Statistics.Correlation.Kendall:numOfTiesBy from math-functions-0.1.5.2

Time bar (total: 1.4s)

start0.0ms (0%)

Memory
0.1MiB live, 0.1MiB allocated; 0ms collecting garbage

analyze0.0ms (0%)

Memory
0.5MiB live, 0.5MiB allocated; 0ms collecting garbage
Algorithm
search
Search
ProbabilityValidUnknownPreconditionInfiniteDomainCan'tIter
0%0%100%0%0%0%0%0
100%100%0%0%0%0%0%1
Compiler

Compiled 7 to 6 computations (14.3% saved)

sample524.0ms (38.7%)

Memory
10.2MiB live, 569.3MiB allocated; 139ms collecting garbage
Samples
263.0ms8 256×0valid
Precisions
Click to see histograms. Total time spent on operations: 69.0ms
ival-mult!: 40.0ms (57.8% of total)
ival-sub!: 28.0ms (40.4% of total)
adjust: 1.0ms (1.4% of total)
Bogosity

preprocess233.0ms (17.2%)

Memory
-3.7MiB live, 37.8MiB allocated; 14ms collecting garbage
Iterations

Useful iterations: 2 (0.0ms)

IterNodesCost
0820
12916
29115
322215
481615
5597415
Stop Event
node-limit
Alt Table
Click to see full alt table
StatusAccuracyProgram
100.0%
(*.f64 x (-.f64 x #s(literal 1 binary64)))
Compiler

Compiled 8 to 8 computations (0% saved)

series2.0ms (0.2%)

Memory
4.0MiB live, 4.0MiB allocated; 0ms collecting garbage
Counts
4 → 9
Calls
Call 1
Inputs
(*.f64 x (-.f64 x #s(literal 1 binary64)))
x
(-.f64 x #s(literal 1 binary64))
#s(literal 1 binary64)
Outputs
#s(approx (* x (- x 1)) #s(hole binary64 (* -1 x)))
#s(approx (* x (- x 1)) #s(hole binary64 (* x (- x 1))))
#s(approx x #s(hole binary64 x))
#s(approx (- x 1) #s(hole binary64 -1))
#s(approx (- x 1) #s(hole binary64 (- x 1)))
#s(approx (* x (- x 1)) #s(hole binary64 (pow x 2)))
#s(approx (* x (- x 1)) #s(hole binary64 (* (pow x 2) (- 1 (/ 1 x)))))
#s(approx (- x 1) #s(hole binary64 (* x (- 1 (/ 1 x)))))
#s(approx (- x 1) #s(hole binary64 (* -1 (* x (- (/ 1 x) 1)))))
Calls

3 calls:

TimeVariablePointExpression
1.0ms
x
@inf
((* x (- x 1)) x (- x 1) 1)
1.0ms
x
@0
((* x (- x 1)) x (- x 1) 1)
1.0ms
x
@-inf
((* x (- x 1)) x (- x 1) 1)

rewrite56.0ms (4.2%)

Memory
30.5MiB live, 74.8MiB allocated; 9ms collecting garbage
Iterations

Useful iterations: 0 (0.0ms)

IterNodesCost
02693
02890
19790
061190
1165390
Stop Event
saturated
iter-limit
node-limit
iter-limit
Counts
13 → 23
Calls
Call 1
Inputs
(*.f64 x (-.f64 x #s(literal 1 binary64)))
x
(-.f64 x #s(literal 1 binary64))
#s(literal 1 binary64)
#s(approx (* x (- x 1)) #s(hole binary64 (* -1 x)))
#s(approx (* x (- x 1)) #s(hole binary64 (* x (- x 1))))
#s(approx x #s(hole binary64 x))
#s(approx (- x 1) #s(hole binary64 -1))
#s(approx (- x 1) #s(hole binary64 (- x 1)))
#s(approx (* x (- x 1)) #s(hole binary64 (pow x 2)))
#s(approx (* x (- x 1)) #s(hole binary64 (* (pow x 2) (- 1 (/ 1 x)))))
#s(approx (- x 1) #s(hole binary64 (* x (- 1 (/ 1 x)))))
#s(approx (- x 1) #s(hole binary64 (* -1 (* x (- (/ 1 x) 1)))))
Outputs
(*.f64 (-.f64 x #s(literal 1 binary64)) x)
(*.f64 x (-.f64 x #s(literal 1 binary64)))
(neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) x))
(neg.f64 (*.f64 x (-.f64 #s(literal 1 binary64) x)))
(fma.f64 (fabs.f64 x) (fabs.f64 x) (neg.f64 x))
(fma.f64 (neg.f64 x) (neg.f64 x) (neg.f64 x))
(fma.f64 x x (neg.f64 x))
(+.f64 (*.f64 x x) (neg.f64 x))
x
(neg.f64 (-.f64 #s(literal 1 binary64) x))
(-.f64 x #s(literal 1 binary64))
(+.f64 #s(literal -1 binary64) x)
(+.f64 x #s(literal -1 binary64))
#s(literal 1 binary64)
#s(approx (* x (- x 1)) (neg.f64 x))
#s(approx (* x (- x 1)) (*.f64 (-.f64 x #s(literal 1 binary64)) x))
#s(approx x x)
#s(approx (- x 1) #s(literal -1 binary64))
#s(approx (- x 1) (-.f64 x #s(literal 1 binary64)))
#s(approx (* x (- x 1)) (*.f64 x x))
#s(approx (* x (- x 1)) (*.f64 (*.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) x)) x) x))
#s(approx (- x 1) (*.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) x)) x))
#s(approx (- x 1) (*.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) x)) x))

eval1.0ms (0.1%)

Memory
4.2MiB live, 4.2MiB allocated; 0ms collecting garbage
Compiler

Compiled 69 to 43 computations (37.7% saved)

prune1.0ms (0.1%)

Memory
3.4MiB live, 3.4MiB allocated; 0ms collecting garbage
Pruning

4 alts after pruning (3 fresh and 1 done)

PrunedKeptTotal
New15318
Fresh000
Picked011
Done000
Total15419
Accuracy
100.0%
Counts
19 → 4
Alt Table
Click to see full alt table
StatusAccuracyProgram
100.0%
(fma.f64 x x (neg.f64 x))
100.0%
(*.f64 x (-.f64 x #s(literal 1 binary64)))
50.6%
#s(approx (* x (- x 1)) (*.f64 x x))
52.0%
#s(approx (* x (- x 1)) (neg.f64 x))
Compiler

Compiled 19 to 19 computations (0% saved)

series2.0ms (0.2%)

Memory
5.6MiB live, 5.7MiB allocated; 0ms collecting garbage
Counts
6 → 5
Calls
Call 1
Inputs
#s(approx (* x (- x 1)) (neg.f64 x))
(neg.f64 x)
x
#s(approx (* x (- x 1)) (*.f64 x x))
(*.f64 x x)
(fma.f64 x x (neg.f64 x))
Outputs
#s(approx (* x (- x 1)) #s(hole binary64 (* -1 x)))
#s(approx (* x (- x 1)) #s(hole binary64 (* x (- x 1))))
#s(approx x #s(hole binary64 x))
#s(approx (* x x) #s(hole binary64 (pow x 2)))
#s(approx (* x (- x 1)) #s(hole binary64 (* (pow x 2) (- 1 (/ 1 x)))))
Calls

3 calls:

TimeVariablePointExpression
1.0ms
x
@-inf
((* x (- x 1)) (neg x) x (* x (- x 1)) (* x x) (+ (* x x) (neg x)))
1.0ms
x
@0
((* x (- x 1)) (neg x) x (* x (- x 1)) (* x x) (+ (* x x) (neg x)))
1.0ms
x
@inf
((* x (- x 1)) (neg x) x (* x (- x 1)) (* x x) (+ (* x x) (neg x)))

rewrite71.0ms (5.3%)

Memory
-16.1MiB live, 88.1MiB allocated; 33ms collecting garbage
Iterations

Useful iterations: 0 (0.0ms)

IterNodesCost
02273
02572
19672
074272
1295772
Stop Event
saturated
iter-limit
node-limit
iter-limit
Counts
11 → 54
Calls
Call 1
Inputs
#s(approx (* x (- x 1)) (neg.f64 x))
(neg.f64 x)
x
#s(approx (* x (- x 1)) (*.f64 x x))
(*.f64 x x)
(fma.f64 x x (neg.f64 x))
#s(approx (* x (- x 1)) #s(hole binary64 (* -1 x)))
#s(approx (* x (- x 1)) #s(hole binary64 (* x (- x 1))))
#s(approx x #s(hole binary64 x))
#s(approx (* x x) #s(hole binary64 (pow x 2)))
#s(approx (* x (- x 1)) #s(hole binary64 (* (pow x 2) (- 1 (/ 1 x)))))
Outputs
#s(approx (* x (- x 1)) (neg.f64 x))
(*.f64 #s(literal -1 binary64) x)
(*.f64 x #s(literal -1 binary64))
(neg.f64 x)
x
#s(approx (* x (- x 1)) (*.f64 x x))
(*.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (fabs.f64 x)))
(*.f64 (neg.f64 (fabs.f64 x)) (neg.f64 (fabs.f64 x)))
(*.f64 (*.f64 (neg.f64 x) x) #s(literal -1 binary64))
(*.f64 (fabs.f64 x) (fabs.f64 x))
(*.f64 #s(literal -1 binary64) (*.f64 x (neg.f64 x)))
(*.f64 (*.f64 x x) #s(literal 1 binary64))
(*.f64 (neg.f64 x) (neg.f64 x))
(*.f64 #s(literal 1 binary64) (*.f64 x x))
(*.f64 x x)
(pow.f64 (fabs.f64 (fabs.f64 x)) #s(literal 2 binary64))
(pow.f64 (neg.f64 (fabs.f64 x)) #s(literal 2 binary64))
(pow.f64 (fabs.f64 x) #s(literal 2 binary64))
(pow.f64 (neg.f64 x) #s(literal 2 binary64))
(pow.f64 x #s(literal 2 binary64))
(neg.f64 (*.f64 x (neg.f64 x)))
(neg.f64 (*.f64 (neg.f64 x) x))
(sqrt.f64 (*.f64 (*.f64 (*.f64 x x) x) x))
(fabs.f64 (neg.f64 (*.f64 x x)))
(fabs.f64 (*.f64 x x))
(*.f64 (+.f64 (neg.f64 x) #s(literal 1 binary64)) (neg.f64 x))
(*.f64 (-.f64 x #s(literal 1 binary64)) x)
(*.f64 x (-.f64 x #s(literal 1 binary64)))
(neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) x))
(neg.f64 (*.f64 x (-.f64 #s(literal 1 binary64) x)))
(neg.f64 (-.f64 x (*.f64 x x)))
(fma.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (fabs.f64 x)) (neg.f64 x))
(fma.f64 (neg.f64 (fabs.f64 x)) (neg.f64 (fabs.f64 x)) (neg.f64 x))
(fma.f64 (*.f64 (neg.f64 x) x) #s(literal -1 binary64) (neg.f64 x))
(fma.f64 (fabs.f64 x) (fabs.f64 x) (neg.f64 x))
(fma.f64 #s(literal -1 binary64) (*.f64 x (neg.f64 x)) (neg.f64 x))
(fma.f64 #s(literal -1 binary64) x (*.f64 x x))
(fma.f64 (*.f64 x x) #s(literal 1 binary64) (neg.f64 x))
(fma.f64 (neg.f64 x) (neg.f64 x) (neg.f64 x))
(fma.f64 #s(literal 1 binary64) (*.f64 x x) (neg.f64 x))
(fma.f64 x #s(literal -1 binary64) (*.f64 x x))
(fma.f64 x x (neg.f64 x))
(-.f64 (*.f64 x x) x)
(-.f64 (neg.f64 x) (*.f64 (neg.f64 (fabs.f64 x)) (fabs.f64 x)))
(-.f64 (neg.f64 x) (neg.f64 (*.f64 x x)))
(-.f64 (neg.f64 x) (*.f64 x (neg.f64 x)))
(-.f64 (neg.f64 x) (*.f64 (neg.f64 x) x))
(+.f64 (*.f64 x x) (neg.f64 x))
(+.f64 (neg.f64 x) (*.f64 x x))
#s(approx (* x (- x 1)) (neg.f64 x))
#s(approx (* x (- x 1)) (*.f64 (-.f64 x #s(literal 1 binary64)) x))
#s(approx x x)
#s(approx (* x x) (*.f64 x x))
#s(approx (* x (- x 1)) (*.f64 (*.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) x)) x) x))

eval4.0ms (0.3%)

Memory
9.1MiB live, 9.1MiB allocated; 0ms collecting garbage
Compiler

Compiled 194 to 103 computations (46.9% saved)

prune203.0ms (15%)

Memory
-40.6MiB live, 5.2MiB allocated; 11ms collecting garbage
Pruning

4 alts after pruning (1 fresh and 3 done)

PrunedKeptTotal
New49150
Fresh000
Picked123
Done011
Total50454
Accuracy
100.0%
Counts
54 → 4
Alt Table
Click to see full alt table
StatusAccuracyProgram
100.0%
(-.f64 (*.f64 x x) x)
100.0%
(*.f64 x (-.f64 x #s(literal 1 binary64)))
50.6%
#s(approx (* x (- x 1)) (*.f64 x x))
52.0%
#s(approx (* x (- x 1)) (neg.f64 x))
Compiler

Compiled 19 to 19 computations (0% saved)

series1.0ms (0.1%)

Memory
3.3MiB live, 3.3MiB allocated; 0ms collecting garbage
Counts
3 → 5
Calls
Call 1
Inputs
(-.f64 (*.f64 x x) x)
(*.f64 x x)
x
Outputs
#s(approx (- (* x x) x) #s(hole binary64 (* -1 x)))
#s(approx (- (* x x) x) #s(hole binary64 (* x (- x 1))))
#s(approx (* x x) #s(hole binary64 (pow x 2)))
#s(approx x #s(hole binary64 x))
#s(approx (- (* x x) x) #s(hole binary64 (* (pow x 2) (- 1 (/ 1 x)))))
Calls

3 calls:

TimeVariablePointExpression
1.0ms
x
@-inf
((- (* x x) x) (* x x) x)
1.0ms
x
@inf
((- (* x x) x) (* x x) x)
0.0ms
x
@0
((- (* x x) x) (* x x) x)

rewrite50.0ms (3.7%)

Memory
23.1MiB live, 23.1MiB allocated; 0ms collecting garbage
Iterations

Useful iterations: 0 (0.0ms)

IterNodesCost
02054
02153
19253
071453
1307053
Stop Event
saturated
iter-limit
node-limit
iter-limit
Counts
8 → 54
Calls
Call 1
Inputs
(-.f64 (*.f64 x x) x)
(*.f64 x x)
x
#s(approx (- (* x x) x) #s(hole binary64 (* -1 x)))
#s(approx (- (* x x) x) #s(hole binary64 (* x (- x 1))))
#s(approx (* x x) #s(hole binary64 (pow x 2)))
#s(approx x #s(hole binary64 x))
#s(approx (- (* x x) x) #s(hole binary64 (* (pow x 2) (- 1 (/ 1 x)))))
Outputs
(*.f64 (+.f64 (neg.f64 x) #s(literal 1 binary64)) (neg.f64 x))
(*.f64 (-.f64 x #s(literal 1 binary64)) x)
(*.f64 x (-.f64 x #s(literal 1 binary64)))
(neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) x))
(neg.f64 (*.f64 x (-.f64 #s(literal 1 binary64) x)))
(neg.f64 (-.f64 x (*.f64 x x)))
(fma.f64 (*.f64 (neg.f64 x) #s(literal -1 binary64)) x (neg.f64 x))
(fma.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (fabs.f64 x)) (neg.f64 x))
(fma.f64 (neg.f64 (fabs.f64 x)) (neg.f64 (fabs.f64 x)) (neg.f64 x))
(fma.f64 (neg.f64 (neg.f64 x)) (neg.f64 (neg.f64 x)) (neg.f64 x))
(fma.f64 (*.f64 (neg.f64 x) x) #s(literal -1 binary64) (neg.f64 x))
(fma.f64 (fabs.f64 x) (fabs.f64 x) (neg.f64 x))
(fma.f64 (neg.f64 x) (neg.f64 x) (neg.f64 x))
(fma.f64 #s(literal 1 binary64) (*.f64 x x) (neg.f64 x))
(fma.f64 #s(literal -1 binary64) (*.f64 x (neg.f64 x)) (neg.f64 x))
(fma.f64 #s(literal -1 binary64) x (*.f64 x x))
(fma.f64 (*.f64 x x) #s(literal 1 binary64) (neg.f64 x))
(fma.f64 x (*.f64 #s(literal -1 binary64) (neg.f64 x)) (neg.f64 x))
(fma.f64 x #s(literal -1 binary64) (*.f64 x x))
(fma.f64 x x (neg.f64 x))
(-.f64 (*.f64 x x) (*.f64 (neg.f64 x) #s(literal -1 binary64)))
(-.f64 (*.f64 x x) (neg.f64 (neg.f64 x)))
(-.f64 (*.f64 x x) x)
(+.f64 (neg.f64 x) (*.f64 x x))
(+.f64 (*.f64 x x) (neg.f64 x))
(*.f64 (*.f64 (neg.f64 x) #s(literal -1 binary64)) x)
(*.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (fabs.f64 x)))
(*.f64 (neg.f64 (fabs.f64 x)) (neg.f64 (fabs.f64 x)))
(*.f64 (neg.f64 (neg.f64 x)) (neg.f64 (neg.f64 x)))
(*.f64 (*.f64 (neg.f64 x) x) #s(literal -1 binary64))
(*.f64 (fabs.f64 x) (fabs.f64 x))
(*.f64 (neg.f64 x) (neg.f64 x))
(*.f64 #s(literal 1 binary64) (*.f64 x x))
(*.f64 #s(literal -1 binary64) (*.f64 x (neg.f64 x)))
(*.f64 (*.f64 x x) #s(literal 1 binary64))
(*.f64 x (*.f64 #s(literal -1 binary64) (neg.f64 x)))
(*.f64 x x)
(pow.f64 (fabs.f64 (fabs.f64 x)) #s(literal 2 binary64))
(pow.f64 (neg.f64 (fabs.f64 x)) #s(literal 2 binary64))
(pow.f64 (neg.f64 (neg.f64 x)) #s(literal 2 binary64))
(pow.f64 (fabs.f64 x) #s(literal 2 binary64))
(pow.f64 (neg.f64 x) #s(literal 2 binary64))
(pow.f64 x #s(literal 2 binary64))
(neg.f64 (*.f64 x (neg.f64 x)))
(neg.f64 (*.f64 (neg.f64 x) x))
(sqrt.f64 (*.f64 (*.f64 (*.f64 x x) x) x))
(fabs.f64 (neg.f64 (*.f64 x x)))
(fabs.f64 (*.f64 x x))
x
#s(approx (- (* x x) x) (neg.f64 x))
#s(approx (- (* x x) x) (-.f64 (*.f64 x x) x))
#s(approx (* x x) (*.f64 x x))
#s(approx x x)
#s(approx (- (* x x) x) (*.f64 (*.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) x)) x) x))

eval4.0ms (0.3%)

Memory
3.8MiB live, 3.8MiB allocated; 0ms collecting garbage
Compiler

Compiled 200 to 105 computations (47.5% saved)

prune3.0ms (0.2%)

Memory
2.1MiB live, 2.1MiB allocated; 0ms collecting garbage
Pruning

4 alts after pruning (0 fresh and 4 done)

PrunedKeptTotal
New50050
Fresh000
Picked011
Done033
Total50454
Accuracy
100.0%
Counts
54 → 4
Alt Table
Click to see full alt table
StatusAccuracyProgram
100.0%
(-.f64 (*.f64 x x) x)
100.0%
(*.f64 x (-.f64 x #s(literal 1 binary64)))
50.6%
#s(approx (* x (- x 1)) (*.f64 x x))
52.0%
#s(approx (* x (- x 1)) (neg.f64 x))
Compiler

Compiled 36 to 32 computations (11.1% saved)

regimes4.0ms (0.3%)

Memory
5.7MiB live, 5.7MiB allocated; 0ms collecting garbage
Counts
5 → 1
Calls
Call 1
Inputs
#s(approx (* x (- x 1)) (neg.f64 x))
#s(approx (* x (- x 1)) (*.f64 x x))
(*.f64 x (-.f64 x #s(literal 1 binary64)))
(-.f64 (*.f64 x x) x)
(fma.f64 x x (neg.f64 x))
Outputs
(fma.f64 x x (neg.f64 x))
Calls

2 calls:

2.0ms
x
2.0ms
(*.f64 x (-.f64 x #s(literal 1 binary64)))
Results
AccuracySegmentsBranch
100.0%1(*.f64 x (-.f64 x #s(literal 1 binary64)))
100.0%1x
Compiler

Compiled 5 to 5 computations (0% saved)

regimes4.0ms (0.3%)

Memory
5.4MiB live, 5.4MiB allocated; 0ms collecting garbage
Counts
4 → 1
Calls
Call 1
Inputs
#s(approx (* x (- x 1)) (neg.f64 x))
#s(approx (* x (- x 1)) (*.f64 x x))
(*.f64 x (-.f64 x #s(literal 1 binary64)))
(-.f64 (*.f64 x x) x)
Outputs
(*.f64 x (-.f64 x #s(literal 1 binary64)))
Calls

2 calls:

2.0ms
(*.f64 x (-.f64 x #s(literal 1 binary64)))
2.0ms
x
Results
AccuracySegmentsBranch
100.0%1(*.f64 x (-.f64 x #s(literal 1 binary64)))
100.0%1x
Compiler

Compiled 5 to 5 computations (0% saved)

regimes110.0ms (8.1%)

Memory
-38.3MiB live, 8.5MiB allocated; 4ms collecting garbage
Counts
2 → 2
Calls
Call 1
Inputs
#s(approx (* x (- x 1)) (neg.f64 x))
#s(approx (* x (- x 1)) (*.f64 x x))
Outputs
#s(approx (* x (- x 1)) (neg.f64 x))
#s(approx (* x (- x 1)) (*.f64 x x))
Calls

2 calls:

108.0ms
x
1.0ms
(*.f64 x (-.f64 x #s(literal 1 binary64)))
Results
AccuracySegmentsBranch
97.9%2(*.f64 x (-.f64 x #s(literal 1 binary64)))
97.9%3x
Compiler

Compiled 5 to 5 computations (0% saved)

regimes2.0ms (0.2%)

Memory
5.3MiB live, 5.3MiB allocated; 0ms collecting garbage
Accuracy

Total -0.0b remaining (-0%)

Threshold costs -0b (-0%)

Counts
1 → 1
Calls
Call 1
Inputs
#s(approx (* x (- x 1)) (neg.f64 x))
Outputs
#s(approx (* x (- x 1)) (neg.f64 x))
Calls

2 calls:

1.0ms
(*.f64 x (-.f64 x #s(literal 1 binary64)))
1.0ms
x
Results
AccuracySegmentsBranch
52.0%1x
52.0%1(*.f64 x (-.f64 x #s(literal 1 binary64)))
Compiler

Compiled 5 to 5 computations (0% saved)

bsearch0.0ms (0%)

Memory
0.4MiB live, 0.4MiB allocated; 0ms collecting garbage
Algorithm
left-value
Steps
TimeLeftRight
0.0ms
0.001486288095574496
28495662.001507487
Compiler

Compiled 7 to 8 computations (-14.3% saved)

derivations7.0ms (0.5%)

Memory
8.4MiB live, 8.4MiB allocated; 0ms collecting garbage
Stop Event
done
Compiler

Compiled 18 to 15 computations (16.7% saved)

preprocess68.0ms (5%)

Memory
7.8MiB live, 50.9MiB allocated; 9ms collecting garbage
Compiler

Compiled 22 to 19 computations (13.6% saved)

end0.0ms (0%)

Memory
0.0MiB live, 0.0MiB allocated; 0ms collecting garbage

Profiling

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