Average Error: 0.0 → 0.0
Time: 25.9s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[0.5 \cdot \left(e^{im} \cdot \cos re + \frac{\cos re}{e^{im}}\right)\]

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. Initial simplification0.0

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

  3. Taylor expanded around -inf 0.0

    \[\leadsto \color{blue}{\left(0.5 \cdot e^{im} + 0.5 \cdot \frac{1}{e^{im}}\right) \cdot \cos re}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2018362 
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

Details

Time bar (total: 14.1s)Debug log

start245.0ms

Algorithm
intervals

setup507.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize31.0ms

Local error

Found 3 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 45
Calls

3 calls. Slowest were:

9.0ms
(* (+ (/ 0.5 (exp im)) (* 0.5 (exp im))) (cos re))
4.0ms
(+ (/ 0.5 (exp im)) (* 0.5 (exp im)))
0.0ms
(/ 0.5 (exp im))

series148.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

76.0ms
(* (+ (/ 0.5 (exp im)) (* 0.5 (exp im))) (cos re))
62.0ms
(+ (/ 0.5 (exp im)) (* 0.5 (exp im)))
10.0ms
(/ 0.5 (exp im))

simplify1.1s

Counts
24 → 54
Calls

24 calls. Slowest were:

432.0ms
(* (+ (pow (/ 0.5 (exp im)) 3) (pow (* 0.5 (exp im)) 3)) (cos re))
124.0ms
(* (- (* (/ 0.5 (exp im)) (/ 0.5 (exp im))) (* (* 0.5 (exp im)) (* 0.5 (exp im)))) (cos re))
116.0ms
(* (* (* (+ (/ 0.5 (exp im)) (* 0.5 (exp im))) (+ (/ 0.5 (exp im)) (* 0.5 (exp im)))) (+ (/ 0.5 (exp im)) (* 0.5 (exp im)))) (* (* (cos re) (cos re)) (cos re)))

prune598.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize10.0ms

Local error

Found 3 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 55
Calls

3 calls. Slowest were:

8.0ms
(+ (/ (cos re) (exp im)) (* (exp im) (cos re)))
3.0ms
(* (exp im) (cos re))
3.0ms
(/ (cos re) (exp im))

series134.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

66.0ms
(+ (/ (cos re) (exp im)) (* (exp im) (cos re)))
35.0ms
(* (exp im) (cos re))
34.0ms
(/ (cos re) (exp im))

simplify446.0ms

Counts
43 → 64
Calls

43 calls. Slowest were:

112.0ms
(/ (* (* (cos re) (cos re)) (cos re)) (* (* (exp im) (exp im)) (exp im)))
83.0ms
(- 1 (+ (* 1/2 (pow re 2)) im))
49.0ms
(* (* (* (exp im) (exp im)) (exp im)) (* (* (cos re) (cos re)) (cos re)))

prune624.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 238
Calls

4 calls. Slowest were:

7.0ms
(/ (/ (cos re) (sqrt (exp im))) (sqrt (exp im)))
4.0ms
(/ (cos re) (sqrt (exp im)))
1.0ms
(sqrt (exp im))

series163.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

73.0ms
(/ (cos re) (sqrt (exp im)))
51.0ms
(/ (/ (cos re) (sqrt (exp im))) (sqrt (exp im)))
19.0ms
(sqrt (exp im))
19.0ms
(sqrt (exp im))

simplify6.4s

Counts
362 → 250
Calls

362 calls. Slowest were:

504.0ms
(/ (* (* (/ (cos re) (sqrt (exp im))) (/ (cos re) (sqrt (exp im)))) (/ (cos re) (sqrt (exp im)))) (* (* (sqrt (exp im)) (sqrt (exp im))) (sqrt (exp im))))
409.0ms
(/ (* (cbrt (/ (cos re) (sqrt (exp im)))) (cbrt (/ (cos re) (sqrt (exp im))))) (sqrt 1))
293.0ms
(/ (/ (* (* (cos re) (cos re)) (cos re)) (* (* (sqrt (exp im)) (sqrt (exp im))) (sqrt (exp im)))) (* (* (sqrt (exp im)) (sqrt (exp im))) (sqrt (exp im))))

prune3.6s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes69.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms