Average Error: 0.0 → 0.0
Time: 7.0s
Precision: 64
Internal Precision: 128
\[e^{re} \cdot \cos im\]
↓
\[\sqrt{e^{re}} \cdot \left(\cos im \cdot \sqrt{e^{re}}\right)\]

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

    \[e^{re} \cdot \cos im\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.0

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

  4. Applied associate-*l*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019021 
(FPCore (re im)
  :name "math.exp on complex, real part"
  (* (exp re) (cos im)))

Details

Time bar (total: 6.2s)Debug log

sample100.0ms

Algorithm
intervals
Results
30.0ms41×body1280valid
25.0ms221×body80valid
21.0ms41×body640valid
6.0ms17×body320valid
3.0ms12×body160valid

simplify4.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(* (exp re) (cos im))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize412.0ms

Local error

Found 1 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
3×add-cube-cbrt
3×associate-*l*
3×associate-*r*
3×add-cbrt-cube
3×*-un-lft-identity
3×pow1
3×add-sqr-sqrt
2×add-exp-log
1×add-log-exp
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
Counts
1 → 17
Calls
1 calls:
Slowest
5.0ms
(* (exp re) (cos im))

series33.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
33.0ms
(* (exp re) (cos im))

simplify84.0ms

Counts
11 → 20
Calls
11 calls:
Slowest
53.0ms
(* (* (* (exp re) (exp re)) (exp re)) (* (* (cos im) (cos im)) (cos im)))
20.0ms
(+ re (+ (* 1/2 (pow re 2)) 1))
2.0ms
(+ re (log (cos im)))
2.0ms
(* (exp re) (cos im))
2.0ms
(* (cos im) (exp re))

prune133.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize10.0ms

Local error

Found 4 expressions with local error:

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

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
13×add-sqr-sqrt
12×sqrt-prod
12×associate-*l*
11×add-cube-cbrt
11×add-exp-log
11×add-cbrt-cube
11×*-un-lft-identity
11×pow1
4×add-log-exp
4×cbrt-unprod
4×prod-exp
4×pow-prod-down
4×associate-*r*
2×*-commutative
2×pow1/2
2×rem-sqrt-square
Counts
4 → 65
Calls
4 calls:
Slowest
17.0ms
(* (sqrt (exp re)) (* (sqrt (exp re)) (cos im)))
4.0ms
(* (sqrt (exp re)) (cos im))
0.0ms
(sqrt (exp re))
0.0ms
(sqrt (exp re))

series113.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
51.0ms
(* (sqrt (exp re)) (cos im))
33.0ms
(* (sqrt (exp re)) (* (sqrt (exp re)) (cos im)))
17.0ms
(sqrt (exp re))
12.0ms
(sqrt (exp re))

simplify899.0ms

Counts
43 → 77
Calls
43 calls:
Slowest
358.0ms
(* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (* (sqrt (exp re)) (cos im)) (* (sqrt (exp re)) (cos im))) (* (sqrt (exp re)) (cos im))))
291.0ms
(* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (cos im) (cos im)) (cos im))))
57.0ms
(* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (cos im) (cos im)) (cos im)))
43.0ms
(+ (* 1/2 re) (+ (* 1/8 (pow re 2)) 1))
24.0ms
(+ re (+ (* 1/2 (pow re 2)) 1))

prune869.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 2 expressions with local error:

2.0b
(log (cos im))
0.0b
(exp (+ re (log (cos im))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×add-log-exp
3×add-exp-log
3×exp-prod
3×log-prod
3×pow1
2×add-cbrt-cube
1×rem-exp-log
1×log-pow
1×exp-sum
1×rem-log-exp
Counts
2 → 24
Calls
2 calls:
Slowest
6.0ms
(exp (+ re (log (cos im))))
1.0ms
(log (cos im))

series154.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
129.0ms
(log (cos im))
25.0ms
(exp (+ re (log (cos im))))

simplify139.0ms

Counts
15 → 30
Calls
15 calls:
Slowest
96.0ms
(- (+ (* 1/2 (pow im 2)) (+ (* 1/12 (pow im 4)) (* 1/45 (pow im 6)))))
21.0ms
(+ re (+ (* 1/2 (pow re 2)) 1))
4.0ms
(exp (+ (log (cos im)) re))
4.0ms
(exp (+ (log (cos im)) re))
3.0ms
(log (* (cbrt (cos im)) (cbrt (cos im))))

prune278.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes87.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample2.9s

Algorithm
intervals
Results
1.2s1518×body1280valid
779.0ms6940×body80valid
580.0ms1241×body640valid
181.0ms629×body320valid
61.0ms324×body160valid