Average Error: 0.0 → 0.0
Time: 25.8s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[e^{im} \cdot \left(0.5 \cdot \cos re\right) + \left(0.5 \cdot \cos re\right) \cdot e^{-im}\]

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. Using strategy rm

  3. Applied distribute-lft-in0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

Details

Time bar (total: 25.3s)Debug log

sample275.0ms

Algorithm
intervals

simplify229.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

228.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 2 expressions with local error:

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

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
2 → 39
Calls

2 calls. Slowest were:

13.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))
7.0ms
(+ (exp (- im)) (exp im))

series137.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

94.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))
43.0ms
(+ (exp (- im)) (exp im))

simplify1.5s

Counts
21 → 45
Calls

21 calls. Slowest were:

441.0ms
(* 0.5 (* (cos re) (+ (exp (- im)) (exp im))))
247.0ms
(* 0.5 (* (+ (exp im) (exp (* -1 im))) (cos re)))
224.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

prune480.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 3 expressions with local error:

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

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
3 → 52
Calls

3 calls. Slowest were:

10.0ms
(+ (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp im)))
8.0ms
(* (* 0.5 (cos re)) (exp (- im)))
5.0ms
(* (* 0.5 (cos re)) (exp im))

series214.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

111.0ms
(+ (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp im)))
53.0ms
(* (* 0.5 (cos re)) (exp im))
50.0ms
(* (* 0.5 (cos re)) (exp (- im)))

simplify2.6s

Counts
28 → 61
Calls

28 calls. Slowest were:

493.0ms
(+ (* 0.5 (* (exp im) (cos re))) (* 0.5 (* (exp (* -1 im)) (cos re))))
445.0ms
(+ (* 0.5 (* (exp (- im)) (cos re))) (* 0.5 (* (exp im) (cos re))))
398.0ms
(* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (exp (- im)) (exp (- im))) (exp (- im))))

prune800.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

0.1b
(cbrt (exp im))
0.1b
(cbrt (exp im))
0.1b
(cbrt (exp im))
0.0b
(* (cbrt (exp im)) (cbrt (exp im)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (exp im)) (cbrt (exp im)))
1.0ms
(cbrt (exp im))
1.0ms
(cbrt (exp im))

series552.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

161.0ms
(cbrt (exp im))
149.0ms
(cbrt (exp im))
139.0ms
(* (cbrt (exp im)) (cbrt (exp im)))
103.0ms
(cbrt (exp im))

simplify284.0ms

Counts
41 → 81
Calls

41 calls. Slowest were:

67.0ms
(+ (* 1/18 (pow im 2)) (+ (* 1/3 im) 1))
62.0ms
(+ (* 1/18 (pow im 2)) (+ (* 1/3 im) 1))
45.0ms
(+ (* 2/9 (pow im 2)) (+ (* 2/3 im) 1))

prune1.1s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.8b
(* (* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im)))) (sqrt (+ (exp (- im)) (exp im))))
0.2b
(* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im))))
0.0b
(sqrt (+ (exp (- im)) (exp im)))
0.0b
(sqrt (+ (exp (- im)) (exp im)))

rewrite88.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

38.0ms
(* (* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im)))) (sqrt (+ (exp (- im)) (exp im))))
25.0ms
(* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im))))
16.0ms
(sqrt (+ (exp (- im)) (exp im)))

series242.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

81.0ms
(* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im))))
69.0ms
(* (* (* 0.5 (cos re)) (sqrt (+ (exp (- im)) (exp im)))) (sqrt (+ (exp (- im)) (exp im))))
48.0ms
(sqrt (+ (exp (- im)) (exp im)))
43.0ms
(sqrt (+ (exp (- im)) (exp im)))

simplify6.5s

Counts
66 → 103
Calls

66 calls. Slowest were:

703.0ms
(* (* (* (* 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)))))
485.0ms
(* 0.5 (* (cos re) (+ (exp (- im)) (exp im))))
375.0ms
(* (* (* 0.5 (cos re)) (sqrt (- (* (exp (- im)) (exp (- im))) (* (exp im) (exp im))))) (sqrt (- (* (exp (- im)) (exp (- im))) (* (exp im) (exp im)))))

prune1.4s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes220.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample8.6s

Algorithm
intervals