Average Error: 0.3 → 0.0
Time: 5.3s
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. Using strategy rm

  3. Applied diff-log0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  (- (log x) (log (log x))))

Details

Time bar (total: 5.0s)Debug log

sample64.0ms

Algorithm
intervals
Results
32.0ms753×body80nan
16.0ms256×body80valid

simplify5.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
5.0ms
(- (log x) (log (log x)))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize12.0ms

Local error

Found 2 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
15×*-un-lft-identity
14×add-sqr-sqrt
12×add-cube-cbrt
12×prod-diff
7×pow1
6×log-prod
5×log-pow
3×associate--r+
3×add-exp-log
3×fma-neg
2×add-log-exp
2×distribute-lft-out--
2×log1p-expm1-u
2×add-cbrt-cube
2×expm1-log1p-u
1×difference-of-squares
1×flip--
1×rem-log-exp
1×diff-log
1×flip3--
1×sub-neg
Counts
2 → 49
Calls
2 calls:
Slowest
5.0ms
(- (log x) (log (log x)))
1.0ms
(log (log x))

series695.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
423.0ms
(- (log x) (log (log x)))
272.0ms
(log (log x))

simplify956.0ms

Counts
44 → 55
Calls
44 calls:
Slowest
255.0ms
(- (log -1) (+ (log (- (log -1) (log (/ -1 x)))) (log (/ -1 x))))
56.0ms
(fma (sqrt (log x)) (sqrt (log x)) (- (* (log (log x)) 1)))
46.0ms
(fma 1 (log x) (- (* (log (log x)) 1)))
43.0ms
(log (- (log -1) (log (/ -1 x))))
42.0ms
(fma (sqrt (log x)) (sqrt (log x)) (- (* (cbrt (log (log x))) (* (cbrt (log (log x))) (cbrt (log (log x)))))))

prune403.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize11.0ms

Local error

Found 1 expressions with local error:

0.3b
(/ x (log x))

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
3×associate-/r*
2×add-cube-cbrt
2×*-un-lft-identity
2×add-sqr-sqrt
1×add-log-exp
1×div-inv
1×log1p-expm1-u
1×add-exp-log
1×add-cbrt-cube
1×frac-2neg
1×clear-num
1×pow1
1×expm1-log1p-u
Counts
1 → 15
Calls
1 calls:
Slowest
1.0ms
(/ x (log x))

series301.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
301.0ms
(/ x (log x))

simplify70.0ms

Counts
6 → 18
Calls
6 calls:
Slowest
42.0ms
(/ x (- (log -1) (log (/ -1 x))))
24.0ms
(* -1 (/ x (log (/ 1 x))))
1.0ms
(cbrt (log x))
1.0ms
(/ x (log x))
1.0ms
(sqrt (log x))

prune134.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes106.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch1.0ms

end0.0ms

sample2.2s

Algorithm
intervals
Results
1.2s24206×body80nan
550.0ms8000×body80valid