Average Error: 0.0 → 0.0
Time: 18.8s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\frac{\cos re \cdot 0.5}{e^{im}} + \left(\cos re \cdot 0.5\right) \cdot e^{im}\]

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. Initial simplification0.0

    \[\leadsto (\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*\]
  3. Using strategy rm

  4. Applied fma-udef0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2018362 +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: 6.8s)Debug log

start272.0ms

Algorithm
intervals

setup261.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize37.0ms

Local error

Found 2 expressions with local error:

0.0b
(/ (* (cos re) 0.5) (exp im))
0.0b
(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

11.0ms
(/ (* (cos re) 0.5) (exp im))
0.0ms
(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))

series146.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

84.0ms
(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))
61.0ms
(/ (* (cos re) 0.5) (exp im))

simplify830.0ms

Counts
18 → 37
Calls

18 calls. Slowest were:

475.0ms
(/ (* (* (* (cos re) 0.5) (* (cos re) 0.5)) (* (cos re) 0.5)) (* (* (exp im) (exp im)) (exp im)))
146.0ms
(- 0.5 (+ (* 0.25 (pow re 2)) (* 0.5 im)))
80.0ms
(- (+ (* 0.5 (pow im 2)) 1.0) (* 0.5 (pow re 2)))

prune550.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 3 expressions with local error:

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

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
3 → 52
Calls

3 calls. Slowest were:

18.0ms
(+ (* (exp im) (* (cos re) 0.5)) (/ (* (cos re) 0.5) (exp im)))
13.0ms
(* (exp im) (* (cos re) 0.5))
11.0ms
(/ (* (cos re) 0.5) (exp im))

series209.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

92.0ms
(+ (* (exp im) (* (cos re) 0.5)) (/ (* (cos re) 0.5) (exp im)))
63.0ms
(* (exp im) (* (cos re) 0.5))
54.0ms
(/ (* (cos re) 0.5) (exp im))

simplify1.3s

Counts
28 → 61
Calls

28 calls. Slowest were:

446.0ms
(/ (* (* (* (cos re) 0.5) (* (cos re) 0.5)) (* (cos re) 0.5)) (* (* (exp im) (exp im)) (exp im)))
277.0ms
(* (* (* (exp im) (exp im)) (exp im)) (* (* (* (cos re) 0.5) (* (cos re) 0.5)) (* (cos re) 0.5)))
129.0ms
(- 0.5 (+ (* 0.25 (pow re 2)) (* 0.5 im)))

prune682.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.0b
(/ (cos re) (sqrt (exp im)))
0.0b
(/ 0.5 (sqrt (exp im)))
0.0b
(sqrt (exp im))
0.0b
(sqrt (exp im))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

4.0ms
(/ (cos re) (sqrt (exp im)))
1.0ms
(sqrt (exp im))
1.0ms
(sqrt (exp im))

series113.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

43.0ms
(/ (cos re) (sqrt (exp im)))
38.0ms
(/ 0.5 (sqrt (exp im)))
19.0ms
(sqrt (exp im))
12.0ms
(sqrt (exp im))

simplify719.0ms

Counts
72 → 99
Calls

72 calls. Slowest were:

171.0ms
(- 1 (+ (* 1/2 (pow re 2)) (* 1/2 im)))
140.0ms
(/ (* (* (cos re) (cos re)) (cos re)) (* (* (sqrt (exp im)) (sqrt (exp im))) (sqrt (exp im))))
68.0ms
(+ (* 1/8 (pow im 2)) (+ (* 1/2 im) 1))

prune1.5s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes114.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms