Metrics for sqrt C (should all be same)

start0.0ms (0%)

Memory: 0.0MiB live, 0.0MiB allocated; 0ms collecting garbage

analyze0.0ms (0%)

Memory

0.5MiB live, 0.5MiB allocated; 0ms collecting garbage

Algorithm

1×

Search

Probability	Valid	Unknown	Precondition	Infinite	Domain	Can't	Iter
0%	0%	100%	0%	0%	0%	0%	0
0%	0%	100%	0%	0%	0%	0%	1
100%	100%	0%	0%	0%	0%	0%	2

Compiler

Compiled 8 to 7 computations (12.5% saved)

sample572.0ms (60.7%)

Memory

5.7MiB live, 474.2MiB allocated; 167ms collecting garbage

Samples

334.0ms

8 256×

valid

Precisions

Click to see histograms. Total time spent on operations: 199.0ms

ival-sqrt: 99.0ms (49.6% of total)

ival-mult!: 99.0ms (49.6% of total)

adjust: 1.0ms (0.5% of total)

Bogosity

explain56.0ms (6%)

Memory

14.6MiB live, 65.3MiB allocated; 12ms collecting garbage

FPErrors

Click to see full error table

Ground Truth	Example	Underpredictions	Example	Subexpression
123	-	2	(-2.5563634837735915e-161)	`(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))`
0	-	0	-	`#s(literal 2 binary64)`
0	-	0	-	`(*.f64 x x)`
0	-	0	-	`(.f64 #s(literal 2 binary64) (.f64 x x))`
0	-	0	-	`x`

Explanations

Click to see full explanations table

Operator	Subexpression	Explanation	Count
`sqrt.f64`	`(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))`	uflow-rescue	62	0
↳	`(.f64 #s(literal 2 binary64) (.f64 x x))`	underflow	62
↳	`(*.f64 x x)`	underflow	62
`sqrt.f64`	`(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))`	oflow-rescue	59	0
↳	`(.f64 #s(literal 2 binary64) (.f64 x x))`	overflow	59
↳	`(*.f64 x x)`	overflow	59

Confusion

	Predicted +	Predicted -
+	121	2
-	0	133

Precision

1.0

Recall

0.983739837398374

Confusion?

	Predicted +	Predicted Maybe	Predicted -
+	121	0	2
-	0	0	133

Precision?

1.0

Recall?

0.983739837398374

Freqs

test

number	freq
0	135
1	121

Total Confusion?

	Predicted +	Predicted Maybe	Predicted -
+	1	0	0
-	0	0	0

Precision?

1.0

Recall?

1.0

Samples

20.0ms

512×

valid

Compiler

Compiled 42 to 19 computations (54.8% saved)

Precisions

Click to see histograms. Total time spent on operations: 9.0ms

ival-mult!: 5.0ms (54.3% of total)

ival-sqrt: 4.0ms (43.4% of total)

adjust: 0.0ms (0% of total)

preprocess77.0ms (8.1%)

Memory

-22.1MiB live, 23.7MiB allocated; 4ms collecting garbage

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	10	19
1	28	19
2	90	19
3	185	19
4	578	19
5	1314	19
6	4975	19

Stop Event

1×	node-limit

Alt Table

Click to see full alt table

Status	Accuracy	Program
▶	55.3%	(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))

Symmetry

(abs x)

Compiler

Compiled 12 to 10 computations (16.7% saved)

series4.0ms (0.4%)

Memory

8.9MiB live, 8.9MiB allocated; 0ms collecting garbage

Counts

5 → 5

Calls

Call 1

Inputs
(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))
(.f64 #s(literal 2 binary64) (.f64 x x))
#s(literal 2 binary64)
(*.f64 x x)
x

Outputs
#s(approx (sqrt (* 2 (* x x))) #s(hole binary64 (* x (sqrt 2))))
#s(approx (* 2 (* x x)) #s(hole binary64 (* 2 (pow x 2))))
#s(approx (* x x) #s(hole binary64 (pow x 2)))
#s(approx x #s(hole binary64 x))
#s(approx (sqrt (* 2 (* x x))) #s(hole binary64 (* -1 (* x (sqrt 2)))))

Calls

3 calls:

Time	Variable		Point	Expression
1.0ms	x	@	inf	((sqrt (* 2 (* x x))) (* 2 (* x x)) 2 (* x x) x)
1.0ms	x	@	0	((sqrt (* 2 (* x x))) (* 2 (* x x)) 2 (* x x) x)
1.0ms	x	@	-inf	((sqrt (* 2 (* x x))) (* 2 (* x x)) 2 (* x x) x)

rewrite41.0ms (4.4%)

Memory

23.3MiB live, 66.1MiB allocated; 10ms collecting garbage

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	19	61
0	22	60
1	50	54
0	171	49

Stop Event

1×	iter-limit
1×	iter-limit
1×	iter-limit
1×	unsound
1×	iter-limit

Counts

10 → 65

Calls

Call 1

Inputs
(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))
(.f64 #s(literal 2 binary64) (.f64 x x))
#s(literal 2 binary64)
(*.f64 x x)
x
#s(approx (sqrt (* 2 (* x x))) #s(hole binary64 (* x (sqrt 2))))
#s(approx (* 2 (* x x)) #s(hole binary64 (* 2 (pow x 2))))
#s(approx (* x x) #s(hole binary64 (pow x 2)))
#s(approx x #s(hole binary64 x))
#s(approx (sqrt (* 2 (* x x))) #s(hole binary64 (* -1 (* x (sqrt 2)))))

Outputs
(.f64 (pow.f64 (.f64 (.f64 x x) #s(literal 2 binary64)) #s(literal 1/4 binary64)) (pow.f64 (.f64 (*.f64 x x) #s(literal 2 binary64)) #s(literal 1/4 binary64)))
(.f64 (sqrt.f64 (.f64 x #s(literal 2 binary64))) (sqrt.f64 x))
(.f64 (pow.f64 (.f64 x #s(literal 2 binary64)) #s(literal 1/2 binary64)) (pow.f64 x #s(literal 1/2 binary64)))
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
(*.f64 x (sqrt.f64 #s(literal 2 binary64)))
(pow.f64 (.f64 (.f64 x x) #s(literal 2 binary64)) #s(literal 1/2 binary64))
(sqrt.f64 (.f64 (.f64 x x) #s(literal 2 binary64)))
(hypot.f64 (neg.f64 (neg.f64 x)) (neg.f64 (neg.f64 x)))
(hypot.f64 (neg.f64 (neg.f64 x)) (exp.f64 (log.f64 x)))
(hypot.f64 (neg.f64 (neg.f64 x)) (neg.f64 x))
(hypot.f64 (neg.f64 (neg.f64 x)) x)
(hypot.f64 (exp.f64 (log.f64 x)) (neg.f64 (neg.f64 x)))
(hypot.f64 (exp.f64 (log.f64 x)) (exp.f64 (log.f64 x)))
(hypot.f64 (exp.f64 (log.f64 x)) (neg.f64 x))
(hypot.f64 (exp.f64 (log.f64 x)) x)
(hypot.f64 (neg.f64 x) (neg.f64 (neg.f64 x)))
(hypot.f64 (neg.f64 x) (exp.f64 (log.f64 x)))
(hypot.f64 (neg.f64 x) (neg.f64 x))
(hypot.f64 (neg.f64 x) x)
(hypot.f64 x (neg.f64 (neg.f64 x)))
(hypot.f64 x (exp.f64 (log.f64 x)))
(hypot.f64 x (neg.f64 x))
(hypot.f64 x x)
(exp.f64 (.f64 (log.f64 (.f64 (*.f64 x x) #s(literal 2 binary64))) #s(literal 1/2 binary64)))
(.f64 (.f64 #s(literal 2 binary64) (neg.f64 x)) (neg.f64 x))
(.f64 (.f64 x #s(literal 2 binary64)) x)
(.f64 (neg.f64 x) (.f64 (neg.f64 x) #s(literal 2 binary64)))
(*.f64 (neg.f64 x) (fma.f64 #s(literal -1 binary64) x (neg.f64 x)))
(.f64 (.f64 x x) #s(literal 2 binary64))
(.f64 x (.f64 x #s(literal 2 binary64)))
(.f64 #s(literal 2 binary64) (.f64 x x))
(/.f64 (-.f64 (.f64 (.f64 x x) (.f64 x x)) (.f64 (.f64 x x) (.f64 x x))) (-.f64 (.f64 x x) (.f64 x x)))
(/.f64 (+.f64 (pow.f64 (.f64 x x) #s(literal 3 binary64)) (pow.f64 (.f64 x x) #s(literal 3 binary64))) (fma.f64 (.f64 x x) (.f64 x x) (-.f64 (.f64 (.f64 x x) (.f64 x x)) (.f64 (.f64 x x) (.f64 x x)))))
(fma.f64 (neg.f64 (neg.f64 x)) (neg.f64 (neg.f64 x)) (*.f64 x x))
(fma.f64 (exp.f64 (log.f64 x)) (exp.f64 (log.f64 x)) (*.f64 x x))
(fma.f64 #s(literal 1 binary64) (.f64 x x) (.f64 #s(literal 1 binary64) (*.f64 x x)))
(fma.f64 (neg.f64 x) (neg.f64 x) (*.f64 x x))
(fma.f64 (.f64 x x) #s(literal 1 binary64) (.f64 (*.f64 x x) #s(literal 1 binary64)))
(fma.f64 x x (*.f64 x x))
(-.f64 (.f64 x x) (.f64 x (neg.f64 x)))
(-.f64 (.f64 x x) (.f64 (neg.f64 (neg.f64 x)) (neg.f64 x)))
(-.f64 (.f64 x x) (.f64 (neg.f64 x) x))
(+.f64 (.f64 #s(literal 1 binary64) (.f64 x x)) (.f64 #s(literal 1 binary64) (.f64 x x)))
(+.f64 (.f64 (.f64 x x) #s(literal 1 binary64)) (.f64 (.f64 x x) #s(literal 1 binary64)))
(+.f64 (.f64 x x) (.f64 x x))
#s(literal 2 binary64)
(*.f64 (neg.f64 (neg.f64 x)) (neg.f64 (neg.f64 x)))
(*.f64 (exp.f64 (log.f64 x)) (exp.f64 (log.f64 x)))
(*.f64 (neg.f64 x) (neg.f64 x))
(*.f64 x x)
(pow.f64 (exp.f64 (log.f64 x)) #s(literal 2 binary64))
(pow.f64 (neg.f64 x) #s(literal 2 binary64))
(pow.f64 (*.f64 x x) #s(literal 1 binary64))
(pow.f64 x #s(literal 2 binary64))
(neg.f64 (*.f64 x (neg.f64 x)))
(neg.f64 (*.f64 (neg.f64 x) x))
(fabs.f64 (*.f64 x x))
(exp.f64 (*.f64 (log.f64 x) #s(literal 2 binary64)))
(+.f64 (cosh.f64 (.f64 (log.f64 x) #s(literal 2 binary64))) (sinh.f64 (.f64 (log.f64 x) #s(literal 2 binary64))))
x
#s(approx (sqrt (* 2 (* x x))) (hypot.f64 x x))
#s(approx (* 2 (* x x)) (.f64 (.f64 x x) #s(literal 2 binary64)))
#s(approx (* x x) (*.f64 x x))
#s(approx x x)
#s(approx (sqrt (* 2 (* x x))) (neg.f64 (hypot.f64 x x)))

eval6.0ms (0.6%)

Memory: 11.9MiB live, 11.9MiB allocated; 0ms collecting garbage
Compiler: Compiled 563 to 158 computations (71.9% saved)

prune3.0ms (0.4%)

Memory

6.2MiB live, 6.2MiB allocated; 0ms collecting garbage

Pruning

4 alts after pruning (4 fresh and 0 done)

	Pruned	Kept	Total
New	63	4	67
Fresh	0	0	0
Picked	1	0	1
Done	0	0	0
Total	64	4	68

Accuracy

100.0%

Counts

68 → 4

Alt Table

Click to see full alt table

Status	Accuracy	Program
▶	100.0%	(hypot.f64 x x)
▶	99.6%	(.f64 (sqrt.f64 (.f64 x #s(literal 2 binary64))) (sqrt.f64 x))
▶	99.4%	(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
▶	55.3%	(sqrt.f64 (.f64 (.f64 x #s(literal 2 binary64)) x))

Compiler

Compiled 20 to 17 computations (15% saved)

series12.0ms (1.3%)

Memory

-35.2MiB live, 10.2MiB allocated; 7ms collecting garbage

Counts

11 → 14

Calls

Call 1

Inputs
(sqrt.f64 (.f64 (.f64 x #s(literal 2 binary64)) x))
(.f64 (.f64 x #s(literal 2 binary64)) x)
(*.f64 x #s(literal 2 binary64))
x
#s(literal 2 binary64)
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
(sqrt.f64 #s(literal 2 binary64))
(.f64 (sqrt.f64 (.f64 x #s(literal 2 binary64))) (sqrt.f64 x))
(sqrt.f64 (*.f64 x #s(literal 2 binary64)))
(sqrt.f64 x)
(hypot.f64 x x)

Outputs
#s(approx (sqrt (* (* x 2) x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (* (* x 2) x) #s(hole binary64 (* 2 (pow x 2))))
#s(approx (* x 2) #s(hole binary64 (* 2 x)))
#s(approx x #s(hole binary64 x))
#s(approx (* (sqrt 2) x) #s(hole binary64 (* x (sqrt 2))))
#s(approx (* (sqrt (* x 2)) (sqrt x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (sqrt (* x 2)) #s(hole binary64 (* (sqrt x) (sqrt 2))))
#s(approx (sqrt x) #s(hole binary64 (sqrt x)))
#s(approx (sqrt (+ (* x x) (* x x))) #s(hole binary64 (* x (sqrt 2))))
#s(approx (sqrt (* (* x 2) x)) #s(hole binary64 (* -1 (* x (sqrt 2)))))
#s(approx (* (sqrt (* x 2)) (sqrt x)) #s(hole binary64 (* -1 (* x (* (sqrt -2) (sqrt -1))))))
#s(approx (sqrt (* x 2)) #s(hole binary64 (* -1 (* (sqrt x) (* (sqrt -2) (sqrt -1))))))
#s(approx (sqrt x) #s(hole binary64 (* -1 (* (sqrt x) (pow (sqrt -1) 2)))))
#s(approx (sqrt (+ (* x x) (* x x))) #s(hole binary64 (* -1 (* x (sqrt 2)))))

Calls

3 calls:

Time	Variable		Point	Expression
7.0ms	x	@	0	((sqrt (* (* x 2) x)) (* (* x 2) x) (* x 2) x 2 (* (sqrt 2) x) (sqrt 2) (* (sqrt (* x 2)) (sqrt x)) (sqrt (* x 2)) (sqrt x) (sqrt (+ (* x x) (* x x))))
3.0ms	x	@	-inf	((sqrt (* (* x 2) x)) (* (* x 2) x) (* x 2) x 2 (* (sqrt 2) x) (sqrt 2) (* (sqrt (* x 2)) (sqrt x)) (sqrt (* x 2)) (sqrt x) (sqrt (+ (* x x) (* x x))))
1.0ms	x	@	inf	((sqrt (* (* x 2) x)) (* (* x 2) x) (* x 2) x 2 (* (sqrt 2) x) (sqrt 2) (* (sqrt (* x 2)) (sqrt x)) (sqrt (* x 2)) (sqrt x) (sqrt (+ (* x x) (* x x))))

Time

Variable

Point

Expression

7.0ms

((sqrt (* (* x 2) x)) (* (* x 2) x) (* x 2) x 2 (* (sqrt 2) x) (sqrt 2) (* (sqrt (* x 2)) (sqrt x)) (sqrt (* x 2)) (sqrt x) (sqrt (+ (* x x) (* x x))))

3.0ms

-inf

((sqrt (* (* x 2) x)) (* (* x 2) x) (* x 2) x 2 (* (sqrt 2) x) (sqrt 2) (* (sqrt (* x 2)) (sqrt x)) (sqrt (* x 2)) (sqrt x) (sqrt (+ (* x x) (* x x))))

1.0ms

inf

((sqrt (* (* x 2) x)) (* (* x 2) x) (* x 2) x 2 (* (sqrt 2) x) (sqrt 2) (* (sqrt (* x 2)) (sqrt x)) (sqrt (* x 2)) (sqrt x) (sqrt (+ (* x x) (* x x))))

rewrite22.0ms (2.3%)

Memory

22.1MiB live, 22.1MiB allocated; 0ms collecting garbage

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	54	181
0	65	144
0	141	144

Stop Event

1×	iter-limit
1×	iter-limit
1×	iter-limit
1×	unsound
1×	iter-limit

Counts

25 → 105

Calls

Call 1

Inputs
(sqrt.f64 (.f64 (.f64 x #s(literal 2 binary64)) x))
(.f64 (.f64 x #s(literal 2 binary64)) x)
(*.f64 x #s(literal 2 binary64))
x
#s(literal 2 binary64)
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
(sqrt.f64 #s(literal 2 binary64))
(.f64 (sqrt.f64 (.f64 x #s(literal 2 binary64))) (sqrt.f64 x))
(sqrt.f64 (*.f64 x #s(literal 2 binary64)))
(sqrt.f64 x)
(hypot.f64 x x)
#s(approx (sqrt (* (* x 2) x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (* (* x 2) x) #s(hole binary64 (* 2 (pow x 2))))
#s(approx (* x 2) #s(hole binary64 (* 2 x)))
#s(approx x #s(hole binary64 x))
#s(approx (* (sqrt 2) x) #s(hole binary64 (* x (sqrt 2))))
#s(approx (* (sqrt (* x 2)) (sqrt x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (sqrt (* x 2)) #s(hole binary64 (* (sqrt x) (sqrt 2))))
#s(approx (sqrt x) #s(hole binary64 (sqrt x)))
#s(approx (sqrt (+ (* x x) (* x x))) #s(hole binary64 (* x (sqrt 2))))
#s(approx (sqrt (* (* x 2) x)) #s(hole binary64 (* -1 (* x (sqrt 2)))))
#s(approx (* (sqrt (* x 2)) (sqrt x)) #s(hole binary64 (* -1 (* x (* (sqrt -2) (sqrt -1))))))
#s(approx (sqrt (* x 2)) #s(hole binary64 (* -1 (* (sqrt x) (* (sqrt -2) (sqrt -1))))))
#s(approx (sqrt x) #s(hole binary64 (* -1 (* (sqrt x) (pow (sqrt -1) 2)))))
#s(approx (sqrt (+ (* x x) (* x x))) #s(hole binary64 (* -1 (* x (sqrt 2)))))

Outputs
(.f64 (sqrt.f64 x) (sqrt.f64 (.f64 #s(literal 2 binary64) x)))
(.f64 (sqrt.f64 (.f64 #s(literal 2 binary64) x)) (sqrt.f64 x))
(pow.f64 (.f64 (.f64 #s(literal 2 binary64) x) x) #s(literal 1/2 binary64))
(sqrt.f64 (.f64 (.f64 #s(literal 2 binary64) x) x))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) (neg.f64 x))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) (fabs.f64 x))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) x)
(hypot.f64 (neg.f64 x) (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 (neg.f64 x) (neg.f64 x))
(hypot.f64 (neg.f64 x) (fabs.f64 x))
(hypot.f64 (neg.f64 x) x)
(hypot.f64 (fabs.f64 x) (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 (fabs.f64 x) (neg.f64 x))
(hypot.f64 (fabs.f64 x) (fabs.f64 x))
(hypot.f64 (fabs.f64 x) x)
(hypot.f64 x (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 x (neg.f64 x))
(hypot.f64 x (fabs.f64 x))
(hypot.f64 x x)
(.f64 (.f64 x x) #s(literal 2 binary64))
(.f64 (.f64 #s(literal 2 binary64) x) x)
(.f64 #s(literal 2 binary64) (.f64 x x))
(.f64 x (.f64 #s(literal 2 binary64) x))
(/.f64 (-.f64 (.f64 (.f64 x x) (.f64 x x)) (.f64 (.f64 x x) (.f64 x x))) (-.f64 (.f64 x x) (.f64 x x)))
(/.f64 (+.f64 (pow.f64 (.f64 x x) #s(literal 3 binary64)) (pow.f64 (.f64 x x) #s(literal 3 binary64))) (fma.f64 (.f64 x x) (.f64 x x) (-.f64 (.f64 (.f64 x x) (.f64 x x)) (.f64 (.f64 x x) (.f64 x x)))))
(fma.f64 (pow.f64 x #s(literal 1 binary64)) (pow.f64 x #s(literal 1 binary64)) (*.f64 x x))
(fma.f64 (neg.f64 x) (neg.f64 x) (*.f64 x x))
(fma.f64 (fabs.f64 x) (fabs.f64 x) (*.f64 x x))
(fma.f64 x x (*.f64 x x))
(-.f64 (.f64 x x) (.f64 (neg.f64 x) x))
(+.f64 (.f64 x x) (.f64 x x))
(*.f64 #s(literal 2 binary64) x)
(*.f64 x #s(literal 2 binary64))
(+.f64 x x)
x
#s(literal 2 binary64)
(.f64 (.f64 x (sqrt.f64 #s(literal -2 binary64))) (sqrt.f64 #s(literal -1 binary64)))
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
(*.f64 x (sqrt.f64 #s(literal 2 binary64)))
(*.f64 (sqrt.f64 #s(literal -1 binary64)) (sqrt.f64 #s(literal -2 binary64)))
(*.f64 (sqrt.f64 #s(literal -2 binary64)) (sqrt.f64 #s(literal -1 binary64)))
(pow.f64 #s(literal 2 binary64) #s(literal 1/2 binary64))
(sqrt.f64 #s(literal 2 binary64))
(.f64 (sqrt.f64 x) (sqrt.f64 (.f64 #s(literal 2 binary64) x)))
(.f64 (sqrt.f64 (.f64 #s(literal 2 binary64) x)) (sqrt.f64 x))
(pow.f64 (.f64 (.f64 #s(literal 2 binary64) x) x) #s(literal 1/2 binary64))
(sqrt.f64 (.f64 (.f64 #s(literal 2 binary64) x) x))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) (neg.f64 x))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) (fabs.f64 x))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) x)
(hypot.f64 (neg.f64 x) (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 (neg.f64 x) (neg.f64 x))
(hypot.f64 (neg.f64 x) (fabs.f64 x))
(hypot.f64 (neg.f64 x) x)
(hypot.f64 (fabs.f64 x) (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 (fabs.f64 x) (neg.f64 x))
(hypot.f64 (fabs.f64 x) (fabs.f64 x))
(hypot.f64 (fabs.f64 x) x)
(hypot.f64 x (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 x (neg.f64 x))
(hypot.f64 x (fabs.f64 x))
(hypot.f64 x x)
(.f64 (.f64 (sqrt.f64 x) (sqrt.f64 #s(literal -2 binary64))) (sqrt.f64 #s(literal -1 binary64)))
(*.f64 (sqrt.f64 x) (sqrt.f64 #s(literal 2 binary64)))
(*.f64 (sqrt.f64 #s(literal 2 binary64)) (sqrt.f64 x))
(pow.f64 (*.f64 #s(literal 2 binary64) x) #s(literal 1/2 binary64))
(sqrt.f64 (*.f64 #s(literal 2 binary64) x))
(pow.f64 x #s(literal 1/2 binary64))
(sqrt.f64 x)
(.f64 (sqrt.f64 x) (sqrt.f64 (.f64 #s(literal 2 binary64) x)))
(.f64 (sqrt.f64 (.f64 #s(literal 2 binary64) x)) (sqrt.f64 x))
(pow.f64 (.f64 (.f64 #s(literal 2 binary64) x) x) #s(literal 1/2 binary64))
(sqrt.f64 (.f64 (.f64 #s(literal 2 binary64) x) x))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) (neg.f64 x))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) (fabs.f64 x))
(hypot.f64 (pow.f64 x #s(literal 1 binary64)) x)
(hypot.f64 (neg.f64 x) (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 (neg.f64 x) (neg.f64 x))
(hypot.f64 (neg.f64 x) (fabs.f64 x))
(hypot.f64 (neg.f64 x) x)
(hypot.f64 (fabs.f64 x) (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 (fabs.f64 x) (neg.f64 x))
(hypot.f64 (fabs.f64 x) (fabs.f64 x))
(hypot.f64 (fabs.f64 x) x)
(hypot.f64 x (pow.f64 x #s(literal 1 binary64)))
(hypot.f64 x (neg.f64 x))
(hypot.f64 x (fabs.f64 x))
(hypot.f64 x x)
#s(approx (sqrt (* (* x 2) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
#s(approx (* (* x 2) x) (.f64 (.f64 #s(literal 2 binary64) x) x))
#s(approx (* x 2) (*.f64 #s(literal 2 binary64) x))
#s(approx x x)
#s(approx (* (sqrt 2) x) (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
#s(approx (sqrt (* (* x 2) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
#s(approx (sqrt (* x 2)) (sqrt.f64 (*.f64 #s(literal 2 binary64) x)))
#s(approx (sqrt x) (sqrt.f64 x))
#s(approx (sqrt (* (* x 2) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))
#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))
#s(approx (sqrt (* x 2)) (neg.f64 (sqrt.f64 (*.f64 #s(literal 2 binary64) x))))
#s(approx (sqrt x) (neg.f64 (*.f64 #s(literal -1 binary64) (sqrt.f64 x))))
#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))

eval6.0ms (0.6%)

Memory: 10.7MiB live, 10.7MiB allocated; 0ms collecting garbage
Compiler: Compiled 593 to 141 computations (76.2% saved)

prune19.0ms (2%)

Memory

-34.3MiB live, 14.7MiB allocated; 18ms collecting garbage

Pruning

5 alts after pruning (3 fresh and 2 done)

	Pruned	Kept	Total
New	76	3	79
Fresh	0	0	0
Picked	2	2	4
Done	0	0	0
Total	78	5	83

Accuracy

100.0%

Counts

83 → 5

Alt Table

Click to see full alt table

Status	Accuracy	Program
✓	100.0%	(hypot.f64 x x)
▶	99.6%	(*.f64 (sqrt.f64 (+.f64 x x)) (sqrt.f64 x))
✓	99.4%	(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
▶	55.3%	(sqrt.f64 (*.f64 (+.f64 x x) x))
▶	2.2%	#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))

Compiler

Compiled 32 to 25 computations (21.9% saved)

series5.0ms (0.5%)

Memory

6.9MiB live, 6.9MiB allocated; 0ms collecting garbage

Counts

12 → 15

Calls

Call 1

Inputs
#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))
(neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
(sqrt.f64 #s(literal 2 binary64))
#s(literal 2 binary64)
x
(sqrt.f64 (*.f64 (+.f64 x x) x))
(*.f64 (+.f64 x x) x)
(+.f64 x x)
(*.f64 (sqrt.f64 (+.f64 x x)) (sqrt.f64 x))
(sqrt.f64 (+.f64 x x))
(sqrt.f64 x)

Outputs
#s(approx (sqrt (* (* x 2) x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (neg (* (sqrt 2) x)) #s(hole binary64 (* -1 (* x (sqrt 2)))))
#s(approx (* (sqrt 2) x) #s(hole binary64 (* x (sqrt 2))))
#s(approx x #s(hole binary64 x))
#s(approx (sqrt (* (+ x x) x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (* (+ x x) x) #s(hole binary64 (* 2 (pow x 2))))
#s(approx (+ x x) #s(hole binary64 (* 2 x)))
#s(approx (* (sqrt (+ x x)) (sqrt x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (sqrt (+ x x)) #s(hole binary64 (* (sqrt x) (sqrt 2))))
#s(approx (sqrt x) #s(hole binary64 (sqrt x)))
#s(approx (sqrt (* (* x 2) x)) #s(hole binary64 (* -1 (* x (sqrt 2)))))
#s(approx (sqrt (* (+ x x) x)) #s(hole binary64 (* -1 (* x (sqrt 2)))))
#s(approx (* (sqrt (+ x x)) (sqrt x)) #s(hole binary64 (* -1 (* x (* (sqrt -2) (sqrt -1))))))
#s(approx (sqrt (+ x x)) #s(hole binary64 (* -1 (* (sqrt x) (* (sqrt -2) (sqrt -1))))))
#s(approx (sqrt x) #s(hole binary64 (* -1 (* (sqrt x) (pow (sqrt -1) 2)))))

Calls

3 calls:

Time	Variable		Point	Expression
2.0ms	x	@	-inf	((sqrt (* (* x 2) x)) (neg (* (sqrt 2) x)) (* (sqrt 2) x) (sqrt 2) 2 x (sqrt (* (+ x x) x)) (* (+ x x) x) (+ x x) (* (sqrt (+ x x)) (sqrt x)) (sqrt (+ x x)) (sqrt x))
1.0ms	x	@	inf	((sqrt (* (* x 2) x)) (neg (* (sqrt 2) x)) (* (sqrt 2) x) (sqrt 2) 2 x (sqrt (* (+ x x) x)) (* (+ x x) x) (+ x x) (* (sqrt (+ x x)) (sqrt x)) (sqrt (+ x x)) (sqrt x))
1.0ms	x	@	0	((sqrt (* (* x 2) x)) (neg (* (sqrt 2) x)) (* (sqrt 2) x) (sqrt 2) 2 x (sqrt (* (+ x x) x)) (* (+ x x) x) (+ x x) (* (sqrt (+ x x)) (sqrt x)) (sqrt (+ x x)) (sqrt x))

Time

Variable

Point

Expression

2.0ms

-inf

((sqrt (* (* x 2) x)) (neg (* (sqrt 2) x)) (* (sqrt 2) x) (sqrt 2) 2 x (sqrt (* (+ x x) x)) (* (+ x x) x) (+ x x) (* (sqrt (+ x x)) (sqrt x)) (sqrt (+ x x)) (sqrt x))

1.0ms

inf

((sqrt (* (* x 2) x)) (neg (* (sqrt 2) x)) (* (sqrt 2) x) (sqrt 2) 2 x (sqrt (* (+ x x) x)) (* (+ x x) x) (+ x x) (* (sqrt (+ x x)) (sqrt x)) (sqrt (+ x x)) (sqrt x))

1.0ms

((sqrt (* (* x 2) x)) (neg (* (sqrt 2) x)) (* (sqrt 2) x) (sqrt 2) 2 x (sqrt (* (+ x x) x)) (* (+ x x) x) (+ x x) (* (sqrt (+ x x)) (sqrt x)) (sqrt (+ x x)) (sqrt x))

rewrite22.0ms (2.3%)

Memory

18.6MiB live, 18.6MiB allocated; 0ms collecting garbage

Iterations

Useful iterations: 0 (0.0ms)

Iter	Nodes	Cost
0	57	213
0	70	193
0	145	193

Stop Event

1×	iter-limit
1×	iter-limit
1×	iter-limit
1×	unsound
1×	iter-limit

Counts

27 → 53

Calls

Call 1

Inputs
#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))
(neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
(sqrt.f64 #s(literal 2 binary64))
#s(literal 2 binary64)
x
(sqrt.f64 (*.f64 (+.f64 x x) x))
(*.f64 (+.f64 x x) x)
(+.f64 x x)
(*.f64 (sqrt.f64 (+.f64 x x)) (sqrt.f64 x))
(sqrt.f64 (+.f64 x x))
(sqrt.f64 x)
#s(approx (sqrt (* (* x 2) x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (neg (* (sqrt 2) x)) #s(hole binary64 (* -1 (* x (sqrt 2)))))
#s(approx (* (sqrt 2) x) #s(hole binary64 (* x (sqrt 2))))
#s(approx x #s(hole binary64 x))
#s(approx (sqrt (* (+ x x) x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (* (+ x x) x) #s(hole binary64 (* 2 (pow x 2))))
#s(approx (+ x x) #s(hole binary64 (* 2 x)))
#s(approx (* (sqrt (+ x x)) (sqrt x)) #s(hole binary64 (* x (sqrt 2))))
#s(approx (sqrt (+ x x)) #s(hole binary64 (* (sqrt x) (sqrt 2))))
#s(approx (sqrt x) #s(hole binary64 (sqrt x)))
#s(approx (sqrt (* (* x 2) x)) #s(hole binary64 (* -1 (* x (sqrt 2)))))
#s(approx (sqrt (* (+ x x) x)) #s(hole binary64 (* -1 (* x (sqrt 2)))))
#s(approx (* (sqrt (+ x x)) (sqrt x)) #s(hole binary64 (* -1 (* x (* (sqrt -2) (sqrt -1))))))
#s(approx (sqrt (+ x x)) #s(hole binary64 (* -1 (* (sqrt x) (* (sqrt -2) (sqrt -1))))))
#s(approx (sqrt x) #s(hole binary64 (* -1 (* (sqrt x) (pow (sqrt -1) 2)))))

Outputs
#s(approx (sqrt (* (+ x x) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) (neg.f64 x)))
(.f64 (.f64 #s(literal -1 binary64) x) (sqrt.f64 #s(literal 2 binary64)))
(*.f64 (neg.f64 (sqrt.f64 #s(literal 2 binary64))) x)
(.f64 #s(literal -1 binary64) (.f64 (sqrt.f64 #s(literal 2 binary64)) x))
(.f64 (.f64 (sqrt.f64 #s(literal 2 binary64)) x) #s(literal -1 binary64))
(*.f64 (sqrt.f64 #s(literal 2 binary64)) (neg.f64 x))
(neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
(.f64 (.f64 x (sqrt.f64 #s(literal -2 binary64))) (sqrt.f64 #s(literal -1 binary64)))
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
(*.f64 x (sqrt.f64 #s(literal 2 binary64)))
(*.f64 (sqrt.f64 #s(literal -1 binary64)) (sqrt.f64 #s(literal -2 binary64)))
(*.f64 (sqrt.f64 #s(literal -2 binary64)) (sqrt.f64 #s(literal -1 binary64)))
(pow.f64 #s(literal 2 binary64) #s(literal 1/2 binary64))
(sqrt.f64 #s(literal 2 binary64))
#s(literal 2 binary64)
x
(.f64 (sqrt.f64 x) (sqrt.f64 (.f64 #s(literal 2 binary64) x)))
(.f64 (sqrt.f64 (.f64 #s(literal 2 binary64) x)) (sqrt.f64 x))
(pow.f64 (.f64 (.f64 #s(literal 2 binary64) x) x) #s(literal 1/2 binary64))
(sqrt.f64 (.f64 (.f64 #s(literal 2 binary64) x) x))
(.f64 (.f64 #s(literal 2 binary64) x) x)
(.f64 x (.f64 #s(literal 2 binary64) x))
(*.f64 #s(literal 2 binary64) x)
(*.f64 x #s(literal 2 binary64))
(/.f64 (-.f64 (.f64 x x) (.f64 x x)) (-.f64 x x))
(/.f64 (+.f64 (pow.f64 x #s(literal 3 binary64)) (pow.f64 x #s(literal 3 binary64))) (fma.f64 x x (-.f64 (.f64 x x) (.f64 x x))))
(+.f64 x x)
(.f64 (sqrt.f64 x) (sqrt.f64 (.f64 #s(literal 2 binary64) x)))
(.f64 (sqrt.f64 (.f64 #s(literal 2 binary64) x)) (sqrt.f64 x))
(pow.f64 (.f64 (.f64 #s(literal 2 binary64) x) x) #s(literal 1/2 binary64))
(sqrt.f64 (.f64 (.f64 #s(literal 2 binary64) x) x))
(.f64 (.f64 (sqrt.f64 x) (sqrt.f64 #s(literal -2 binary64))) (sqrt.f64 #s(literal -1 binary64)))
(*.f64 (sqrt.f64 x) (sqrt.f64 #s(literal 2 binary64)))
(*.f64 (sqrt.f64 #s(literal 2 binary64)) (sqrt.f64 x))
(pow.f64 (*.f64 #s(literal 2 binary64) x) #s(literal 1/2 binary64))
(sqrt.f64 (*.f64 #s(literal 2 binary64) x))
(pow.f64 x #s(literal 1/2 binary64))
(sqrt.f64 x)
#s(approx (sqrt (* (+ x x) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
#s(approx (neg (* (sqrt 2) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) (neg.f64 x)))
#s(approx (* (sqrt 2) x) (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
#s(approx x x)
#s(approx (sqrt (* (+ x x) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
#s(approx (* (+ x x) x) (.f64 (.f64 x x) #s(literal 2 binary64)))
#s(approx (+ x x) (*.f64 #s(literal 2 binary64) x))
#s(approx (sqrt (* (+ x x) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) x))
#s(approx (sqrt (+ x x)) (sqrt.f64 (*.f64 #s(literal 2 binary64) x)))
#s(approx (sqrt x) (sqrt.f64 x))
#s(approx (sqrt (* (+ x x) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) (neg.f64 x)))
#s(approx (sqrt (* (+ x x) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) (neg.f64 x)))
#s(approx (sqrt (* (+ x x) x)) (*.f64 (sqrt.f64 #s(literal 2 binary64)) (neg.f64 x)))
#s(approx (sqrt (+ x x)) (neg.f64 (sqrt.f64 (*.f64 #s(literal 2 binary64) x))))
#s(approx (sqrt x) (neg.f64 (*.f64 (sqrt.f64 x) #s(literal -1 binary64))))

eval4.0ms (0.4%)

Memory: 6.9MiB live, 6.9MiB allocated; 0ms collecting garbage
Compiler: Compiled 643 to 147 computations (77.1% saved)

prune15.0ms (1.6%)

Memory

-35.6MiB live, 9.3MiB allocated; 8ms collecting garbage

Pruning

5 alts after pruning (0 fresh and 5 done)

	Pruned	Kept	Total
New	62	0	62
Fresh	0	0	0
Picked	0	3	3
Done	0	2	2
Total	62	5	67

Accuracy

100.0%

Counts

67 → 5

Alt Table

Click to see full alt table

Status	Accuracy	Program
✓	100.0%	(hypot.f64 x x)
✓	99.6%	(*.f64 (sqrt.f64 (+.f64 x x)) (sqrt.f64 x))
✓	99.4%	(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
✓	55.3%	(sqrt.f64 (*.f64 (+.f64 x x) x))
✓	2.2%	#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))

Compiler

Compiled 83 to 50 computations (39.8% saved)

regimes10.0ms (1.1%)

Memory

15.2MiB live, 15.2MiB allocated; 0ms collecting garbage

Counts

8 → 1

Calls

Call 1

Inputs
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))
(sqrt.f64 (*.f64 (+.f64 x x) x))
(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))
(sqrt.f64 (.f64 (.f64 x #s(literal 2 binary64)) x))
(*.f64 (sqrt.f64 (+.f64 x x)) (sqrt.f64 x))
(.f64 (sqrt.f64 (.f64 x #s(literal 2 binary64))) (sqrt.f64 x))
(hypot.f64 x x)

Outputs
(hypot.f64 x x)

Calls

4 calls:

3.0ms	x
2.0ms	(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))
2.0ms	(.f64 #s(literal 2 binary64) (.f64 x x))
2.0ms	(*.f64 x x)

Results

Accuracy	Segments	Branch
100.0%	1	`(*.f64 x x)`
100.0%	1	`(.f64 #s(literal 2 binary64) (.f64 x x))`
100.0%	1	`(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))`
100.0%	1	`x`

Compiler

Compiled 15 to 12 computations (20% saved)

regimes9.0ms (1%)

Memory

13.9MiB live, 13.9MiB allocated; 0ms collecting garbage

Counts

7 → 1

Calls

Call 1

Inputs
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))
(sqrt.f64 (*.f64 (+.f64 x x) x))
(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))
(sqrt.f64 (.f64 (.f64 x #s(literal 2 binary64)) x))
(*.f64 (sqrt.f64 (+.f64 x x)) (sqrt.f64 x))
(.f64 (sqrt.f64 (.f64 x #s(literal 2 binary64))) (sqrt.f64 x))

Outputs
(*.f64 (sqrt.f64 (+.f64 x x)) (sqrt.f64 x))

Calls

4 calls:

2.0ms	x
2.0ms	(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))
2.0ms	(.f64 #s(literal 2 binary64) (.f64 x x))
2.0ms	(*.f64 x x)

Results

Accuracy	Segments	Branch
99.6%	1	`(*.f64 x x)`
99.6%	1	`(.f64 #s(literal 2 binary64) (.f64 x x))`
99.6%	1	`(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))`
99.6%	1	`x`

Compiler

Compiled 15 to 12 computations (20% saved)

regimes7.0ms (0.8%)

Memory

12.5MiB live, 12.5MiB allocated; 0ms collecting garbage

Accuracy

Total -62.2b remaining (-15458.8%)

Threshold costs -62.2b (-15458.8%)

Counts

5 → 1

Calls

Call 1

Inputs
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)
#s(approx (sqrt (* (* x 2) x)) (neg.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) x)))
(sqrt.f64 (*.f64 (+.f64 x x) x))
(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))
(sqrt.f64 (.f64 (.f64 x #s(literal 2 binary64)) x))

Outputs
(*.f64 (sqrt.f64 #s(literal 2 binary64)) x)

Calls

4 calls:

2.0ms	x
2.0ms	(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))
2.0ms	(.f64 #s(literal 2 binary64) (.f64 x x))
2.0ms	(*.f64 x x)

Results

Accuracy	Segments	Branch
99.4%	1	`(*.f64 x x)`
99.4%	1	`(.f64 #s(literal 2 binary64) (.f64 x x))`
99.4%	1	`(sqrt.f64 (.f64 #s(literal 2 binary64) (.f64 x x)))`
99.4%	1	`x`

Compiler

Compiled 15 to 12 computations (20% saved)

derivations35.0ms (3.8%)

Memory

-33.5MiB live, 10.3MiB allocated; 6ms collecting garbage

Stop Event

1×

done

Compiler

Compiled 14 to 9 computations (35.7% saved)

preprocess15.0ms (1.6%)

Memory: 35.3MiB live, 35.3MiB allocated; 0ms collecting garbage
Compiler: Compiled 68 to 50 computations (26.5% saved)

end0.0ms (0%)

Memory: 0.0MiB live, 0.0MiB allocated; 0ms collecting garbage

start0.0ms (0%)

analyze0.0ms (0%)

sample572.0ms (60.7%)

explain56.0ms (6%)

preprocess77.0ms (8.1%)

series4.0ms (0.4%)

rewrite41.0ms (4.4%)

eval6.0ms (0.6%)

prune3.0ms (0.4%)

series12.0ms (1.3%)

rewrite22.0ms (2.3%)

eval6.0ms (0.6%)

prune19.0ms (2%)

series5.0ms (0.5%)

rewrite22.0ms (2.3%)

eval4.0ms (0.4%)

prune15.0ms (1.6%)

regimes10.0ms (1.1%)

regimes9.0ms (1%)

regimes7.0ms (0.8%)

derivations35.0ms (3.8%)

preprocess15.0ms (1.6%)

end0.0ms (0%)

Profiling