Average Error: 63.0 → 0
Time: 11.7s
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 2019022 +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: 11.5s)Debug log

sample228.0ms

Algorithm
intervals
Results
122.0ms108×body1280valid
60.0ms91×body640valid
20.0ms43×body320valid
8.0ms256×pre80true
4.0ms14×body160valid

simplify172.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
172.0ms
(- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1)

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 61.9b

localize36.0ms

Local error

Found 3 expressions with local error:

5.8b
(- (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)

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
12×*-un-lft-identity
12×add-sqr-sqrt
10×add-cube-cbrt
9×prod-diff
5×add-log-exp
4×fma-udef
3×log1p-expm1-u
3×add-exp-log
3×fma-neg
3×add-cbrt-cube
3×pow1
3×expm1-log1p-u
1×difference-of-squares
1×distribute-lft-out--
1×associate--r+
1×flip--
1×associate--l+
1×diff-log
1×flip3--
1×sub-neg
Counts
3 → 50
Calls
3 calls:
Slowest
6.0ms
(- (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)

series402.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
259.0ms
(fma n (log n) 1)
96.0ms
(- (fma n (log1p n) (log1p n)) (fma n (log n) 1))
47.0ms
(fma n (log1p n) (log1p n))

simplify5.1s

Counts
34 → 59
Calls
34 calls:
Slowest
897.0ms
(- (+ n (* 1/2 (pow n 2))) (+ (* n (log n)) 1))
571.0ms
(- (+ (* n (log -1)) (+ 1 (+ (log -1) (* 1/2 (/ 1 n))))) (+ (* n (log (/ -1 n))) (log (/ -1 n))))
424.0ms
(- (+ (log -1) (* 1/2 (/ 1 n))) (+ (* 1/6 (/ 1 (pow n 2))) (log (/ -1 n))))
309.0ms
(- (+ 1 (* 1/2 (/ 1 n))) (+ (* n (log (/ 1 n))) (log (/ 1 n))))
287.0ms
(- (+ n (* 1/2 (pow n 2))) (* 1/6 (pow n 3)))

prune299.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize21.0ms

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
Rules
4×associate-/l*
3×div-inv
3×add-cube-cbrt
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×log1p-expm1-u
2×add-exp-log
2×add-cbrt-cube
2×frac-2neg
2×clear-num
2×pow1
2×expm1-log1p-u
1×associate-/l/
Counts
2 → 29
Calls
2 calls:
Slowest
4.0ms
(/ (/ -1/6 n) n)
0.0ms
(/ -1/6 n)

series27.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
16.0ms
(/ (/ -1/6 n) n)
11.0ms
(/ -1/6 n)

simplify24.0ms

Counts
8 → 35
Calls
8 calls:
Slowest
5.0ms
(/ -1/6 (pow n 2))
5.0ms
(/ -1/6 (pow n 2))
5.0ms
(/ -1/6 (pow n 2))
2.0ms
(* (cbrt (/ -1/6 n)) (cbrt (/ -1/6 n)))
1.0ms
(sqrt (/ -1/6 n))

prune250.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample4.9s

Algorithm
intervals
Results
2.8s3629×body1280valid
1.2s2610×body640valid
500.0ms1341×body320valid
182.0ms8000×pre80true
81.0ms420×body160valid