Details

analyze393.0ms (3.4%)

Algorithm

1×

Search

True	Other	False	Iter
0%	99.9%	0.1%	0
0%	99.9%	0.1%	1
0%	50%	50%	2
25%	25%	50%	3
25%	25%	50%	4
25%	25%	50%	5
25%	21.9%	53.2%	6
25%	18.7%	56.3%	7
25%	15.6%	59.4%	8
25%	13.3%	61.8%	9
25%	11.3%	63.7%	10
25%	10%	65.1%	11
25%	8.9%	66.1%	12
25%	8.1%	66.9%	13
25%	7.6%	67.4%	14

Compiler

Compiled 19 to 16 computations (15.8% saved)

sample3.4s (28.8%)

Results

1.6s	8256×	body	128	valid
1.0s	1021×	body	1024	invalid
350.0ms	549×	body	512	invalid
122.0ms	695×	body	128	invalid
120.0ms	287×	body	256	invalid

Compiler

Compiled 56 to 47 computations (16.1% saved)

simplify63.0ms (0.5%)

Algorithm

1×	egg-herbie

Rules

672×	`associate-/l*_binary64`
625×	`associate-/r*_binary64`
461×	`div-sub_binary64`
445×	`distribute-frac-neg_binary64`
288×	`distribute-rgt-neg-out_binary64`

Counts

1 → 2

Iterations

Useful iterations: 5 (0.0ms)

Iter	Nodes	Cost
0	14	25
1	32	22
2	65	22
3	129	22
4	280	22
5	634	20
6	1332	20
7	3121	20
8	4914	20
9	4550	20

prune8.0ms (0.1%)

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

0.5b

Counts

3 → 2

Alt Table

Status	Error	Program
▶	0.6b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)) (sqrt.f64 k))
	0.6b	(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))

Compiler

Compiled 97 to 77 computations (20.6% saved)

localize20.0ms (0.2%)

Local error

Found 3 expressions with local error:

New	Error	Program
✓	0.1b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)) (sqrt.f64 k))
✓	0.2b	(.f64 (.f64 2 (PI.f64)) n)
✓	0.5b	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2))

series842.0ms (7.2%)

Counts

3 → 48

Calls

3 calls:

517.0ms	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)) (sqrt.f64 k))
321.0ms	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2))
5.0ms	(.f64 (.f64 2 (PI.f64)) n)

rewrite24.0ms (0.2%)

Algorithm

1×	rewrite-expression-head

Rules

36×	`times-frac_binary64`
26×	`add-exp-log_binary64`
26×	`*-un-lft-identity_binary64`
26×	`add-sqr-sqrt_binary64`
26×	`add-cube-cbrt_binary64`

Counts

3 → 108

Calls

3 calls:

5.0ms	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)) (sqrt.f64 k))
4.0ms	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2))
4.0ms	(.f64 (.f64 2 (PI.f64)) n)

simplify138.0ms (1.2%)

Algorithm

1×	egg-herbie

Rules

405×	`distribute-rgt-in_binary64`
399×	`distribute-lft-in_binary64`
243×	`fma-neg_binary64`
211×	`associate-/l*_binary64`
162×	`associate-l_binary64`

Counts

156 → 168

Iterations

Useful iterations: 2 (0.0ms)

Iter	Nodes	Cost
0	213	4057
1	428	3712
2	1390	3636
3	4506	3636
4	4960	3636

prune297.0ms (2.5%)

Pruning

7 alts after pruning (7 fresh and 0 done)

	Pruned	Kept	Total
New	161	7	168
Fresh	1	0	1
Picked	1	0	1
Done	0	0	0
Total	163	7	170

Error

0.0b

Counts

170 → 7

Alt Table

Status	Error	Program
	16.5b	(/.f64 (.f64 (pow.f64 (.f64 2 (PI.f64)) (fma.f64 k -1/2 1/2)) (pow.f64 n (fma.f64 k -1/2 1/2))) (sqrt.f64 k))
	1.1b	(.f64 (.f64 (cbrt.f64 (/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)) (sqrt.f64 k))) (cbrt.f64 (/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)) (sqrt.f64 k)))) (cbrt.f64 (/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)) (sqrt.f64 k))))
	20.9b	(/.f64 (.f64 (sqrt.f64 (.f64 n (PI.f64))) (-.f64 (fma.f64 1/8 (.f64 (.f64 k k) (.f64 (sqrt.f64 2) (pow.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) 2))) (sqrt.f64 2)) (.f64 (sqrt.f64 2) (+.f64 (.f64 1/2 (.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) k)) (.f64 1/48 (.f64 (pow.f64 k 3) (pow.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) 3))))))) (sqrt.f64 k))
	16.7b	(.f64 (/.f64 (pow.f64 (.f64 2 (PI.f64)) (fma.f64 k -1/2 1/2)) (fabs.f64 (cbrt.f64 k))) (/.f64 (pow.f64 n (fma.f64 k -1/2 1/2)) (sqrt.f64 (cbrt.f64 k))))
	0.6b	(.f64 (pow.f64 (/.f64 (.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2)) (sqrt.f64 (/.f64 1 k)))
▶	0.5b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (.f64 k -1/2)) (/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 n (*.f64 2 (PI.f64))))))
	16.9b	(.f64 (/.f64 (pow.f64 (.f64 2 (PI.f64)) (fma.f64 k -1/2 1/2)) (*.f64 (cbrt.f64 (sqrt.f64 k)) (cbrt.f64 (sqrt.f64 k)))) (/.f64 (pow.f64 n (fma.f64 k -1/2 1/2)) (cbrt.f64 (sqrt.f64 k))))

Compiler

Compiled 4341 to 2834 computations (34.7% saved)

localize11.0ms (0.1%)

Local error

Found 4 expressions with local error:

New	Error	Program
	0.2b	(.f64 (.f64 2 (PI.f64)) n)
✓	0.3b	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (*.f64 k -1/2))
✓	0.4b	(sqrt.f64 (.f64 n (.f64 2 (PI.f64))))
✓	0.4b	(/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))))

series473.0ms (4%)

Counts

3 → 24

Calls

3 calls:

416.0ms	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (*.f64 k -1/2))
37.0ms	(/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))))
19.0ms	(sqrt.f64 (.f64 n (.f64 2 (PI.f64))))

rewrite41.0ms (0.4%)

Algorithm

1×	rewrite-expression-head

Rules

24×	`times-frac_binary64`
23×	`sqrt-prod_binary64`
20×	`*-un-lft-identity_binary64`
20×	`add-sqr-sqrt_binary64`
20×	`add-cube-cbrt_binary64`

Counts

3 → 82

Calls

3 calls:

15.0ms	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (*.f64 k -1/2))
11.0ms	(/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))))
5.0ms	(sqrt.f64 (.f64 n (.f64 2 (PI.f64))))

simplify162.0ms (1.4%)

Algorithm

1×	egg-herbie

Rules

411×	`associate-l_binary64`
399×	`associate-r_binary64`
362×	`distribute-rgt-in_binary64`
357×	`distribute-lft-in_binary64`
345×	`exp-prod_binary64`

Counts

106 → 103

Iterations

Useful iterations: 3 (0.0ms)

Iter	Nodes	Cost
0	160	2242
1	303	2093
2	855	2081
3	4547	2077
4	5184	2077

prune273.0ms (2.3%)

Pruning

7 alts after pruning (6 fresh and 1 done)

	Pruned	Kept	Total
New	129	2	131
Fresh	2	4	6
Picked	0	1	1
Done	0	0	0
Total	131	7	138

Error

0.0b

Counts

138 → 7

Alt Table

Status	Error	Program
	20.9b	(/.f64 (.f64 (sqrt.f64 (.f64 n (PI.f64))) (-.f64 (fma.f64 1/8 (.f64 (.f64 k k) (.f64 (sqrt.f64 2) (pow.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) 2))) (sqrt.f64 2)) (.f64 (sqrt.f64 2) (+.f64 (.f64 1/2 (.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) k)) (.f64 1/48 (.f64 (pow.f64 k 3) (pow.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) 3))))))) (sqrt.f64 k))
	16.7b	(.f64 (/.f64 (pow.f64 (.f64 2 (PI.f64)) (fma.f64 k -1/2 1/2)) (fabs.f64 (cbrt.f64 k))) (/.f64 (pow.f64 n (fma.f64 k -1/2 1/2)) (sqrt.f64 (cbrt.f64 k))))
	0.6b	(.f64 (pow.f64 (/.f64 (.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2)) (sqrt.f64 (/.f64 1 k)))
▶	0.5b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (.f64 k -1/2)) (/.f64 1 (/.f64 (sqrt.f64 (.f64 n (*.f64 2 (PI.f64)))) (sqrt.f64 k))))
	0.6b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (.f64 k -1/2)) (.f64 (/.f64 1 (sqrt.f64 n)) (/.f64 (sqrt.f64 k) (sqrt.f64 (*.f64 2 (PI.f64))))))
✓	0.5b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (.f64 k -1/2)) (/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 n (*.f64 2 (PI.f64))))))
	16.9b	(.f64 (/.f64 (pow.f64 (.f64 2 (PI.f64)) (fma.f64 k -1/2 1/2)) (*.f64 (cbrt.f64 (sqrt.f64 k)) (cbrt.f64 (sqrt.f64 k)))) (/.f64 (pow.f64 n (fma.f64 k -1/2 1/2)) (cbrt.f64 (sqrt.f64 k))))

Compiler

Compiled 3784 to 2490 computations (34.2% saved)

localize14.0ms (0.1%)

Local error

Found 4 expressions with local error:

New	Error	Program
	0.2b	(.f64 (.f64 2 (PI.f64)) n)
	0.3b	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (*.f64 k -1/2))
✓	0.3b	(/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 k))
	0.4b	(sqrt.f64 (.f64 n (.f64 2 (PI.f64))))

series55.0ms (0.5%)

Counts

1 → 0

Calls

1 calls:

54.0ms

(/.f64 (sqrt.f64 (*.f64 n (*.f64 2 (PI.f64)))) (sqrt.f64 k))

rewrite18.0ms (0.2%)

Algorithm

1×	rewrite-expression-head

Rules

24×	`times-frac_binary64`
22×	`sqrt-prod_binary64`
18×	`*-un-lft-identity_binary64`
18×	`add-sqr-sqrt_binary64`
18×	`add-cube-cbrt_binary64`

Counts

1 → 49

Calls

1 calls:

11.0ms

(/.f64 (sqrt.f64 (*.f64 n (*.f64 2 (PI.f64)))) (sqrt.f64 k))

simplify171.0ms (1.5%)

Algorithm

1×	egg-herbie

Rules

736×	`times-frac_binary64`
580×	`div-sub_binary64`
535×	`associate-/l*_binary64`
477×	`associate-/l/_binary64`
332×	`distribute-rgt-in_binary64`

Counts

49 → 53

Iterations

Useful iterations: 1 (0.0ms)

Iter	Nodes	Cost
0	82	1265
1	129	1203
2	278	1203
3	782	1203
4	2454	1203
5	2368	1203
6	2840	1203
7	3397	1203
8	3102	1203
9	3836	1203
10	4692	1203

prune254.0ms (2.2%)

Pruning

6 alts after pruning (5 fresh and 1 done)

	Pruned	Kept	Total
New	132	1	133
Fresh	1	4	5
Picked	1	0	1
Done	0	1	1
Total	134	6	140

Error

0.0b

Counts

140 → 6

Alt Table

Status	Error	Program
	0.6b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (.f64 k -1/2)) (/.f64 1 (/.f64 (sqrt.f64 n) (/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 2 (PI.f64)))))))
	16.9b	(.f64 (/.f64 (pow.f64 (.f64 2 (PI.f64)) (fma.f64 k -1/2 1/2)) (*.f64 (cbrt.f64 (sqrt.f64 k)) (cbrt.f64 (sqrt.f64 k)))) (/.f64 (pow.f64 n (fma.f64 k -1/2 1/2)) (cbrt.f64 (sqrt.f64 k))))
	20.9b	(/.f64 (.f64 (sqrt.f64 (.f64 n (PI.f64))) (-.f64 (fma.f64 1/8 (.f64 (.f64 k k) (.f64 (sqrt.f64 2) (pow.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) 2))) (sqrt.f64 2)) (.f64 (sqrt.f64 2) (+.f64 (.f64 1/2 (.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) k)) (.f64 1/48 (.f64 (pow.f64 k 3) (pow.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) 3))))))) (sqrt.f64 k))
	16.7b	(.f64 (/.f64 (pow.f64 (.f64 2 (PI.f64)) (fma.f64 k -1/2 1/2)) (fabs.f64 (cbrt.f64 k))) (/.f64 (pow.f64 n (fma.f64 k -1/2 1/2)) (sqrt.f64 (cbrt.f64 k))))
▶	0.6b	(.f64 (pow.f64 (/.f64 (.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2)) (sqrt.f64 (/.f64 1 k)))
✓	0.5b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (.f64 k -1/2)) (/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 n (*.f64 2 (PI.f64))))))

Compiler

Compiled 4062 to 2765 computations (31.9% saved)

localize25.0ms (0.2%)

Local error

Found 4 expressions with local error:

New	Error	Program
✓	0.1b	(sqrt.f64 (/.f64 1 k))
✓	0.2b	(.f64 (pow.f64 (/.f64 (.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2)) (sqrt.f64 (/.f64 1 k)))
✓	0.3b	(/.f64 (*.f64 (PI.f64) -2) (/.f64 -1 n))
✓	0.5b	(pow.f64 (/.f64 (*.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2))

series578.0ms (4.9%)

Counts

4 → 48

Calls

4 calls:

290.0ms	(.f64 (pow.f64 (/.f64 (.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2)) (sqrt.f64 (/.f64 1 k)))
280.0ms	(pow.f64 (/.f64 (*.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2))
5.0ms	(sqrt.f64 (/.f64 1 k))
3.0ms	(/.f64 (*.f64 (PI.f64) -2) (/.f64 -1 n))

rewrite56.0ms (0.5%)

Algorithm

1×	rewrite-expression-head

Rules

101×	`times-frac_binary64`
78×	`add-sqr-sqrt_binary64`
55×	`*-un-lft-identity_binary64`
51×	`add-cube-cbrt_binary64`
50×	`add-exp-log_binary64`

Counts

4 → 187

Calls

4 calls:

17.0ms	(.f64 (pow.f64 (/.f64 (.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2)) (sqrt.f64 (/.f64 1 k)))
14.0ms	(sqrt.f64 (/.f64 1 k))
6.0ms	(pow.f64 (/.f64 (*.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2))
5.0ms	(/.f64 (*.f64 (PI.f64) -2) (/.f64 -1 n))

simplify173.0ms (1.5%)

Algorithm

1×	egg-herbie

Rules

522×	`times-frac_binary64`
329×	`distribute-rgt-in_binary64`
320×	`distribute-lft-in_binary64`
186×	`sqr-pow_binary64`
176×	`associate-l_binary64`

Counts

235 → 297

Iterations

Useful iterations: 2 (0.0ms)

Iter	Nodes	Cost
0	310	5290
1	685	4762
2	2107	4734
3	4973	4734
4	4944	4734

prune432.0ms (3.7%)

Pruning

6 alts after pruning (4 fresh and 2 done)

	Pruned	Kept	Total
New	296	1	297
Fresh	1	3	4
Picked	0	1	1
Done	0	1	1
Total	297	6	303

Error

0.0b

Counts

303 → 6

Alt Table

Status	Error	Program
	0.5b	(.f64 (.f64 (pow.f64 (/.f64 (PI.f64) (/.f64 1 (sqrt.f64 n))) (fma.f64 k -1/2 1/2)) (pow.f64 (/.f64 -2 (/.f64 -1 (sqrt.f64 n))) (fma.f64 k -1/2 1/2))) (sqrt.f64 (/.f64 1 k)))
✓	0.6b	(.f64 (pow.f64 (/.f64 (.f64 (PI.f64) -2) (/.f64 -1 n)) (fma.f64 k -1/2 1/2)) (sqrt.f64 (/.f64 1 k)))
	0.6b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (.f64 k -1/2)) (/.f64 1 (/.f64 (sqrt.f64 n) (/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 2 (PI.f64)))))))
	16.9b	(.f64 (/.f64 (pow.f64 (.f64 2 (PI.f64)) (fma.f64 k -1/2 1/2)) (*.f64 (cbrt.f64 (sqrt.f64 k)) (cbrt.f64 (sqrt.f64 k)))) (/.f64 (pow.f64 n (fma.f64 k -1/2 1/2)) (cbrt.f64 (sqrt.f64 k))))
	20.9b	(/.f64 (.f64 (sqrt.f64 (.f64 n (PI.f64))) (-.f64 (fma.f64 1/8 (.f64 (.f64 k k) (.f64 (sqrt.f64 2) (pow.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) 2))) (sqrt.f64 2)) (.f64 (sqrt.f64 2) (+.f64 (.f64 1/2 (.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) k)) (.f64 1/48 (.f64 (pow.f64 k 3) (pow.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) 3))))))) (sqrt.f64 k))
✓	0.5b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (.f64 k -1/2)) (/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 n (*.f64 2 (PI.f64))))))

Compiler

Compiled 7799 to 5883 computations (24.6% saved)

regimes3.6s (31.2%)

Accuracy

Total 0.3b remaining (88.1%)

Threshold costs 0.3b (88.1%)

Counts

406 → 1

Compiler

Compiled 45606 to 35902 computations (21.3% saved)

simplify2.0ms (0%)

Algorithm

1×	egg-herbie

Rules

4×	`*-commutative_binary64`

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	15	32
1	19	32
2	19	32

end153.0ms (1.3%)

Compiler: Compiled 225 to 174 computations (22.7% saved)

Details

analyze393.0ms (3.4%)

sample3.4s (28.8%)

simplify63.0ms (0.5%)

prune8.0ms (0.1%)

localize20.0ms (0.2%)

series842.0ms (7.2%)

rewrite24.0ms (0.2%)

simplify138.0ms (1.2%)

prune297.0ms (2.5%)

localize11.0ms (0.1%)

series473.0ms (4%)

rewrite41.0ms (0.4%)

simplify162.0ms (1.4%)

prune273.0ms (2.3%)

localize14.0ms (0.1%)

series55.0ms (0.5%)

rewrite18.0ms (0.2%)

simplify171.0ms (1.5%)

prune254.0ms (2.2%)

localize25.0ms (0.2%)

series578.0ms (4.9%)

rewrite56.0ms (0.5%)

simplify173.0ms (1.5%)

prune432.0ms (3.7%)

regimes3.6s (31.2%)

simplify2.0ms (0%)

end153.0ms (1.3%)

Profiling