Average Error: 0.3 → 0.0
Time: 12.4s
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 2019005 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  (- (log x) (log (log x))))

Details

Time bar (total: 11.7s)Debug log

sample66.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

3.0ms
(- (log x) (log (log x)))

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize10.0ms

Local error

Found 2 expressions with local error:

0.2b
(- (log x) (log (log x)))
0.0b
(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))

series834.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

521.0ms
(- (log x) (log (log x)))
313.0ms
(log (log x))

simplify899.0ms

Counts
44 → 55
Calls

44 calls. Slowest were:

242.0ms
(- (log -1) (+ (log (- (log -1) (log (/ -1 x)))) (log (/ -1 x))))
48.0ms
(fma (sqrt (log x)) (sqrt (log x)) (- (* (log (log x)) 1)))
47.0ms
(fma 1 (log x) (- (* (log (log x)) 1)))

prune481.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize6.0ms

Local error

Found 2 expressions with local error:

0.3b
(/ x (log x))
0.0b
(log (/ x (log x)))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

1.0ms
(log (/ x (log x)))
1.0ms
(/ x (log x))

series629.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

355.0ms
(log (/ x (log x)))
274.0ms
(/ x (log x))

simplify695.0ms

Counts
15 → 37
Calls

15 calls. Slowest were:

560.0ms
(- (log (/ -1 (- (log -1) (log (/ -1 x))))) (log (/ -1 x)))
46.0ms
(- (log (/ -1 (log (/ 1 x)))) (log (/ 1 x)))
27.0ms
(/ x (- (log -1) (log (/ -1 x))))

prune291.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (cbrt (/ x (log x))) (cbrt (/ x (log x))))
0.5b
(cbrt (/ x (log x)))
0.5b
(cbrt (/ x (log x)))
0.5b
(cbrt (/ x (log x)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (/ x (log x))) (cbrt (/ x (log x))))
1.0ms
(cbrt (/ x (log x)))
1.0ms
(cbrt (/ x (log x)))

series1.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

481.0ms
(cbrt (/ x (log x)))
457.0ms
(cbrt (/ x (log x)))
455.0ms
(* (cbrt (/ x (log x))) (cbrt (/ x (log x))))
435.0ms
(cbrt (/ x (log x)))

simplify2.8s

Counts
53 → 92
Calls

53 calls. Slowest were:

632.0ms
(* (exp (* 1/3 (- (log (/ 1 (pow (- (log -1) (log (/ -1 x))) 2))) (* 2 (log (/ -1 x)))))) (pow (cbrt -1) 2))
307.0ms
(exp (* 1/3 (- (log (/ -1 (log (/ 1 x)))) (log (/ 1 x)))))
281.0ms
(exp (* 1/3 (- (log (/ -1 (log (/ 1 x)))) (log (/ 1 x)))))

prune877.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes36.0ms

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