Details

analyze343.0ms (3.6%)

Algorithm

1×

Search

True	Other	False	Iter
0%	99.9%	0.1%	0
0%	99.9%	0.1%	1
0%	50%	50%	2
0%	50%	50%	3
0%	50%	50%	4
12.5%	37.5%	50%	5
15.6%	31.2%	53.2%	6
17.2%	25%	57.9%	7
18%	20.3%	61.8%	8
18.3%	16.4%	65.3%	9
18.5%	13.7%	67.8%	10
18.6%	11.5%	69.9%	11
18.7%	10%	71.3%	12
18.7%	8.9%	72.4%	13
18.7%	8.2%	73.1%	14

Compiler

Compiled 19 to 16 computations (15.8% saved)

sample3.1s (32.7%)

Results

1.3s	8256×	body	128	valid
1.1s	1388×	body	1024	invalid
359.0ms	659×	body	512	invalid
182.0ms	1212×	body	128	invalid
118.0ms	376×	body	256	invalid

Compiler

Compiled 56 to 47 computations (16.1% saved)

simplify47.0ms (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

0.3b

Counts

3 → 2

Alt Table

Status	Error	Program
	0.4b	(.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))
▶	0.4b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)) (sqrt.f64 k))

Compiler

Compiled 97 to 77 computations (20.6% saved)

localize9.0ms (0.1%)

Local error

Found 3 expressions with local error:

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

series543.0ms (5.7%)

Counts

3 → 48

Calls

3 calls:

302.0ms	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)) (sqrt.f64 k))
238.0ms	(pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2))
3.0ms	(.f64 (.f64 2 (PI.f64)) n)

rewrite22.0ms (0.2%)

Algorithm

1×	rewrite-expression-head

Rules

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

Counts

3 → 108

Calls

3 calls:

6.0ms	(/.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)

simplify122.0ms (1.3%)

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

prune207.0ms (2.2%)

Pruning

8 alts after pruning (8 fresh and 0 done)

	Pruned	Kept	Total
New	160	8	168
Fresh	1	0	1
Picked	1	0	1
Done	0	0	0
Total	162	8	170

Error

Counts

170 → 8

Alt Table

Status	Error	Program
	24.7b	(/.f64 (pow.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (sqrt.f64 (fma.f64 k -1/2 1/2))) (sqrt.f64 (fma.f64 k -1/2 1/2))) (sqrt.f64 k))
	0.5b	(/.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (/.f64 (fma.f64 k -1/2 1/2) 2)) (/.f64 (sqrt.f64 k) (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (/.f64 (fma.f64 k -1/2 1/2) 2))))
▶	0.3b	(.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 k)))
	0.4b	(.f64 (sqrt.f64 (/.f64 1 k)) (pow.f64 (.f64 n (*.f64 2 (PI.f64))) (fma.f64 k -1/2 1/2)))
	0.3b	(/.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))))))
	0.4b	(.f64 (/.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (sqrt.f64 (sqrt.f64 k))) (/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 (sqrt.f64 k))))
	20.5b	(/.f64 (.f64 (sqrt.f64 (.f64 n (PI.f64))) (+.f64 (sqrt.f64 2) (.f64 (sqrt.f64 2) (-.f64 (.f64 1/8 (.f64 (.f64 k k) (pow.f64 (log.f64 (.f64 n (.f64 2 (PI.f64)))) 2))) (.f64 1/2 (.f64 k (log.f64 (.f64 n (.f64 2 (PI.f64)))))))))) (sqrt.f64 k))
	1.1b	(/.f64 (.f64 (cbrt.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2))) (cbrt.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)))) (/.f64 (sqrt.f64 k) (cbrt.f64 (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)))))

Compiler

Compiled 4341 to 2842 computations (34.5% saved)

localize9.0ms (0.1%)

Local error

Found 4 expressions with local error:

New	Error	Program
✓	0.0b	(.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 k)))
✓	0.1b	(sqrt.f64 (.f64 n (.f64 2 (PI.f64))))
✓	0.2b	(.f64 n (.f64 2 (PI.f64)))
✓	0.3b	(/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 k))

series141.0ms (1.5%)

Counts

4 → 12

Calls

4 calls:

78.0ms	(.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 k)))
49.0ms	(/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 k))
12.0ms	(sqrt.f64 (.f64 n (.f64 2 (PI.f64))))
3.0ms	(.f64 n (.f64 2 (PI.f64)))

rewrite37.0ms (0.4%)

Algorithm

1×	rewrite-expression-head

Rules

52×	`times-frac_binary64`
51×	`add-sqr-sqrt_binary64`
43×	`sqrt-prod_binary64`
40×	`add-exp-log_binary64`
38×	`*-un-lft-identity_binary64`

Counts

4 → 145

Calls

4 calls:

13.0ms	(.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 k)))
6.0ms	(/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 k))
4.0ms	(.f64 n (.f64 2 (PI.f64)))
3.0ms	(sqrt.f64 (.f64 n (.f64 2 (PI.f64))))

simplify177.0ms (1.9%)

Algorithm

1×	egg-herbie

Rules

482×	`associate-/l*_binary64`
441×	`*-commutative_binary64`
366×	`fma-def_binary64`
350×	`associate-*r/_binary64`
319×	`associate-*l/_binary64`

Counts

157 → 209

Iterations

Useful iterations: 2 (0.0ms)

Iter	Nodes	Cost
0	182	4123
1	362	3862
2	1318	3834
3	3693	3834
4	4499	3834
5	4953	3834
6	4942	3834

prune289.0ms (3.1%)

Pruning

7 alts after pruning (7 fresh and 0 done)

	Pruned	Kept	Total
New	237	2	239
Fresh	2	5	7
Picked	1	0	1
Done	0	0	0
Total	240	7	247

Error

Counts

247 → 7

Alt Table

Status	Error	Program
	9.1b	(.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (sqrt.f64 (/.f64 (.f64 n (.f64 2 (PI.f64))) k)))
	24.7b	(/.f64 (pow.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (sqrt.f64 (fma.f64 k -1/2 1/2))) (sqrt.f64 (fma.f64 k -1/2 1/2))) (sqrt.f64 k))
	0.4b	(.f64 (sqrt.f64 (/.f64 1 k)) (pow.f64 (.f64 n (*.f64 2 (PI.f64))) (fma.f64 k -1/2 1/2)))
▶	0.3b	(/.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))))))
	0.4b	(.f64 (/.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (sqrt.f64 (sqrt.f64 k))) (/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 (sqrt.f64 k))))
	0.7b	(.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (/.f64 (sqrt.f64 (.f64 (.f64 (cbrt.f64 n) (cbrt.f64 n)) (.f64 (cbrt.f64 n) (.f64 2 (PI.f64))))) (sqrt.f64 k)))
	1.1b	(/.f64 (.f64 (cbrt.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2))) (cbrt.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)))) (/.f64 (sqrt.f64 k) (cbrt.f64 (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)))))

Compiler

Compiled 7020 to 4330 computations (38.3% saved)

localize14.0ms (0.2%)

Local error

Found 4 expressions with local error:

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

series116.0ms (1.2%)

Counts

2 → 0

Calls

2 calls:

80.0ms	(/.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))))))
36.0ms	(/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))))

rewrite55.0ms (0.6%)

Algorithm

1×	rewrite-expression-head

Rules

308×	`times-frac_binary64`
138×	`*-un-lft-identity_binary64`
138×	`add-sqr-sqrt_binary64`
138×	`add-cube-cbrt_binary64`
130×	`sqrt-prod_binary64`

Counts

2 → 247

Calls

2 calls:

14.0ms	(/.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))))))
6.0ms	(/.f64 (sqrt.f64 k) (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))))

simplify350.0ms (3.7%)

Algorithm

1×	egg-herbie

Rules

711×	`associate-/r*_binary64`
530×	`*-commutative_binary64`
467×	`associate-r_binary64`
297×	`associate-/r/_binary64`
256×	`fma-neg_binary64`

Counts

247 → 324

Iterations

Useful iterations: 2 (0.0ms)

Iter	Nodes	Cost
0	336	12416
1	667	12123
2	2307	11895
3	3774	11895
4	4911	11895
5	4969	11895

prune474.0ms (5%)

Pruning

7 alts after pruning (6 fresh and 1 done)

	Pruned	Kept	Total
New	381	1	382
Fresh	1	5	6
Picked	0	1	1
Done	0	0	0
Total	382	7	389

Error

Counts

389 → 7

Alt Table

Status	Error	Program
	9.1b	(.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (sqrt.f64 (/.f64 (.f64 n (.f64 2 (PI.f64))) k)))
	24.7b	(/.f64 (pow.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (sqrt.f64 (fma.f64 k -1/2 1/2))) (sqrt.f64 (fma.f64 k -1/2 1/2))) (sqrt.f64 k))
▶	0.4b	(.f64 (sqrt.f64 (/.f64 1 k)) (pow.f64 (.f64 n (*.f64 2 (PI.f64))) (fma.f64 k -1/2 1/2)))
✓	0.3b	(/.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))))))
	0.4b	(.f64 (/.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (sqrt.f64 (sqrt.f64 k))) (/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 (sqrt.f64 k))))
	0.7b	(/.f64 (/.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (fabs.f64 (cbrt.f64 k))) (/.f64 (sqrt.f64 (cbrt.f64 k)) (sqrt.f64 (.f64 n (*.f64 2 (PI.f64))))))
	1.1b	(/.f64 (.f64 (cbrt.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2))) (cbrt.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)))) (/.f64 (sqrt.f64 k) (cbrt.f64 (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)))))

Compiler

Compiled 13468 to 8590 computations (36.2% saved)

localize11.0ms (0.1%)

Local error

Found 4 expressions with local error:

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

series571.0ms (6%)

Counts

3 → 36

Calls

3 calls:

322.0ms	(.f64 (sqrt.f64 (/.f64 1 k)) (pow.f64 (.f64 n (*.f64 2 (PI.f64))) (fma.f64 k -1/2 1/2)))
243.0ms	(pow.f64 (.f64 n (.f64 2 (PI.f64))) (fma.f64 k -1/2 1/2))
6.0ms	(sqrt.f64 (/.f64 1 k))

rewrite23.0ms (0.2%)

Algorithm

1×	rewrite-expression-head

Rules

38×	`add-sqr-sqrt_binary64`
32×	`sqrt-prod_binary64`
24×	`times-frac_binary64`
23×	`*-un-lft-identity_binary64`
21×	`add-exp-log_binary64`

Counts

3 → 99

Calls

3 calls:

10.0ms	(.f64 (sqrt.f64 (/.f64 1 k)) (pow.f64 (.f64 n (*.f64 2 (PI.f64))) (fma.f64 k -1/2 1/2)))
5.0ms	(pow.f64 (.f64 n (.f64 2 (PI.f64))) (fma.f64 k -1/2 1/2))
2.0ms	(sqrt.f64 (/.f64 1 k))

simplify168.0ms (1.8%)

Algorithm

1×	egg-herbie

Rules

865×	`fma-def_binary64`
313×	`*-commutative_binary64`
250×	`log-prod_binary64`
240×	`distribute-rgt-in_binary64`
239×	`fma-neg_binary64`

Counts

135 → 135

Iterations

Useful iterations: 3 (0.0ms)

Iter	Nodes	Cost
0	160	3158
1	334	2826
2	961	2798
3	2624	2790
4	4690	2790
5	4968	2790
6	4992	2790
7	4965	2790

prune208.0ms (2.2%)

Pruning

7 alts after pruning (5 fresh and 2 done)

	Pruned	Kept	Total
New	165	0	165
Fresh	0	5	5
Picked	0	1	1
Done	0	1	1
Total	165	7	172

Error

Counts

172 → 7

Alt Table

Status	Error	Program
	9.1b	(.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (sqrt.f64 (/.f64 (.f64 n (.f64 2 (PI.f64))) k)))
	24.7b	(/.f64 (pow.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (sqrt.f64 (fma.f64 k -1/2 1/2))) (sqrt.f64 (fma.f64 k -1/2 1/2))) (sqrt.f64 k))
✓	0.4b	(.f64 (sqrt.f64 (/.f64 1 k)) (pow.f64 (.f64 n (*.f64 2 (PI.f64))) (fma.f64 k -1/2 1/2)))
✓	0.3b	(/.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))))))
	0.4b	(.f64 (/.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (sqrt.f64 (sqrt.f64 k))) (/.f64 (sqrt.f64 (.f64 n (.f64 2 (PI.f64)))) (sqrt.f64 (sqrt.f64 k))))
	0.7b	(/.f64 (/.f64 (pow.f64 (.f64 n (.f64 2 (PI.f64))) (.f64 k -1/2)) (fabs.f64 (cbrt.f64 k))) (/.f64 (sqrt.f64 (cbrt.f64 k)) (sqrt.f64 (.f64 n (*.f64 2 (PI.f64))))))
	1.1b	(/.f64 (.f64 (cbrt.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2))) (cbrt.f64 (pow.f64 (.f64 (.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)))) (/.f64 (sqrt.f64 k) (cbrt.f64 (pow.f64 (.f64 (*.f64 2 (PI.f64)) n) (fma.f64 k -1/2 1/2)))))

Compiler

Compiled 4020 to 2953 computations (26.5% saved)

regimes2.3s (24.1%)

Accuracy

Total 0.3b remaining (78.9%)

Threshold costs 0.3b (78.9%)

Counts

377 → 1

Compiler

Compiled 40852 to 30824 computations (24.5% saved)

simplify3.0ms (0%)

Algorithm

1×	egg-herbie

Rules

5×	`*-commutative_binary64`
1×	`1-exp_binary64`

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	17	35
1	24	35
2	23	35

end139.0ms (1.5%)

Compiler: Compiled 220 to 169 computations (23.2% saved)

Details

analyze343.0ms (3.6%)

sample3.1s (32.7%)

simplify47.0ms (0.5%)

prune8.0ms (0.1%)

localize9.0ms (0.1%)

series543.0ms (5.7%)

rewrite22.0ms (0.2%)

simplify122.0ms (1.3%)

prune207.0ms (2.2%)

localize9.0ms (0.1%)

series141.0ms (1.5%)

rewrite37.0ms (0.4%)

simplify177.0ms (1.9%)

prune289.0ms (3.1%)

localize14.0ms (0.2%)

series116.0ms (1.2%)

rewrite55.0ms (0.6%)

simplify350.0ms (3.7%)

prune474.0ms (5%)

localize11.0ms (0.1%)

series571.0ms (6%)

rewrite23.0ms (0.2%)

simplify168.0ms (1.8%)

prune208.0ms (2.2%)

regimes2.3s (24.1%)

simplify3.0ms (0%)

end139.0ms (1.5%)

Profiling