Average Error: 0.5 → 0.5
Time: 28.9s
Precision: 64
Internal Precision: 128
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

  2. Initial simplification0.5

    \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\]
  3. Using strategy rm

  4. Applied associate-*l/0.5

    \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}}\]

  5. Final simplification0.5

    \[\leadsto \frac{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))

Details

Time bar (total: 18.6s)Debug log

start348.0ms

Algorithm
intervals

setup513.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize44.0ms

Local error

Found 3 expressions with local error:

0.7b
(/ (cos th) (sqrt 2))
0.3b
(* (/ (cos th) (sqrt 2)) (+ (* a1 a1) (* a2 a2)))
0.0b
(+ (* a1 a1) (* a2 a2))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
3 → 74
Calls

3 calls. Slowest were:

13.0ms
(* (/ (cos th) (sqrt 2)) (+ (* a1 a1) (* a2 a2)))
7.0ms
(/ (cos th) (sqrt 2))
2.0ms
(+ (* a1 a1) (* a2 a2))

series305.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

245.0ms
(* (/ (cos th) (sqrt 2)) (+ (* a1 a1) (* a2 a2)))
34.0ms
(/ (cos th) (sqrt 2))
25.0ms
(+ (* a1 a1) (* a2 a2))

simplify2.8s

Counts
64 → 83
Calls

64 calls. Slowest were:

508.0ms
(+ (* (* a1 a1) (* a1 a1)) (- (* (* a2 a2) (* a2 a2)) (* (* a1 a1) (* a2 a2))))
401.0ms
(* (cos th) (- (* (* a1 a1) (* a1 a1)) (* (* a2 a2) (* a2 a2))))
355.0ms
(* (sqrt 2) (- (* a1 a1) (* a2 a2)))

prune1.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 3 expressions with local error:

0.5b
(/ (* (cos th) (+ (* a1 a1) (* a2 a2))) (sqrt 2))
0.2b
(* (cos th) (+ (* a1 a1) (* a2 a2)))
0.0b
(+ (* a1 a1) (* a2 a2))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
3 → 55
Calls

3 calls. Slowest were:

16.0ms
(/ (* (cos th) (+ (* a1 a1) (* a2 a2))) (sqrt 2))
11.0ms
(* (cos th) (+ (* a1 a1) (* a2 a2)))
4.0ms
(+ (* a1 a1) (* a2 a2))

series336.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

183.0ms
(/ (* (cos th) (+ (* a1 a1) (* a2 a2))) (sqrt 2))
128.0ms
(* (cos th) (+ (* a1 a1) (* a2 a2)))
25.0ms
(+ (* a1 a1) (* a2 a2))

simplify2.0s

Counts
38 → 64
Calls

38 calls. Slowest were:

482.0ms
(+ (* (* a1 a1) (* a1 a1)) (- (* (* a2 a2) (* a2 a2)) (* (* a1 a1) (* a2 a2))))
453.0ms
(* (cos th) (- (* (* a1 a1) (* a1 a1)) (* (* a2 a2) (* a2 a2))))
232.0ms
(/ (* (* (* (cos th) (cos th)) (cos th)) (* (* (+ (* a1 a1) (* a2 a2)) (+ (* a1 a1) (* a2 a2))) (+ (* a1 a1) (* a2 a2)))) (* (* (sqrt 2) (sqrt 2)) (sqrt 2)))

prune867.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize5.0ms

Local error

Found 3 expressions with local error:

0.4b
(/ (+ (* a1 a1) (* a2 a2)) (sqrt 2))
0.1b
(* (cos th) (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2)))
0.0b
(+ (* a1 a1) (* a2 a2))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
3 → 70
Calls

3 calls. Slowest were:

8.0ms
(/ (+ (* a1 a1) (* a2 a2)) (sqrt 2))
7.0ms
(* (cos th) (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2)))
2.0ms
(+ (* a1 a1) (* a2 a2))

series263.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

181.0ms
(* (cos th) (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2)))
56.0ms
(/ (+ (* a1 a1) (* a2 a2)) (sqrt 2))
25.0ms
(+ (* a1 a1) (* a2 a2))

simplify1.7s

Counts
58 → 79
Calls

58 calls. Slowest were:

359.0ms
(/ (* (* (+ (* a1 a1) (* a2 a2)) (+ (* a1 a1) (* a2 a2))) (+ (* a1 a1) (* a2 a2))) (* (* (sqrt 2) (sqrt 2)) (sqrt 2)))
264.0ms
(* (* (* (cos th) (cos th)) (cos th)) (* (* (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2)) (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2))) (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2))))
181.0ms
(- (* (* a1 a1) (* a1 a1)) (* (* a2 a2) (* a2 a2)))

prune1.2s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

5.8b
(sqrt (+ (* a1 a1) (* a2 a2)))
5.8b
(sqrt (+ (* a1 a1) (* a2 a2)))
0.5b
(/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2))
0.3b
(* (* (cos th) (/ (sqrt (+ (* a1 a1) (* a2 a2))) 1)) (/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2)))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 114
Calls

4 calls. Slowest were:

16.0ms
(* (* (cos th) (/ (sqrt (+ (* a1 a1) (* a2 a2))) 1)) (/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2)))
5.0ms
(/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2))
2.0ms
(sqrt (+ (* a1 a1) (* a2 a2)))

series333.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

187.0ms
(* (* (cos th) (/ (sqrt (+ (* a1 a1) (* a2 a2))) 1)) (/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2)))
77.0ms
(/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2))
38.0ms
(sqrt (+ (* a1 a1) (* a2 a2)))
31.0ms
(sqrt (+ (* a1 a1) (* a2 a2)))

simplify4.2s

Counts
110 → 126
Calls

110 calls. Slowest were:

838.0ms
(* (* (* (* (cos th) (/ (sqrt (+ (* a1 a1) (* a2 a2))) 1)) (* (cos th) (/ (sqrt (+ (* a1 a1) (* a2 a2))) 1))) (* (cos th) (/ (sqrt (+ (* a1 a1) (* a2 a2))) 1))) (* (* (/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2)) (/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2))) (/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2))))
348.0ms
(+ (+ (log (cos th)) (log (/ (sqrt (+ (* a1 a1) (* a2 a2))) 1))) (- (log (sqrt (+ (* a1 a1) (* a2 a2)))) (log (sqrt 2))))
249.0ms
(+ (+ (log (cos th)) (log (/ (sqrt (+ (* a1 a1) (* a2 a2))) 1))) (log (/ (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt 2))))

prune2.0s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes503.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch4.0ms