Details

analyze105.0ms (1.2%)

Algorithm

1×

Search

True	Other	False	Iter
0%	12.3%	87.7%	0
0%	12.3%	87.7%	1
6.2%	6.2%	87.7%	2
6.2%	6.2%	87.7%	3
9.2%	3.1%	87.7%	4
9.2%	3.1%	87.7%	5
10.8%	1.5%	87.7%	6
10.8%	1.5%	87.7%	7
11.5%	0.8%	87.7%	8
11.5%	0.8%	87.7%	9
11.9%	0.4%	87.7%	10
11.9%	0.4%	87.7%	11
12.1%	0.2%	87.7%	12
12.1%	0.2%	87.7%	13
12.1%	0.1%	87.8%	14

Compiler

Compiled 42 to 32 computations (23.8% saved)

sample2.1s (24%)

Results

2.0s

8256×

body

128

valid

Compiler

Compiled 111 to 85 computations (23.4% saved)

simplify48.0ms (0.6%)

Algorithm

1×	egg-herbie

Rules

940×	`fma-neg_binary32`
929×	`fma-def_binary32`
740×	`times-frac_binary32`
329×	`div-sub_binary32`
317×	`associate-l_binary32`

Counts

1 → 2

Iterations

Useful iterations: 3 (0.0ms)

Iter	Nodes	Cost
0	19	40
1	45	35
2	107	35
3	264	33
4	605	33
5	2500	33
6	3754	33
7	5049	33

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

0.4b

Counts

3 → 2

Alt Table

Status	Error	Program
	0.6b	(/.f32 1 (+.f32 (+.f32 1 c) (.f32 (.f32 (/.f32 1 (sqrt.f32 (PI.f32))) (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) cosTheta)) (exp.f32 (*.f32 (neg.f32 cosTheta) cosTheta)))))
▶	0.5b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))

Status

Error

Program

0.6b

(/.f32 1 (+.f32 (+.f32 1 c) (*.f32 (*.f32 (/.f32 1 (sqrt.f32 (PI.f32))) (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) cosTheta)) (exp.f32 (*.f32 (neg.f32 cosTheta) cosTheta)))))

▶

0.5b

(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))

Compiler

Compiled 148 to 113 computations (23.6% saved)

localize61.0ms (0.7%)

Local error

Found 4 expressions with local error:

New	Error	Program
✓	0.1b	(sqrt.f32 (fma.f32 cosTheta -2 1))
✓	0.3b	(/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))
✓	0.3b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))
✓	0.4b	(.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))

series96.0ms (1.1%)

Counts

4 → 38

Calls

4 calls:

59.0ms	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))
20.0ms	(/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))
9.0ms	(.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))
6.0ms	(sqrt.f32 (fma.f32 cosTheta -2 1))

rewrite93.0ms (1.1%)

Algorithm

2×	batch-egg-rewrite

Rules

315×	`log1p-udef_binary32`
180×	`add-sqr-sqrt_binary32`
172×	`log1p-expm1-u_binary32`
172×	`expm1-log1p-u_binary32`
169×	`add-log-exp_binary32`

Counts

4 → 107

Calls

4 calls:

90.0ms	(sqrt.f32 (fma.f32 cosTheta -2 1))
90.0ms	(/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))
90.0ms	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))
90.0ms	(.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	17	79
1	367	75
2	4816	75
3	5237	75
0	0	0
1	0	0

simplify90.0ms (1%)

Algorithm

1×	egg-herbie

Rules

582×	`times-frac_binary32`
355×	`cancel-sign-sub-inv_binary32`
325×	`fma-neg_binary32`
246×	`fma-def_binary32`
222×	`associate-r_binary32`

Counts

145 → 153

Iterations

Useful iterations: 2 (0.0ms)

Iter	Nodes	Cost
0	212	2281
1	696	2147
2	2682	2095
3	4678	2095
4	5070	2095

prune211.0ms (2.4%)

Pruning

12 alts after pruning (12 fresh and 0 done)

	Pruned	Kept	Total
New	141	12	153
Fresh	1	0	1
Picked	1	0	1
Done	0	0	0
Total	143	12	155

Error

0.0b

Counts

155 → 12

Alt Table

Status	Error	Program
	0.9b	(.f32 (pow.f32 (sqrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) -1) (pow.f32 (sqrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) -1))
	0.8b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (pow.f32 (cbrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 3))))
	0.8b	(/.f32 1 (+.f32 (+.f32 1 c) (pow.f32 (sqrt.f32 (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 2)))
	0.9b	(.f32 (pow.f32 (pow.f32 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) 2) -1) (pow.f32 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) -1))
	13.7b	(/.f32 1 (+.f32 (+.f32 1 c) (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (.f32 (PI.f32) (pow.f32 (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)) 2))))))
	13.8b	(.f32 (/.f32 1 (-.f32 (pow.f32 (+.f32 1 c) 2) (/.f32 (fma.f32 cosTheta -2 1) (.f32 (PI.f32) (pow.f32 (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)) 2))))) (-.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))
	12.7b	(.f32 (/.f32 1 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (cbrt.f32 (pow.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) -2)))
	1.3b	(fma.f32 (sqrt.f32 (PI.f32)) cosTheta (.f32 (.f32 cosTheta cosTheta) (-.f32 (sqrt.f32 (PI.f32)) (*.f32 (+.f32 1 c) (PI.f32)))))
▶	0.5b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))
	0.7b	(/.f32 1 (+.f32 (+.f32 1 c) (.f32 (/.f32 1 (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))
	0.7b	(pow.f32 (pow.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) -1/2) 2)
	1.0b	(pow.f32 (/.f32 1 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) 3)

Compiler

Compiled 5864 to 3694 computations (37% saved)

localize74.0ms (0.9%)

Local error

Found 4 expressions with local error:

New	Error	Program
✓	0.1b	(-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1)
✓	0.3b	(/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))
✓	0.3b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))
	0.4b	(.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))

series225.0ms (2.6%)

Counts

3 → 26

Calls

3 calls:

112.0ms

(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))

87.0ms

(/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))

26.0ms

(-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1)

rewrite124.0ms (1.4%)

Algorithm

2×	batch-egg-rewrite

Rules

358×	`prod-diff_binary32`
203×	`add-sqr-sqrt_binary32`
197×	`expm1-log1p-u_binary32`
196×	`log1p-expm1-u_binary32`
191×	`add-log-exp_binary32`

Counts

3 → 158

Calls

3 calls:

115.0ms

(-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1)

115.0ms

(/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))

115.0ms

(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	20	64
1	414	60
2	4775	60
3	5536	60
0	0	0
1	0	0

simplify85.0ms (1%)

Algorithm

1×	egg-herbie

Rules

582×	`times-frac_binary32`
355×	`cancel-sign-sub-inv_binary32`
340×	`fma-neg_binary32`
217×	`fma-def_binary32`
212×	`associate-r_binary32`

Counts

184 → 197

Iterations

Useful iterations: 3 (0.0ms)

Iter	Nodes	Cost
0	196	2068
1	644	1938
2	2546	1829
3	4826	1827
4	5057	1827

prune553.0ms (6.4%)

Pruning

10 alts after pruning (10 fresh and 0 done)

	Pruned	Kept	Total
New	217	5	222
Fresh	6	5	11
Picked	1	0	1
Done	0	0	0
Total	224	10	234

Error

0.0b

Counts

234 → 10

Alt Table

Status	Error	Program
	0.8b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (pow.f32 (cbrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 3))))
	13.8b	(.f32 (/.f32 1 (-.f32 (pow.f32 (+.f32 1 c) 2) (/.f32 (fma.f32 cosTheta -2 1) (.f32 (PI.f32) (pow.f32 (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)) 2))))) (-.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))
	1.0b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (fma.f32 (pow.f32 (cbrt.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 2) (cbrt.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1)))) -1) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))
	1.0b	(/.f32 1 (+.f32 (+.f32 1 c) (+.f32 (fma.f32 (sqrt.f32 (/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (sqrt.f32 (/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (neg.f32 (.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 1))) (fma.f32 (neg.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) 1 (.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 1)))))
	1.0b	(pow.f32 (/.f32 1 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) 3)
	0.9b	(.f32 (pow.f32 (pow.f32 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) 2) -1) (pow.f32 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) -1))
	12.7b	(.f32 (/.f32 1 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (cbrt.f32 (pow.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) -2)))
	1.3b	(fma.f32 (sqrt.f32 (PI.f32)) cosTheta (.f32 (.f32 cosTheta cosTheta) (-.f32 (sqrt.f32 (PI.f32)) (*.f32 (+.f32 1 c) (PI.f32)))))
▶	0.7b	(/.f32 1 (+.f32 (+.f32 1 c) (.f32 (/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (cbrt.f32 (sqrt.f32 (fma.f32 cosTheta -2 1))) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))
	0.7b	(/.f32 1 (+.f32 (+.f32 1 c) (+.f32 (fma.f32 1 (/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) (neg.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (fma.f32 (neg.f32 (sqrt.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (sqrt.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))

Compiler

Compiled 12998 to 6655 computations (48.8% saved)

localize91.0ms (1%)

Local error

Found 4 expressions with local error:

New	Error	Program
✓	0.3b	(/.f32 1 (+.f32 (+.f32 1 c) (.f32 (/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (cbrt.f32 (sqrt.f32 (fma.f32 cosTheta -2 1))) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))
✓	0.3b	(/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))
	0.4b	(.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))
✓	0.5b	(.f32 (/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (cbrt.f32 (sqrt.f32 (fma.f32 cosTheta -2 1))) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))

series239.0ms (2.7%)

Counts

3 → 21

Calls

3 calls:

165.0ms

(/.f32 1 (+.f32 (+.f32 1 c) (*.f32 (/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (cbrt.f32 (sqrt.f32 (fma.f32 cosTheta -2 1))) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))

37.0ms

(/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))

36.0ms

(*.f32 (/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (cbrt.f32 (sqrt.f32 (fma.f32 cosTheta -2 1))) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))

rewrite78.0ms (0.9%)

Algorithm

2×	batch-egg-rewrite

Rules

224×	`add-sqr-sqrt_binary32`
214×	`log1p-expm1-u_binary32`
214×	`expm1-log1p-u_binary32`
209×	`add-cbrt-cube_binary32`
209×	`add-log-exp_binary32`

Counts

3 → 88

Calls

3 calls:

75.0ms

(/.f32 1 (+.f32 (+.f32 1 c) (*.f32 (/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (cbrt.f32 (sqrt.f32 (fma.f32 cosTheta -2 1))) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))

75.0ms

(/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))

75.0ms

(*.f32 (/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (cbrt.f32 (sqrt.f32 (fma.f32 cosTheta -2 1))) (sqrt.f32 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	22	105
1	467	83
2	4886	83
3	5121	83
0	0	0
1	0	0

simplify122.0ms (1.4%)

Algorithm

1×	egg-herbie

Rules

582×	`times-frac_binary32`
319×	`fma-neg_binary32`
316×	`cancel-sign-sub-inv_binary32`
211×	`fma-def_binary32`
208×	`associate-r_binary32`

Counts

109 → 121

Iterations

Useful iterations: 3 (0.0ms)

Iter	Nodes	Cost
0	189	2027
1	619	1897
2	2460	1778
3	4770	1776
4	4981	1776

prune320.0ms (3.7%)

Pruning

10 alts after pruning (9 fresh and 1 done)

	Pruned	Kept	Total
New	171	0	171
Fresh	0	9	9
Picked	0	1	1
Done	0	0	0
Total	171	10	181

Error

0.0b

Counts

181 → 10

Alt Table

Status	Error	Program
	0.8b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (pow.f32 (cbrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 3))))
	13.8b	(.f32 (/.f32 1 (-.f32 (pow.f32 (+.f32 1 c) 2) (/.f32 (fma.f32 cosTheta -2 1) (.f32 (PI.f32) (pow.f32 (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)) 2))))) (-.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))
	1.0b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (fma.f32 (pow.f32 (cbrt.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 2) (cbrt.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1)))) -1) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))
	1.0b	(/.f32 1 (+.f32 (+.f32 1 c) (+.f32 (fma.f32 (sqrt.f32 (/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (sqrt.f32 (/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (neg.f32 (.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 1))) (fma.f32 (neg.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) 1 (.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 1)))))
	1.0b	(pow.f32 (/.f32 1 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) 3)
	0.9b	(.f32 (pow.f32 (pow.f32 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) 2) -1) (pow.f32 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) -1))
	12.7b	(.f32 (/.f32 1 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (cbrt.f32 (pow.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) -2)))
	1.3b	(fma.f32 (sqrt.f32 (PI.f32)) cosTheta (.f32 (.f32 cosTheta cosTheta) (-.f32 (sqrt.f32 (PI.f32)) (*.f32 (+.f32 1 c) (PI.f32)))))
✓	0.7b	(/.f32 1 (+.f32 (+.f32 1 c) (.f32 (/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (cbrt.f32 (sqrt.f32 (fma.f32 cosTheta -2 1))) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))
▶	0.7b	(/.f32 1 (+.f32 (+.f32 1 c) (+.f32 (fma.f32 1 (/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) (neg.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (fma.f32 (neg.f32 (sqrt.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (sqrt.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))

Compiler

Compiled 8132 to 4490 computations (44.8% saved)

localize101.0ms (1.2%)

Local error

Found 4 expressions with local error:

New	Error	Program
✓	0.3b	(/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))
✓	0.3b	(/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))
	0.4b	(.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))
✓	30.3b	(fma.f32 (neg.f32 (sqrt.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (sqrt.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))

series115.0ms (1.3%)

Counts

3 → 19

Calls

3 calls:

45.0ms

(/.f32 1 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))

42.0ms

(fma.f32 (neg.f32 (sqrt.f32 (/.f32 1 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (sqrt.f32 (/.f32 1 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 1 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))

27.0ms

(/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))

rewrite119.0ms (1.4%)

Algorithm

2×	batch-egg-rewrite

Rules

583×	`prod-diff_binary32`
180×	`add-sqr-sqrt_binary32`
170×	`log1p-expm1-u_binary32`
170×	`expm1-log1p-u_binary32`
169×	`add-log-exp_binary32`

Counts

3 → 103

Calls

3 calls:

115.0ms

(/.f32 1 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))

115.0ms

(/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))

115.0ms

(fma.f32 (neg.f32 (sqrt.f32 (/.f32 1 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (sqrt.f32 (/.f32 1 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 1 (*.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	18	107
1	377	77
2	4363	77
3	5323	77
0	0	0
1	0	0

simplify84.0ms (1%)

Algorithm

1×	egg-herbie

Rules

723×	`fma-def_binary32`
445×	`fma-neg_binary32`
408×	`cancel-sign-sub-inv_binary32`
368×	`distribute-rgt-neg-in_binary32`
326×	`sub-neg_binary32`

Counts

122 → 121

Iterations

Useful iterations: 3 (0.0ms)

Iter	Nodes	Cost
0	40	480
1	115	419
2	318	379
3	995	345
4	2876	345
5	4352	345
6	4864	345
7	5103	345

prune672.0ms (7.7%)

Pruning

10 alts after pruning (8 fresh and 2 done)

	Pruned	Kept	Total
New	366	0	366
Fresh	0	8	8
Picked	0	1	1
Done	0	1	1
Total	366	10	376

Error

0.0b

Counts

376 → 10

Alt Table

Status	Error	Program
	0.8b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (-.f32 (exp.f32 (log1p.f32 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 1) (pow.f32 (cbrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 3))))
	13.8b	(.f32 (/.f32 1 (-.f32 (pow.f32 (+.f32 1 c) 2) (/.f32 (fma.f32 cosTheta -2 1) (.f32 (PI.f32) (pow.f32 (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)) 2))))) (-.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))
	1.0b	(/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (fma.f32 (pow.f32 (cbrt.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1)))) 2) (cbrt.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1)))) -1) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))
	1.0b	(/.f32 1 (+.f32 (+.f32 1 c) (+.f32 (fma.f32 (sqrt.f32 (/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (sqrt.f32 (/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (neg.f32 (.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 1))) (fma.f32 (neg.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) 1 (.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) 1)))))
	1.0b	(pow.f32 (/.f32 1 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) 3)
	0.9b	(.f32 (pow.f32 (pow.f32 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) 2) -1) (pow.f32 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) -1))
	12.7b	(.f32 (/.f32 1 (cbrt.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (cbrt.f32 (pow.f32 (+.f32 (+.f32 1 c) (/.f32 (sqrt.f32 (/.f32 (fma.f32 cosTheta -2 1) (PI.f32))) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) -2)))
	1.3b	(fma.f32 (sqrt.f32 (PI.f32)) cosTheta (.f32 (.f32 cosTheta cosTheta) (-.f32 (sqrt.f32 (PI.f32)) (*.f32 (+.f32 1 c) (PI.f32)))))
✓	0.7b	(/.f32 1 (+.f32 (+.f32 1 c) (.f32 (/.f32 (cbrt.f32 (fma.f32 cosTheta -2 1)) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 (cbrt.f32 (sqrt.f32 (fma.f32 cosTheta -2 1))) (sqrt.f32 (.f32 (sqrt.f32 (PI.f32)) (*.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))
✓	0.7b	(/.f32 1 (+.f32 (+.f32 1 c) (+.f32 (fma.f32 1 (/.f32 (+.f32 1 (sqrt.f32 (fma.f32 cosTheta -2 1))) (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))) (neg.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (fma.f32 (neg.f32 (sqrt.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta)))))) (sqrt.f32 (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))) (/.f32 1 (.f32 (sqrt.f32 (PI.f32)) (.f32 cosTheta (pow.f32 (exp.f32 cosTheta) cosTheta))))))))

Compiler

Compiled 29468 to 12265 computations (58.4% saved)

regimes2.5s (29.1%)

Accuracy

Total 0.4b remaining (90.4%)

Threshold costs 0.4b (90.4%)

Counts

255 → 1

Compiler

Compiled 57480 to 39656 computations (31% saved)

simplify3.0ms (0%)

Algorithm

1×	egg-herbie

Rules

2×	`+-commutative_binary32`
2×	`*-commutative_binary32`
1×	`1-exp_binary32`

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	20	4294967295
1	26	4294967295
2	25	4294967295

end369.0ms (4.2%)

Compiler: Compiled 505 to 331 computations (34.5% saved)

Details

analyze105.0ms (1.2%)

sample2.1s (24%)

simplify48.0ms (0.6%)

prune8.0ms (0.1%)

localize61.0ms (0.7%)

series96.0ms (1.1%)

rewrite93.0ms (1.1%)

simplify90.0ms (1%)

prune211.0ms (2.4%)

localize74.0ms (0.9%)

series225.0ms (2.6%)

rewrite124.0ms (1.4%)

simplify85.0ms (1%)

prune553.0ms (6.4%)

localize91.0ms (1%)

series239.0ms (2.7%)

rewrite78.0ms (0.9%)

simplify122.0ms (1.4%)

prune320.0ms (3.7%)

localize101.0ms (1.2%)

series115.0ms (1.3%)

rewrite119.0ms (1.4%)

simplify84.0ms (1%)

prune672.0ms (7.7%)

regimes2.5s (29.1%)

simplify3.0ms (0%)

end369.0ms (4.2%)

Profiling