Average Error: 0.0 → 0.0
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right)\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]

  2. Final simplification0.0

    \[\leadsto \left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right)\]

Reproduce

herbie shell --seed 2018365 
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

Details

Time bar (total: 1.2m)Debug log

start329.0ms

Algorithm
intervals

setup450.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize40.0ms

Local error

Found 1 expressions with local error:

0.0b
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
1 → 20
Calls

1 calls. Slowest were:

24.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

series76.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

76.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

simplify1.2s

Counts
14 → 23
Calls

14 calls. Slowest were:

357.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))
259.0ms
(* 0.5 (* (cos re) (+ (exp (- im)) (exp im))))
199.0ms
(* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (+ (exp (- im)) (exp im)) (+ (exp (- im)) (exp im))) (+ (exp (- im)) (exp im))))

prune270.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

0.3b
(cbrt (* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
0.3b
(* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
0.2b
(* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
0.0b
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

rewrite402.0ms

Algorithm
rewrite-expression-head
Counts
4 → 189
Calls

4 calls. Slowest were:

197.0ms
(* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
108.0ms
(cbrt (* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
61.0ms
(* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

series372.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

124.0ms
(* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
107.0ms
(cbrt (* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
81.0ms
(* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))
60.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

simplify57.3s

Counts
205 → 201
Calls

205 calls. Slowest were:

959.0ms
(* (* (* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (+ (exp (- im)) (exp im)) (+ (exp (- im)) (exp im))) (+ (exp (- im)) (exp im)))) (* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))) (* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
948.0ms
(* (* (* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (+ (exp (- im)) (exp im)) (+ (exp (- im)) (exp im))) (+ (exp (- im)) (exp im))))) (* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (+ (exp (- im)) (exp im)) (+ (exp (- im)) (exp im))) (+ (exp (- im)) (exp im)))))
921.0ms
(* (* (* (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))) (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))) (* (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))) (* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (+ (exp (- im)) (exp im)) (+ (exp (- im)) (exp im))) (+ (exp (- im)) (exp im)))))

prune3.4s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize7.0ms

Local error

Found 4 expressions with local error:

0.7b
(* (* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im)))) (sqrt (+ (exp (- im)) (exp im))))
0.3b
(* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im))))
0.0b
(sqrt (+ (exp (- im)) (exp im)))
0.0b
(sqrt (+ (exp (- im)) (exp im)))

rewrite96.0ms

Algorithm
rewrite-expression-head
Counts
4 → 81
Calls

4 calls. Slowest were:

41.0ms
(* (* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im)))) (sqrt (+ (exp (- im)) (exp im))))
24.0ms
(* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im))))
17.0ms
(sqrt (+ (exp (- im)) (exp im)))

series279.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

97.0ms
(* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im))))
83.0ms
(* (* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im)))) (sqrt (+ (exp (- im)) (exp im))))
56.0ms
(sqrt (+ (exp (- im)) (exp im)))
43.0ms
(sqrt (+ (exp (- im)) (exp im)))

simplify6.2s

Counts
64 → 93
Calls

64 calls. Slowest were:

629.0ms
(* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (sqrt (+ (exp (- im)) (exp im))) (sqrt (+ (exp (- im)) (exp im)))) (sqrt (+ (exp (- im)) (exp im)))))
411.0ms
(- (+ (sqrt 2) (+ (* 1/2 (/ (pow im 2) (sqrt 2))) (* 1/24 (/ (pow im 4) (sqrt 2))))) (* 1/8 (/ (pow im 4) (pow (sqrt 2) 3))))
375.0ms
(* 0.5 (* (cos re) (+ (exp (- im)) (exp im))))

prune1.6s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes283.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms