Details

analyze103.0ms (0.5%)

Algorithm

1×

Search

Probability	Valid	Unknown	Precondition	Infinite	Domain	Can't	Iter
0%	0%	99.9%	0.1%	0%	0%	0%	0
0%	0%	99.9%	0.1%	0%	0%	0%	1
0%	0%	50%	0.1%	0%	50%	0%	2
0%	0%	50%	0.1%	0%	50%	0%	3
0%	0%	50%	0.1%	0%	50%	0%	4
25%	12.5%	37.5%	0.1%	0%	50%	0%	5
26.7%	12.5%	34.3%	0.1%	0%	53.1%	0%	6
37%	15.6%	26.5%	0.1%	1.6%	54.6%	1.6%	7
40.8%	15.6%	22.6%	0.1%	2.3%	57%	2.3%	8
49.4%	17.2%	17.6%	0.1%	3.5%	58.1%	3.5%	9
53.3%	17.2%	15%	0.1%	4.1%	59.5%	4.1%	10
59.5%	18%	12.2%	0.1%	4.8%	60.2%	4.8%	11
62.5%	18%	10.8%	0.1%	5.1%	60.9%	5.1%	12

Compiler

Compiled 19 to 16 computations (15.8% saved)

sample2.6s (12.3%)

Results

1.3s	8256×	body	256	valid
729.0ms	1384×	body	1024	invalid
224.0ms	709×	body	512	invalid
199.0ms	1597×	body	256	invalid
94.0ms	534×	body	256	infinite
58.0ms	249×	body	256	unsamplable

Bogosity

preprocess4.3s (20.7%)

Algorithm

2×

egglog

Calls

Call 1

Inputs
0
1

Outputs
0
1

Call 2

Inputs
(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))
(.f64 (/.f64 1 (sqrt.f64 n)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) k) (/.f64 (-.f64 1 n) 2)))

Outputs
(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))
(/.f64 (pow.f64 (*.f64 n (+.f64 (PI.f64) (PI.f64))) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
(.f64 (/.f64 1 (sqrt.f64 n)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) k) (/.f64 (-.f64 1 n) 2)))
(/.f64 (pow.f64 (*.f64 k (+.f64 (PI.f64) (PI.f64))) (neg.f64 (fma.f64 n 1/2 -1/2))) (sqrt.f64 n))

Compiler

Compiled 20 to 17 computations (15% saved)

simplify1.8s (8.5%)

Algorithm

1×

egglog

Counts

1 → 2

Calls

Call 1

Inputs
(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))

Outputs
(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))
(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))

eval1.0ms (0%)

Compiler: Compiled 32 to 23 computations (28.1% saved)

prune3.0ms (0%)

Pruning

3 alts after pruning (3 fresh and 0 done)

	Kept	Total
New	2	2
Fresh	1	1
Picked	0	0
Done	0	0
Total	3	3

Error

0.4b

Counts

3 → 2

Alt Table

Click to see full alt table

Status	Error	Program
▶	0.5b	(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
▶	0.5b	(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))

Compiler

Compiled 34 to 27 computations (20.6% saved)

localize43.0ms (0.2%)

Local error

Found 4 expressions with local error:

New	Error	Program
✓	0.2b	(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))
✓	0.2b	(/.f64 1 (sqrt.f64 k))
✓	0.3b	(.f64 (.f64 2 (PI.f64)) n)
✓	0.3b	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2))

Compiler

Compiled 56 to 38 computations (32.1% saved)

series288.0ms (1.4%)

Counts

4 → 40

Calls

18 calls:

Time	Variable		Point	Expression
42.0ms	n	@	0	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2))
31.0ms	n	@	-inf	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2))
10.0ms	n	@	inf	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2))
5.0ms	n	@	inf	(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))
3.0ms	k	@	-inf	(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))

Compiler

Compiled 12380 to 8820 computations (28.8% saved)

rewrite2.5s (12%)

Algorithm

1×	batch-egg-rewrite

Rules

66×	`egg-rr`
60×	`egg-rr`
49×	`egg-rr`
21×	`egg-rr`

Counts

4 → 196

Calls

Call 1

Inputs
(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2))
(.f64 (.f64 2 (PI.f64)) n)
(/.f64 1 (sqrt.f64 k))
(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))

Outputs
((#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 1 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 1)))) (#(struct:change #<rule egg-rr> (2) ((x .f64 -1 (neg.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))))) (#(struct:change #<rule egg-rr> (2) ((x -.f64 0 (neg.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))))) (#(struct:change #<rule egg-rr> (2) ((x /.f64 (neg.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) -1)))) (#(struct:change #<rule egg-rr> (2) ((x /.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) -1) -1)))) (#(struct:change #<rule egg-rr> (2) ((x /.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 1)))) (#(struct:change #<rule egg-rr> (2) ((x pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))) (#(struct:change #<rule egg-rr> (2) ((x pow.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) 3)))) (#(struct:change #<rule egg-rr> (2) ((x pow.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 3) 1/3)))) (#(struct:change #<rule egg-rr> (2) ((x pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 1)))) (#(struct:change #<rule egg-rr> (2) ((x cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 3))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (neg.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (pow.f64 (neg.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))) 3))))) (#(struct:change #<rule egg-rr> (2) ((x if (!=.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 0) (pow.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) -1) -1) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))))) (#(struct:change #<rule egg-rr> (2) ((x if (!=.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) 0) (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (fma.f64 k 1/2 -1/2)) -1) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))))) (#(struct:change #<rule egg-rr> (2) ((x if (!=.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) 0) (/.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (/.f64 1 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)))) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))))) (#(struct:change #<rule egg-rr> (2) ((x if (!=.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) 0) (/.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) (/.f64 1 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))))) (#(struct:change #<rule egg-rr> (2) ((x if (!=.f64 1 0) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))))))
((#(struct:change #<rule egg-rr> (2) ((x .f64 (+.f64 (PI.f64) (PI.f64)) n)))) (#(struct:change #<rule egg-rr> (2) ((x .f64 n (+.f64 (PI.f64) (PI.f64)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n)) (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) 2)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) 2)) (cbrt.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 (+.f64 (PI.f64) (PI.f64))) (.f64 (cbrt.f64 (.f64 4 (.f64 (PI.f64) (PI.f64)))) n))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (PI.f64) (+.f64 n n))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 n) (.f64 (cbrt.f64 (pow.f64 n 2)) (+.f64 (PI.f64) (PI.f64))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (.f64 n (cbrt.f64 (+.f64 (PI.f64) (PI.f64)))) (cbrt.f64 (.f64 4 (.f64 (PI.f64) (PI.f64)))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (+.f64 n n) (PI.f64))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (.f64 (cbrt.f64 (.f64 4 (.f64 (PI.f64) (PI.f64)))) n) (cbrt.f64 (+.f64 (PI.f64) (PI.f64))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (.f64 (cbrt.f64 (pow.f64 n 2)) (+.f64 (PI.f64) (PI.f64))) (cbrt.f64 n))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 (.f64 4 (.f64 (PI.f64) (PI.f64)))) (.f64 n (cbrt.f64 (+.f64 (PI.f64) (PI.f64)))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 (pow.f64 n 2)) (.f64 (+.f64 (PI.f64) (PI.f64)) (cbrt.f64 n)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (.f64 (+.f64 (PI.f64) (PI.f64)) (cbrt.f64 n)) (cbrt.f64 (pow.f64 n 2)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 2 (.f64 (PI.f64) n))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (.f64 (PI.f64) n) 2)))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 2) (.f64 (cbrt.f64 4) (.f64 (PI.f64) n)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (.f64 (cbrt.f64 4) (.f64 (PI.f64) n)) (cbrt.f64 2))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (sqrt.f64 2) (.f64 (sqrt.f64 2) (.f64 (PI.f64) n)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (.f64 (sqrt.f64 2) (.f64 (PI.f64) n)) (sqrt.f64 2))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 1 (.f64 (+.f64 (PI.f64) (PI.f64)) n))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) 1)))) (#(struct:change #<rule egg-rr> (2) ((x .f64 -1 (.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) -1))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 -1 (.f64 -2 (.f64 (PI.f64) n)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 -1 (neg.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 -1 (.f64 (.f64 (PI.f64) n) -2))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (+.f64 (cbrt.f64 (pow.f64 (.f64 (PI.f64) n) 2)) (-.f64 (cbrt.f64 (pow.f64 (.f64 (PI.f64) n) 2)) (cbrt.f64 (pow.f64 (.f64 (PI.f64) n) 2)))) (+.f64 (cbrt.f64 (.f64 (PI.f64) n)) (cbrt.f64 (.f64 (PI.f64) n))))))) (#(struct:change #<rule egg-rr> (2) ((x -.f64 (.f64 (PI.f64) n) (neg.f64 (.f64 (PI.f64) n)))))) (#(struct:change #<rule egg-rr> (2) ((x -.f64 0 (.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) -1))))) (#(struct:change #<rule egg-rr> (2) ((x -.f64 0 (.f64 -2 (.f64 (PI.f64) n)))))) (#(struct:change #<rule egg-rr> (2) ((x -.f64 0 (neg.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n)))))) (#(struct:change #<rule egg-rr> (2) ((x -.f64 0 (.f64 (.f64 (PI.f64) n) -2))))) (#(struct:change #<rule egg-rr> (2) ((x /.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) 1)))) (#(struct:change #<rule egg-rr> (2) ((x pow.f64 (cbrt.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n)) 3)))) (#(struct:change #<rule egg-rr> (2) ((x pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) 3) 1/3)))) (#(struct:change #<rule egg-rr> (2) ((x pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) 1)))) (#(struct:change #<rule egg-rr> (2) ((x cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) 3))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) -1))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (.f64 (.f64 (PI.f64) n) -2))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (neg.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n)))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (.f64 -2 (.f64 (PI.f64) n)))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (+.f64 (neg.f64 (.f64 (PI.f64) n)) (neg.f64 (.f64 (PI.f64) n))))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (pow.f64 (neg.f64 (cbrt.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n))) 3))))) (#(struct:change #<rule egg-rr> (2) ((x if (!=.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) 0) (pow.f64 (/.f64 1 (.f64 (+.f64 (PI.f64) (PI.f64)) n)) -1) (.f64 (+.f64 (PI.f64) (PI.f64)) n))))) (#(struct:change #<rule egg-rr> (2) ((x if (!=.f64 (.f64 (PI.f64) n) 0) (/.f64 (+.f64 (pow.f64 (.f64 (PI.f64) n) 3) (pow.f64 (.f64 (PI.f64) n) 3)) (+.f64 (pow.f64 (.f64 (PI.f64) n) 2) (-.f64 (pow.f64 (.f64 (PI.f64) n) 2) (pow.f64 (.f64 (PI.f64) n) 2)))) (.f64 (+.f64 (PI.f64) (PI.f64)) n))))) (#(struct:change #<rule 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simplify5.1s (24.6%)

Algorithm

1×

egglog

Counts

236 → 242

Calls

Call 1

Inputs
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 n) (log.f64 (*.f64 2 (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 n) (log.f64 (*.f64 2 (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 n) (log.f64 (*.f64 2 (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 n) (log.f64 (*.f64 2 (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64)))))))
(pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2)
(+.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (.f64 -1/2 (.f64 k (.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (log.f64 (.f64 2 (*.f64 n (PI.f64))))))))
(+.f64 (.f64 1/2 (.f64 (pow.f64 k 2) (.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (pow.f64 (.f64 -1/2 (log.f64 (.f64 2 (.f64 n (PI.f64))))) 2)))) (+.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (.f64 -1/2 (.f64 k (.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (log.f64 (.f64 2 (*.f64 n (PI.f64)))))))))
(+.f64 (.f64 1/2 (.f64 (pow.f64 k 2) (.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (pow.f64 (.f64 -1/2 (log.f64 (.f64 2 (.f64 n (PI.f64))))) 2)))) (+.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (+.f64 (.f64 -1/2 (.f64 k (.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (log.f64 (.f64 2 (.f64 n (PI.f64))))))) (.f64 1/6 (.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (.f64 (pow.f64 (.f64 -1/2 (log.f64 (.f64 2 (*.f64 n (PI.f64))))) 3) (pow.f64 k 3)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (log.f64 (.f64 2 (.f64 n (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (log.f64 (.f64 2 (.f64 n (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (log.f64 (.f64 2 (.f64 n (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (log.f64 (.f64 2 (.f64 n (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (+.f64 1 (.f64 -1 k)) (log.f64 (.f64 2 (*.f64 n (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (+.f64 1 (.f64 -1 k)) (log.f64 (.f64 2 (*.f64 n (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (+.f64 1 (.f64 -1 k)) (log.f64 (.f64 2 (*.f64 n (PI.f64)))))))
(exp.f64 (.f64 1/2 (.f64 (+.f64 1 (.f64 -1 k)) (log.f64 (.f64 2 (*.f64 n (PI.f64)))))))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(.f64 2 (.f64 n (PI.f64)))
(/.f64 (exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 (*.f64 2 (PI.f64))) (log.f64 n))))) (sqrt.f64 k))
(/.f64 (exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 (*.f64 2 (PI.f64))) (log.f64 n))))) (sqrt.f64 k))
(/.f64 (exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 (*.f64 2 (PI.f64))) (log.f64 n))))) (sqrt.f64 k))
(/.f64 (exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 (*.f64 2 (PI.f64))) (log.f64 n))))) (sqrt.f64 k))
(/.f64 (exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64))))))) (sqrt.f64 k))
(/.f64 (exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64))))))) (sqrt.f64 k))
(/.f64 (exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64))))))) (sqrt.f64 k))
(/.f64 (exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64))))))) (sqrt.f64 k))
(.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (cbrt.f64 (pow.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)))
(.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) (cbrt.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))
(.f64 1 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))
(.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 1)
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(.f64 (/.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) (sqrt.f64 k)) (cbrt.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))
(.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) (/.f64 (cbrt.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (sqrt.f64 k)))
(.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (/.f64 (cbrt.f64 (pow.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) (sqrt.f64 k)))
(.f64 (/.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (sqrt.f64 k)) (cbrt.f64 (pow.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)))
(.f64 (pow.f64 (sqrt.f64 k) -1/2) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 (sqrt.f64 k))))
(.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 (sqrt.f64 k))) (pow.f64 (sqrt.f64 k) -1/2))
(.f64 (cbrt.f64 (pow.f64 (sqrt.f64 k) -2)) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (cbrt.f64 (sqrt.f64 k))))
(.f64 (neg.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (/.f64 -1 (sqrt.f64 k)))
(.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (cbrt.f64 (sqrt.f64 k))) (cbrt.f64 (pow.f64 (sqrt.f64 k) -2)))
(.f64 1 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 1)
(.f64 -1 (.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) -1))
(.f64 -1 (.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (/.f64 -1 (sqrt.f64 k))))
(.f64 -1 (neg.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))))
(.f64 -1 (.f64 (/.f64 -1 (sqrt.f64 k)) (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))
(.f64 -1 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (neg.f64 (sqrt.f64 k))))
(-.f64 0 (.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) -1))
(-.f64 0 (.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (/.f64 -1 (sqrt.f64 k))))
(-.f64 0 (neg.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))))
(-.f64 0 (.f64 (/.f64 -1 (sqrt.f64 k)) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))
(-.f64 0 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (neg.f64 (sqrt.f64 k))))
(/.f64 (/.f64 (neg.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (cbrt.f64 (sqrt.f64 k))) (neg.f64 (pow.f64 (sqrt.f64 k) 2/3)))
(/.f64 (/.f64 (neg.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (sqrt.f64 (sqrt.f64 k))) (neg.f64 (sqrt.f64 (sqrt.f64 k))))
(/.f64 (neg.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (neg.f64 (sqrt.f64 k)))
(/.f64 (/.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (cbrt.f64 (sqrt.f64 k))) (cbrt.f64 (pow.f64 (sqrt.f64 k) 2/3))) (cbrt.f64 (pow.f64 (sqrt.f64 k) (.f64 2 2/3))))
(/.f64 (neg.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (cbrt.f64 (sqrt.f64 k)))) (neg.f64 (pow.f64 (sqrt.f64 k) 2/3)))
(/.f64 (neg.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 (sqrt.f64 k)))) (neg.f64 (sqrt.f64 (sqrt.f64 k))))
(/.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (cbrt.f64 (sqrt.f64 k))) (pow.f64 (sqrt.f64 k) 2/3))
(/.f64 (.f64 (cbrt.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))) (cbrt.f64 (pow.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2))) (pow.f64 (sqrt.f64 k) 2/3))
(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
(/.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (pow.f64 (sqrt.f64 k) 2/3)) (cbrt.f64 (sqrt.f64 k)))
(/.f64 (neg.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (pow.f64 (sqrt.f64 k) 2/3))) (neg.f64 (cbrt.f64 (sqrt.f64 k))))
(/.f64 (.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (cbrt.f64 (pow.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 2))) (cbrt.f64 (sqrt.f64 k)))
(/.f64 (.f64 (cbrt.f64 (pow.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 2)) (cbrt.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))) (cbrt.f64 (sqrt.f64 k)))
(/.f64 (/.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 (sqrt.f64 k))) (cbrt.f64 (sqrt.f64 (sqrt.f64 k)))) (cbrt.f64 (sqrt.f64 k)))
(/.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 (sqrt.f64 k))) (sqrt.f64 (sqrt.f64 k)))
(/.f64 (.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (/.f64 (cbrt.f64 (pow.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) (sqrt.f64 (sqrt.f64 k)))) (sqrt.f64 (sqrt.f64 k)))
(/.f64 (.f64 (pow.f64 (sqrt.f64 k) -1/2) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (sqrt.f64 (sqrt.f64 k)))
(/.f64 (.f64 (/.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (sqrt.f64 (sqrt.f64 k))) (cbrt.f64 (pow.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2))) (sqrt.f64 (sqrt.f64 k)))
(/.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 1)
(/.f64 (/.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 (sqrt.f64 k))) (sqrt.f64 (sqrt.f64 (sqrt.f64 k)))) (sqrt.f64 (sqrt.f64 (sqrt.f64 k))))
(pow.f64 (cbrt.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))) 3)
(pow.f64 (pow.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 3) 1/3)
(pow.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 1)
(cbrt.f64 (pow.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 3))
(neg.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (neg.f64 (sqrt.f64 k))))
(neg.f64 (.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) -1))
(neg.f64 (.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (/.f64 -1 (sqrt.f64 k))))
(neg.f64 (neg.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))))
(neg.f64 (.f64 (/.f64 -1 (sqrt.f64 k)) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))
(neg.f64 (pow.f64 (neg.f64 (cbrt.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))) 3))
(if (!=.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) 0) (/.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (/.f64 (sqrt.f64 k) (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)))) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 0) (/.f64 1 (/.f64 (sqrt.f64 k) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))) (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 0) (pow.f64 (/.f64 1 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))) -1) (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 -1 0) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) 0) (/.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) (/.f64 (sqrt.f64 k) (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 0) (/.f64 -1 (/.f64 (neg.f64 (sqrt.f64 k)) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))) (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 1 0) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))

Outputs
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 n) (log.f64 (*.f64 2 (PI.f64)))))))
(pow.f64 (exp.f64 (*.f64 1/2 (-.f64 1 k))) (+.f64 (log.f64 n) (log.f64 (+.f64 (PI.f64) (PI.f64)))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 n) (log.f64 (*.f64 2 (PI.f64)))))))
(pow.f64 (exp.f64 (*.f64 1/2 (-.f64 1 k))) (+.f64 (log.f64 n) (log.f64 (+.f64 (PI.f64) (PI.f64)))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 n) (log.f64 (*.f64 2 (PI.f64)))))))
(pow.f64 (exp.f64 (*.f64 1/2 (-.f64 1 k))) (+.f64 (log.f64 n) (log.f64 (+.f64 (PI.f64) (PI.f64)))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (log.f64 n) (log.f64 (*.f64 2 (PI.f64)))))))
(pow.f64 (exp.f64 (*.f64 1/2 (-.f64 1 k))) (+.f64 (log.f64 n) (log.f64 (+.f64 (PI.f64) (PI.f64)))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64)))))))
(pow.f64 (exp.f64 (*.f64 1/2 (-.f64 1 k))) (-.f64 (log.f64 (+.f64 (PI.f64) (PI.f64))) (log.f64 (pow.f64 n -1))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64)))))))
(pow.f64 (exp.f64 (*.f64 1/2 (-.f64 1 k))) (-.f64 (log.f64 (+.f64 (PI.f64) (PI.f64))) (log.f64 (pow.f64 n -1))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64)))))))
(pow.f64 (exp.f64 (*.f64 1/2 (-.f64 1 k))) (-.f64 (log.f64 (+.f64 (PI.f64) (PI.f64))) (log.f64 (pow.f64 n -1))))
(exp.f64 (.f64 1/2 (.f64 (-.f64 1 k) (+.f64 (.f64 -1 (log.f64 (/.f64 1 n))) (log.f64 (.f64 2 (PI.f64)))))))
(pow.f64 (exp.f64 (*.f64 1/2 (-.f64 1 k))) (-.f64 (log.f64 (+.f64 (PI.f64) (PI.f64))) (log.f64 (pow.f64 n -1))))
(pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2)
(sqrt.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n))
(+.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (.f64 -1/2 (.f64 k (.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (log.f64 (.f64 2 (*.f64 n (PI.f64))))))))
(fma.f64 (.f64 k (.f64 (sqrt.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n)) (log.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n)))) -1/2 (sqrt.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n)))
(+.f64 (.f64 1/2 (.f64 (pow.f64 k 2) (.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (pow.f64 (.f64 -1/2 (log.f64 (.f64 2 (.f64 n (PI.f64))))) 2)))) (+.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (.f64 -1/2 (.f64 k (.f64 (pow.f64 (.f64 2 (.f64 n (PI.f64))) 1/2) (log.f64 (.f64 2 (*.f64 n (PI.f64)))))))))
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(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
(neg.f64 (.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (/.f64 -1 (sqrt.f64 k))))
(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
(neg.f64 (neg.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))))
(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
(neg.f64 (.f64 (/.f64 -1 (sqrt.f64 k)) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))
(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
(neg.f64 (pow.f64 (neg.f64 (cbrt.f64 (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))) 3))
(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
(if (!=.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) 0) (/.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) (/.f64 (sqrt.f64 k) (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)))) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 0) (/.f64 1 (/.f64 (sqrt.f64 k) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))) (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 0) (pow.f64 (/.f64 (sqrt.f64 k) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) -1) (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 0) (pow.f64 (/.f64 1 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))) -1) (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) 0) (/.f64 1 (pow.f64 (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) -1)) (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 -1 0) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) 0) (/.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 2)) (/.f64 (sqrt.f64 k) (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))))) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 0) (/.f64 -1 (/.f64 (neg.f64 (sqrt.f64 k)) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))))) (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) 0) (pow.f64 (/.f64 (sqrt.f64 k) (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))) -1) (/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))
(if (!=.f64 1 0) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)) (/.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k)))

localize41.0ms (0.2%)

Local error

Found 3 expressions with local error:

New	Error	Program
✓	0.3b	(*.f64 (+.f64 (PI.f64) (PI.f64)) n)
✓	0.3b	(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
✓	0.3b	(pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))

New

Error

Program

✓

0.3b

(*.f64 (+.f64 (PI.f64) (PI.f64)) n)

✓

0.3b

(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))

✓

0.3b

(pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))

Compiler

Compiled 51 to 26 computations (49% saved)

series136.0ms (0.7%)

Counts

3 → 40

Calls

15 calls:

Time	Variable		Point	Expression
3.0ms	k	@	inf	(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
2.0ms	k	@	-inf	(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
1.0ms	n	@	0	(pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))
1.0ms	n	@	inf	(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
1.0ms	n	@	-inf	(pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))

Compiler

Compiled 9990 to 7055 computations (29.4% saved)

rewrite2.0s (9.4%)

Algorithm

1×	batch-egg-rewrite

Rules

64×	`egg-rr`
42×	`egg-rr`
32×	`egg-rr`

Counts

3 → 138

Calls

Call 1

Inputs
(pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))
(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))
(*.f64 (+.f64 (PI.f64) (PI.f64)) n)

Inputs

(pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2)))

(/.f64 (pow.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 k 1/2 -1/2))) (sqrt.f64 k))

(*.f64 (+.f64 (PI.f64) (PI.f64)) n)

Outputs
((#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))) (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))) 2)))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))) 2)) (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (neg.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))))) (neg.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))) 2))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (+.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))) 2)) (+.f64 0 (.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))) 0))) (-.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))) 0))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (+.f64 0 (+.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))) 2)) (.f64 0 (neg.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))))))) (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 (+.f64 0 (-.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))) 2)) 0)) (+.f64 0 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 -1 (neg.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))))))) (#(struct:change #<rule egg-rr> (2) ((x .f64 1 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))))))) (#(struct:change #<rule 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n) (neg.f64 (fma.f64 1/2 k -1/2))) 3) 1/3)))) (#(struct:change #<rule egg-rr> (2) ((x pow.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))) 3)))) (#(struct:change #<rule egg-rr> (2) ((x pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))) 1)))) (#(struct:change #<rule egg-rr> (2) ((x cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))) 3))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (neg.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))))))) (#(struct:change #<rule egg-rr> (2) ((x neg.f64 (-.f64 (neg.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))) 0))))) (#(struct:change #<rule egg-rr> (2) ((x if (!=.f64 (cbrt.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2)))) 0) (/.f64 (cbrt.f64 (pow.f64 (pow.f64 (.f64 (+.f64 (PI.f64) (PI.f64)) n) (neg.f64 (fma.f64 1/2 k -1/2))) 2)) (/.f64 1 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Outputs

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simplify2.0s (9.5%)

Algorithm

1×

egglog

Details

analyze103.0ms (0.5%)

sample2.6s (12.3%)

preprocess4.3s (20.7%)

simplify1.8s (8.5%)

eval1.0ms (0%)

prune3.0ms (0%)

localize43.0ms (0.2%)

series288.0ms (1.4%)

rewrite2.5s (12%)

simplify5.1s (24.6%)

localize41.0ms (0.2%)

series136.0ms (0.7%)

rewrite2.0s (9.4%)

simplify2.0s (9.5%)

end0.0ms (0%)

Profiling