Average Error: 0.3 → 0.0
Time: 5.8s
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: 5.8s)Debug log

start111.0ms

Algorithm
intervals

setup30.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize10.0ms

Local error

Found 2 expressions with local error:

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

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 49
Calls

2 calls. Slowest were:

5.0ms
(- (log x) (log (log x)))
1.0ms
(log (log x))

series696.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

446.0ms
(- (log x) (log (log x)))
250.0ms
(log (log x))

simplify989.0ms

Counts
44 → 55
Calls

44 calls. Slowest were:

189.0ms
(- (log -1) (+ (log (- (log -1) (log (/ -1 x)))) (log (/ -1 x))))
60.0ms
(fma (sqrt (log x)) (sqrt (log x)) (- (* (log (log x)) 1)))
53.0ms
(fma 1 (log x) (- (* (log (log x)) 1)))

prune471.0ms

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 → 15
Calls

1 calls. Slowest were:

1.0ms
(/ x (log x))

series279.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

279.0ms
(/ x (log x))

simplify58.0ms

Counts
6 → 18
Calls

6 calls. Slowest were:

29.0ms
(/ x (- (log -1) (log (/ -1 x))))
24.0ms
(* -1 (/ x (log (/ 1 x))))
1.0ms
(/ x (log x))

prune162.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

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