Average Error: 0.3 → 0.0
Time: 4.5s
Precision: 64
Internal Precision: 128
\[\log x - \log \left(\log x\right)\]
↓
\[\log \left(\frac{x}{\log x}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.3

    \[\log x - \log \left(\log x\right)\]

  2. Initial simplification0.3

    \[\leadsto \log x - \log \left(\log x\right)\]
  3. Using strategy rm

  4. Applied diff-log0.0

    \[\leadsto \color{blue}{\log \left(\frac{x}{\log x}\right)}\]

  5. Final simplification0.0

    \[\leadsto \log \left(\frac{x}{\log x}\right)\]

Runtime

Time bar (total: 4.5s)Debug log

start126.0ms

Algorithm
intervals

setup20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize19.0ms

Local error

Found 2 expressions with local error:

0.3b
(- (log x) (log (log x)))
0.1b
(log (log x))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 30
Calls

2 calls. Slowest were:

8.0ms
(- (log x) (log (log x)))
1.0ms
(log (log x))

series783.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

487.0ms
(- (log x) (log (log x)))
295.0ms
(log (log x))

simplify362.0ms

Counts
18 → 36
Calls

18 calls. Slowest were:

269.0ms
(- (log -1) (+ (log (- (log -1) (log (/ -1 x)))) (log (/ -1 x))))
34.0ms
(log (- (log -1) (log (/ -1 x))))
29.0ms
(- (+ (log (/ 1 x)) (log (- (log (/ 1 x))))))

prune268.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize6.0ms

Local error

Found 1 expressions with local error:

0.3b
(/ x (log x))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 13
Calls

1 calls. Slowest were:

1.0ms
(/ x (log x))

series294.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

294.0ms
(/ x (log x))

simplify67.0ms

Counts
6 → 16
Calls

6 calls. Slowest were:

31.0ms
(* -1 (/ x (log (/ 1 x))))
30.0ms
(/ x (- (log -1) (log (/ -1 x))))
2.0ms
(cbrt (log x))

prune176.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018359 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  (- (log x) (log (log x))))