Average Error: 43.2 → 43.2
Time: 27.7s
Precision: 64
Internal Precision: 1344
\[\Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0 i\right)}\right))\]
↓
\[\Im(\left(\frac{\left(\left(e^{\left(-\left(xre + xim i\right)\right)}\right) + \left(e^{\left(xre + xim i\right)}\right)\right)}{\left(2 + 0 i\right)}\right))\]

Error

Bits error versus xre

Bits error versus xim

Derivation

  1. Initial program 43.2

    \[\Im(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0 i\right)}\right))\]

  2. Final simplification43.2

    \[\leadsto \Im(\left(\frac{\left(\left(e^{\left(-\left(xre + xim i\right)\right)}\right) + \left(e^{\left(xre + xim i\right)}\right)\right)}{\left(2 + 0 i\right)}\right))\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (xre xim)
  :name "exp with complex power imaginary part (p55)"
  (im (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0))))

Details

Time bar (total: 23.5s)Debug log

sample776.0ms

Algorithm
halfpoints

simplify12.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

11.0ms
(im (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))

prune33.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 43.8b

localize184.0ms

Local error

Found 4 expressions with local error:

43.8b
(+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim))))
0.0b
(exp.c (complex xre xim))
0.0b
(exp.c (neg.c (complex xre xim)))
0.0b
(/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 2
Calls

4 calls. Slowest were:

3.0ms
(+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim))))
2.0ms
(/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0))
0.0ms
(exp.c (complex xre xim))

series0.0ms

Counts
4 → 4
Calls

4 calls. Slowest were:

0.0ms
(+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim))))
0.0ms
(exp.c (complex xre xim))
0.0ms
(/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0))
0.0ms
(exp.c (neg.c (complex xre xim)))

simplify31.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

17.0ms
(im (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))
5.0ms
(im (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))
5.0ms
(im (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))

prune49.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 43.8b

end0.0ms

sample22.4s

Algorithm
halfpoints