 1 n))))
244.0ms
(* (* (* (+ n 1) (+ n 1)) (+ n 1)) (* (* (log (+ n 1)) (log (+ n 1))) (log (+ n 1))))

prune526.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize37.0ms

Local error

Found 2 expressions with local error:

0.4b
(/ 1/6 (* n n))
0.0b
(- (+ (+ 1 (log n)) (- (/ 1/2 n) (/ 1/6 (* n n)))) 1)

rewrite84.0ms

Algorithm
rewrite-expression-head
Rules
3×add-sqr-sqrt
2×add-log-exp
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×*-un-lft-identity
2×pow1
1×flip--
1×div-inv
1×associate--l+
1×difference-of-sqr-1
1×flip3--
1×frac-2neg
1×sub-neg
1×associate-/r*
1×associate-+r-
1×clear-num
1×associate--l-
Counts
2 → 25
Calls
2 calls:
Slowest
83.0ms
(- (+ (+ 1 (log n)) (- (/ 1/2 n) (/ 1/6 (* n n)))) 1)
0.0ms
(/ 1/6 (* n n))

series43.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
31.0ms
(- (+ (+ 1 (log n)) (- (/ 1/2 n) (/ 1/6 (* n n)))) 1)
12.0ms
(/ 1/6 (* n n))

simplify648.0ms

Counts
9 → 31
Calls
9 calls:
Slowest
231.0ms
(- (+ (log -1) (* 1/2 (/ 1 n))) (+ (* 1/6 (/ 1 (pow n 2))) (log (/ -1 n))))
189.0ms
(+ (sqrt (+ (+ 1 (log n)) (- (/ 1/2 n) (/ 1/6 (* n n))))) 1)
109.0ms
(- (* 1/2 (/ 1 n)) (+ (* 1/6 (/ 1 (pow n 2))) (log (/ 1 n))))
97.0ms
(- (+ (log n) (* 1/2 (/ 1 n))) (* 1/6 (/ 1 (pow n 2))))
8.0ms
(+ (+ 1 (log n)) (/ 1/2 n))

prune318.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 2 expressions with local error:

0.3b
(/ -1/6 n)
0.1b
(/ (/ -1/6 n) n)

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
4×associate-/l*
3×div-inv
3×add-cube-cbrt
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×add-exp-log
2×add-cbrt-cube
2×frac-2neg
2×clear-num
2×pow1
1×associate-/l/
Counts
2 → 25
Calls
2 calls:
Slowest
2.0ms
(/ (/ -1/6 n) n)
0.0ms
(/ -1/6 n)

series22.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
14.0ms
(/ (/ -1/6 n) n)
7.0ms
(/ -1/6 n)

simplify18.0ms

Counts
8 → 31
Calls
8 calls:
Slowest
3.0ms
(/ -1/6 (pow n 2))
3.0ms
(/ -1/6 (pow n 2))
3.0ms
(/ -1/6 (pow n 2))
2.0ms
(* (cbrt (/ -1/6 n)) (cbrt (/ -1/6 n)))
2.0ms
(sqrt (/ -1/6 n))

prune256.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample5.7s

Algorithm
intervals
Results
2.9s3666×body1280valid
1.8s2607×body640valid
535.0ms1341×body320valid
211.0ms8000×pre80true
159.0ms386×body160valid