Average Error: 0.0 → 0.0
Time: 12.9s
Precision: 64
Internal Precision: 128
\[e^{re} \cdot \sin im\]
↓
\[e^{re} \cdot \sin im\]

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

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

  3. Applied add-sqr-sqrt0.0

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

  4. Applied associate-*l*0.0

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

  5. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\sin im \cdot e^{re}}\]

  6. Final simplification0.0

    \[\leadsto e^{re} \cdot \sin im\]

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  (* (exp re) (sin im)))

Details

Time bar (total: 12.4s)Debug log

sample126.0ms

Algorithm
intervals

simplify4.0ms

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

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 1 expressions with local error:

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

rewrite8.0ms

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×log1p-expm1-u
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×expm1-log1p-u
Counts
1 → 19
Calls
1 calls:
Slowest
8.0ms
(* (exp re) (sin im))

series93.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
93.0ms
(* (exp re) (sin im))

simplify758.0ms

Counts
11 → 22
Calls
11 calls:
Slowest
450.0ms
(* (* (* (exp re) (exp re)) (exp re)) (* (* (sin im) (sin im)) (sin im)))
293.0ms
(+ (* 1/2 (* (pow re 2) im)) (+ (* re im) im))
2.0ms
(* (exp re) (sin im))
2.0ms
(+ re (log (sin im)))
2.0ms
(* (sin im) (exp re))

prune188.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

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

rewrite17.0ms

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×log1p-expm1-u
4×cbrt-unprod
4×prod-exp
4×pow-prod-down
4×associate-*r*
4×expm1-log1p-u
2×*-commutative
2×pow1/2
2×rem-sqrt-square
Counts
4 → 73
Calls
4 calls:
Slowest
11.0ms
(* (sqrt (exp re)) (* (sqrt (exp re)) (sin im)))
4.0ms
(* (sqrt (exp re)) (sin im))
1.0ms
(sqrt (exp re))
1.0ms
(sqrt (exp re))

series138.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
64.0ms
(* (sqrt (exp re)) (sin im))
50.0ms
(* (sqrt (exp re)) (* (sqrt (exp re)) (sin im)))
13.0ms
(sqrt (exp re))
11.0ms
(sqrt (exp re))

simplify1.8s

Counts
43 → 85
Calls
43 calls:
Slowest
554.0ms
(* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (* (sqrt (exp re)) (sin im)) (* (sqrt (exp re)) (sin im))) (* (sqrt (exp re)) (sin im))))
377.0ms
(+ (* 1/2 (* (pow re 2) im)) (+ (* re im) im))
361.0ms
(* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (sin im) (sin im)) (sin im))))
232.0ms
(+ (* 1/8 (* (pow re 2) im)) (+ (* 1/2 (* re im)) im))
71.0ms
(* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (sin im) (sin im)) (sin im)))

prune876.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize6.0ms

Local error

Found 1 expressions with local error:

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

rewrite9.0ms

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×log1p-expm1-u
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×expm1-log1p-u
Counts
1 → 19
Calls
1 calls:
Slowest
9.0ms
(* (sin im) (exp re))

series36.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
36.0ms
(* (sin im) (exp re))

simplify331.0ms

Counts
11 → 22
Calls
11 calls:
Slowest
256.0ms
(- (+ (* re im) im) (* 1/6 (pow im 3)))
53.0ms
(* (* (* (sin im) (sin im)) (sin im)) (* (* (exp re) (exp re)) (exp re)))
4.0ms
(* (sin im) (exp re))
4.0ms
(+ (log (sin im)) re)
4.0ms
(* (sin im) (exp re))

prune164.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite87.0ms

Algorithm
rewrite-expression-head
Rules
18×sqrt-prod
18×add-exp-log
18×add-cbrt-cube
18×pow1
15×associate-*l*
13×add-sqr-sqrt
11×add-cube-cbrt
11×*-un-lft-identity
9×cbrt-unprod
9×prod-exp
9×pow-prod-down
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
2×*-commutative
2×associate-*r*
2×pow1/2
2×rem-sqrt-square
Counts
4 → 80
Calls
4 calls:
Slowest
56.0ms
(* (sqrt (exp re)) (* (sqrt (sqrt (exp re))) (* (sqrt (sqrt (exp re))) (sin im))))
26.0ms
(* (sqrt (sqrt (exp re))) (* (sqrt (sqrt (exp re))) (sin im)))
2.0ms
(sqrt (exp re))
1.0ms
(sqrt (exp re))

series161.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
78.0ms
(* (sqrt (sqrt (exp re))) (* (sqrt (sqrt (exp re))) (sin im)))
48.0ms
(* (sqrt (exp re)) (* (sqrt (sqrt (exp re))) (* (sqrt (sqrt (exp re))) (sin im))))
18.0ms
(sqrt (exp re))
17.0ms
(sqrt (exp re))

simplify2.8s

Counts
50 → 92
Calls
50 calls:
Slowest
568.0ms
(* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (* (sqrt (sqrt (exp re))) (sqrt (sqrt (exp re)))) (sqrt (sqrt (exp re)))) (* (* (* (sqrt (sqrt (exp re))) (sqrt (sqrt (exp re)))) (sqrt (sqrt (exp re)))) (* (* (sin im) (sin im)) (sin im)))))
451.0ms
(* (* (* (sqrt (exp re)) (sqrt (exp re))) (sqrt (exp re))) (* (* (* (sqrt (sqrt (exp re))) (* (sqrt (sqrt (exp re))) (sin im))) (* (sqrt (sqrt (exp re))) (* (sqrt (sqrt (exp re))) (sin im)))) (* (sqrt (sqrt (exp re))) (* (sqrt (sqrt (exp re))) (sin im)))))
389.0ms
(* (* (* (sqrt (sqrt (exp re))) (sqrt (sqrt (exp re)))) (sqrt (sqrt (exp re)))) (* (* (* (sqrt (sqrt (exp re))) (sin im)) (* (sqrt (sqrt (exp re))) (sin im))) (* (sqrt (sqrt (exp re))) (sin im))))
334.0ms
(* (* (* (sqrt (sqrt (exp re))) (sqrt (sqrt (exp re)))) (sqrt (sqrt (exp re)))) (* (* (* (sqrt (sqrt (exp re))) (sqrt (sqrt (exp re)))) (sqrt (sqrt (exp re)))) (* (* (sin im) (sin im)) (sin im))))
293.0ms
(+ (* 1/2 (* (pow re 2) im)) (+ (* re im) im))

prune1.0s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes128.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample3.7s

Algorithm
intervals