Details

analyze3.6s (13.8%)

Algorithm

1×	search
2×	egg-herbie

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	2392	341
1	2392	341
0	5	5
1	5	5

Search

True	Other	False	Iter
0%	99.8%	0.2%	0
0%	99.8%	0.2%	1
0%	99.8%	0.2%	2
0%	99.8%	0.2%	3
0%	99.8%	0.2%	4
0%	99.8%	0.2%	5
0%	99.8%	0.2%	6
0%	99.8%	0.2%	7
0%	99.8%	0.2%	8
0%	99.8%	0.2%	9
0%	99.8%	0.2%	10
0.8%	99%	0.2%	11
1.2%	98.6%	0.2%	12
1.8%	98%	0.2%	13
2%	97.7%	0.2%	14

Symmetry

(sort lambda1 lambda2)

(sort phi1 phi2)

Compiler

Compiled 20 to 18 computations (10% saved)

sample229.0ms (0.9%)

Algorithm

1×

intervals

Results

157.0ms	179×	body	1024	valid
27.0ms	53×	body	512	valid
6.0ms	4×	body	2048	valid
4.0ms	12×	body	256	valid
1.0ms	8×	body	128	valid

Compiler

Compiled 39 to 40 computations (-2.6% saved)

simplify11.0ms (0%)

Algorithm

1×	egg-herbie

Counts

1 → 1

Iterations

Useful iterations: 0 (6.0ms)

Iter	Nodes	Cost
0	96	31
1	96	31

prune9.0ms (0%)

Pruning

2 alts after pruning (2 fresh and 0 done)

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

Error

16.7b

Counts

2 → 1

Compiler

Compiled 38 to 34 computations (10.5% saved)

localize20.0ms (0.1%)

Local error

Found 4 expressions with local error:

0.2b	(*.f64 (sin.f64 phi1) (sin.f64 phi2))
0.2b	(.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) R)
0.6b	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))
3.9b	(cos.f64 (-.f64 lambda1 lambda2))

rewrite97.0ms (0.4%)

Algorithm

1×	rewrite-expression-head

Error

2.5b

Rules

13×	`add-sqr-sqrt_binary64_1464`
9×	`add-cube-cbrt_binary64_1477` `*-un-lft-identity_binary64_1442`
8×	`pow1_binary64_1503` `add-exp-log_binary64_1480` `add-cbrt-cube_binary64_1478`
6×	`associate-l_binary64_1383` `associate-r_binary64_1382`
4×	`cos-sum_binary64_1576` `add-log-exp_binary64_1481`
3×	`cancel-sign-sub-inv_binary64_1408`
2×	`pow-prod-down_binary64_1513` `prod-exp_binary64_1491` `cbrt-unprod_binary64_1475` `unswap-sqr_binary64_1410` `*-commutative_binary64_1373`
1×	`cos-diff_binary64_1579` `sub-neg_binary64_1435` `acos-asin_binary64_1623` `sin-mult_binary64_1597`

Counts

4 → 57

Calls

4 calls:

5.0ms	(*.f64 (sin.f64 phi1) (sin.f64 phi2))
5.0ms	(cos.f64 (-.f64 lambda1 lambda2))
4.0ms	(.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) R)
1.0ms	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

Compiler

Compiled 1667 to 309 computations (81.5% saved)

series581.0ms (2.2%)

Error

15.7b

Counts

4 → 51

Calls

4 calls:

257.0ms	(.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) R)
121.0ms	(*.f64 (sin.f64 phi1) (sin.f64 phi2))
99.0ms	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))
55.0ms	(cos.f64 (-.f64 lambda1 lambda2))

Compiler

Compiled 2959 to 1776 computations (40% saved)

simplify105.0ms (0.4%)

Algorithm

1×	egg-herbie

Counts

108 → 108

Iterations

Useful iterations: 0 (78.0ms)

Iter	Nodes	Cost
0	4932	2421
1	4932	2421

prune138.0ms (0.5%)

Pruning

13 alts after pruning (13 fresh and 0 done)

	Pruned	Kept	Total
New	95	13	108
Fresh	0	0	0
Picked	1	0	1
Done	0	0	0
Total	96	13	109

Error

2.5b

Counts

109 → 13

Compiler

Compiled 3008 to 684 computations (77.3% saved)

localize25.0ms (0.1%)

Local error

Found 4 expressions with local error:

0.2b	(.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (.f64 (sin.f64 lambda1) (sin.f64 lambda2))))
0.2b	(*.f64 (sin.f64 lambda1) (sin.f64 lambda2))
0.2b	(.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (.f64 (sin.f64 lambda1) (sin.f64 lambda2)))))) R)
0.6b	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))))))

rewrite135.0ms (0.5%)

Algorithm

1×	rewrite-expression-head

Error

2.3b

Rules

13×	`pow1_binary64_1503` `add-exp-log_binary64_1480` `add-cbrt-cube_binary64_1478` `add-sqr-sqrt_binary64_1464`
9×	`add-cube-cbrt_binary64_1477` `-un-lft-identity_binary64_1442` `associate-r*_binary64_1382`
7×	`associate-l_binary64_1383`
6×	`cos-mult_binary64_1596`
5×	`pow-prod-down_binary64_1513` `prod-exp_binary64_1491` `cbrt-unprod_binary64_1475`
4×	`add-log-exp_binary64_1481`
3×	`-commutative_binary64_1373` `sin-mult_binary64_1597` `frac-times_binary64_1452` `associate-r/_binary64_1384`
2×	`unswap-sqr_binary64_1410` `frac-add_binary64_1450` `flip3-+_binary64_1445` `flip-+_binary64_1416`
1×	`acos-asin_binary64_1623` `distribute-rgt-in_binary64_1392` `distribute-lft-in_binary64_1391` `associate-*l/_binary64_1385`

Counts

4 → 72

Calls

4 calls:

11.0ms	(.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (.f64 (sin.f64 lambda1) (sin.f64 lambda2))))
5.0ms	(*.f64 (sin.f64 lambda1) (sin.f64 lambda2))
3.0ms	(.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (.f64 (sin.f64 lambda1) (sin.f64 lambda2)))))) R)
1.0ms	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))))))

Compiler

Compiled 2888 to 452 computations (84.3% saved)

series1.4s (5.3%)

Error

2.4b

Counts

4 → 59

Calls

4 calls:

701.0ms	(.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (.f64 (sin.f64 lambda1) (sin.f64 lambda2))))
358.0ms	(.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (.f64 (sin.f64 lambda1) (sin.f64 lambda2)))))) R)
159.0ms	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))))))
110.0ms	(*.f64 (sin.f64 lambda1) (sin.f64 lambda2))

Compiler

Compiled 5537 to 2820 computations (49.1% saved)

simplify95.0ms (0.4%)

Algorithm

1×	egg-herbie

Counts

131 → 131

Iterations

Useful iterations: 0 (48.0ms)

Iter	Nodes	Cost
0	5046	4666

prune176.0ms (0.7%)

Pruning

14 alts after pruning (14 fresh and 0 done)

	Pruned	Kept	Total
New	120	11	131
Fresh	9	3	12
Picked	1	0	1
Done	0	0	0
Total	130	14	144

Error

2.3b

Counts

144 → 14

Compiler

Compiled 4972 to 940 computations (81.1% saved)

localize27.0ms (0.1%)

Local error

Found 4 expressions with local error:

0.2b	(*.f64 (sin.f64 phi1) (sin.f64 phi2))
0.2b	(.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 lambda1))) (.f64 (sin.f64 lambda2) (.f64 (sin.f64 lambda1) (.f64 (cos.f64 phi1) (cos.f64 phi2))))))) R)
0.3b	(.f64 (sin.f64 lambda2) (.f64 (sin.f64 lambda1) (*.f64 (cos.f64 phi1) (cos.f64 phi2))))
0.6b	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 lambda1))) (.f64 (sin.f64 lambda2) (.f64 (sin.f64 lambda1) (*.f64 (cos.f64 phi1) (cos.f64 phi2)))))))

rewrite109.0ms (0.4%)

Algorithm

1×	rewrite-expression-head

Error

2.3b

Rules

17×	`pow1_binary64_1503` `add-exp-log_binary64_1480` `add-cbrt-cube_binary64_1478`
13×	`add-sqr-sqrt_binary64_1464`
9×	`add-cube-cbrt_binary64_1477` `-un-lft-identity_binary64_1442` `associate-l*_binary64_1383`
8×	`pow-prod-down_binary64_1513` `prod-exp_binary64_1491` `cbrt-unprod_binary64_1475`
7×	`associate-r_binary64_1382`
4×	`add-log-exp_binary64_1481`
3×	`*-commutative_binary64_1373`
2×	`associate-*r/_binary64_1384` `unswap-sqr_binary64_1410`
1×	`acos-asin_binary64_1623` `cos-mult_binary64_1596` `sin-mult_binary64_1597`

Counts

4 → 67

Calls

4 calls:

9.0ms	(.f64 (sin.f64 lambda2) (.f64 (sin.f64 lambda1) (*.f64 (cos.f64 phi1) (cos.f64 phi2))))
5.0ms	(*.f64 (sin.f64 phi1) (sin.f64 phi2))
3.0ms	(.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 lambda1))) (.f64 (sin.f64 lambda2) (.f64 (sin.f64 lambda1) (.f64 (cos.f64 phi1) (cos.f64 phi2))))))) R)
1.0ms	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 lambda1))) (.f64 (sin.f64 lambda2) (.f64 (sin.f64 lambda1) (*.f64 (cos.f64 phi1) (cos.f64 phi2)))))))

Compiler

Compiled 3017 to 373 computations (87.6% saved)

series1.2s (4.5%)

Error

2.3b

Counts

4 → 54

Calls

4 calls:

464.0ms	(.f64 (sin.f64 lambda2) (.f64 (sin.f64 lambda1) (*.f64 (cos.f64 phi1) (cos.f64 phi2))))
398.0ms	(.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 lambda1))) (.f64 (sin.f64 lambda2) (.f64 (sin.f64 lambda1) (.f64 (cos.f64 phi1) (cos.f64 phi2))))))) R)
158.0ms	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (+.f64 (.f64 (cos.f64 lambda2) (.f64 (.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 lambda1))) (.f64 (sin.f64 lambda2) (.f64 (sin.f64 lambda1) (*.f64 (cos.f64 phi1) (cos.f64 phi2)))))))
121.0ms	(*.f64 (sin.f64 phi1) (sin.f64 phi2))

Compiler

Compiled 5272 to 2585 computations (51% saved)

simplify102.0ms (0.4%)

Algorithm

1×	egg-herbie

Counts

121 → 121

Iterations

Useful iterations: 0 (63.0ms)

Iter	Nodes	Cost
0	4937	3499
1	4937	3499

prune147.0ms (0.6%)

Pruning

14 alts after pruning (13 fresh and 1 done)

	Pruned	Kept	Total
New	121	0	121
Fresh	0	13	13
Picked	0	1	1
Done	0	0	0
Total	121	14	135

Error

2.3b

Counts

135 → 14

Compiler

Compiled 4754 to 789 computations (83.4% saved)

localize30.0ms (0.1%)

Local error

Found 4 expressions with local error:

0.2b	(.f64 (log.f64 (exp.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (.f64 (cos.f64 lambda2) (cos.f64 lambda1))))))))) R)
0.4b	(exp.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))))))))
0.4b	(log.f64 (exp.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))))))))
0.6b	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))))))

rewrite95.0ms (0.4%)

Algorithm

1×	rewrite-expression-head

Error

2.3b

Rules

12×	`add-sqr-sqrt_binary64_1464`
10×	`add-cube-cbrt_binary64_1477` `*-un-lft-identity_binary64_1442`
9×	`exp-prod_binary64_1494`
8×	`pow1_binary64_1503` `log-pow_binary64_1531`
7×	`associate-l_binary64_1383`
6×	`add-exp-log_binary64_1480` `add-cbrt-cube_binary64_1478`
5×	`add-log-exp_binary64_1481`
3×	`acos-asin_binary64_1623` `log-prod_binary64_1528` `associate-r_binary64_1382`
2×	`exp-diff_binary64_1490`
1×	`log-div_binary64_1529` `rem-log-exp_binary64_1483` `rem-exp-log_binary64_1482` `pow-prod-down_binary64_1513` `prod-exp_binary64_1491` `cbrt-unprod_binary64_1475` `unswap-sqr_binary64_1410` `*-commutative_binary64_1373`

Counts

4 → 58

Calls

4 calls:

6.0ms	(.f64 (log.f64 (exp.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (.f64 (cos.f64 lambda2) (cos.f64 lambda1))))))))) R)
2.0ms	(log.f64 (exp.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))))))))
2.0ms	(exp.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))))))))
1.0ms	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))))))

Compiler

Compiled 2764 to 221 computations (92% saved)

series3.2s (12.1%)

Error

2.3b

Counts

4 → 51

Calls

4 calls:

1.6s	(.f64 (log.f64 (exp.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (.f64 (cos.f64 lambda2) (cos.f64 lambda1))))))))) R)
1.1s	(log.f64 (exp.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))))))))
275.0ms	(exp.f64 (acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))))))))
170.0ms	(acos.f64 (+.f64 (.f64 (sin.f64 phi1) (sin.f64 phi2)) (.f64 (cos.f64 phi1) (.f64 (cos.f64 phi2) (+.f64 (.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))))))

Compiler

Compiled 5202 to 2575 computations (50.5% saved)

simplify152.0ms (0.6%)

Algorithm

1×	egg-herbie

Counts

109 → 109

Iterations

Useful iterations: 0 (103.0ms)

Iter	Nodes	Cost
0	4866	4840
1	4866	4840

prune112.0ms (0.4%)

Pruning

14 alts after pruning (13 fresh and 1 done)

	Pruned	Kept	Total
New	104	5	109
Fresh	4	8	12
Picked	1	0	1
Done	0	1	1
Total	109	14	123

Error

2.3b

Counts

123 → 14

Compiler

Compiled 3764 to 197 computations (94.8% saved)

regimes3.9s (14.8%)

Accuracy

Total 0.4b remaining (10.3%)

Threshold costs 0.4b (10.3%)

Compiler

Compiled 58168 to 33090 computations (43.1% saved)

bsearch0.0ms (0%)

simplify27.0ms (0.1%)

Algorithm

1×	egg-herbie

Iterations

Useful iterations: 0 (1.0ms)

Iter	Nodes	Cost
0	33	53
1	33	53

Remove

(sort phi1 phi2)

(sort lambda1 lambda2)

Compiler

Compiled 128 to 92 computations (28.1% saved)

end0.0ms (0%)

sample10.5s (40.2%)

Algorithm

1×

intervals

Results

4.9s	5538×	body	1024	valid
757.0ms	1397×	body	512	valid
547.0ms	275×	body	2048	valid
177.0ms	470×	body	256	valid
60.0ms	320×	body	128	valid
2.0ms	2×	body	1024	invalid

Compiler

Compiled 5370 to 3059 computations (43% saved)

Details

analyze3.6s (13.8%)

sample229.0ms (0.9%)

simplify11.0ms (0%)

prune9.0ms (0%)

localize20.0ms (0.1%)

rewrite97.0ms (0.4%)

series581.0ms (2.2%)

simplify105.0ms (0.4%)

prune138.0ms (0.5%)

localize25.0ms (0.1%)

rewrite135.0ms (0.5%)

series1.4s (5.3%)

simplify95.0ms (0.4%)

prune176.0ms (0.7%)

localize27.0ms (0.1%)

rewrite109.0ms (0.4%)

series1.2s (4.5%)

simplify102.0ms (0.4%)

prune147.0ms (0.6%)

localize30.0ms (0.1%)

rewrite95.0ms (0.4%)

series3.2s (12.1%)

simplify152.0ms (0.6%)

prune112.0ms (0.4%)

regimes3.9s (14.8%)

bsearch0.0ms (0%)

simplify27.0ms (0.1%)

end0.0ms (0%)

sample10.5s (40.2%)

Profiling