Average Error: 43.0 → 43.0
Time: 27.1s
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 43.0

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

  2. Final simplification43.0

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

Reproduce

herbie shell --seed 2018365 +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: 1.2s)Debug log

start913.0ms

Algorithm
halfpoints

setup26.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 44.2b

localize172.0ms

Local error

Found 3 expressions with local error:

44.2b
(+.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)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 1
Calls

3 calls. Slowest were:

5.0ms
(+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim))))
0.0ms
(exp.c (complex xre xim))
0.0ms
(exp.c (neg.c (complex xre xim)))

series1.0ms

Counts
3 → 3
Calls

3 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
(exp.c (neg.c (complex xre xim)))

simplify26.0ms

Counts
3 → 4
Calls

3 calls. Slowest were:

10.0ms
(im (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))
9.0ms
(im (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))
6.0ms
(im (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))

prune32.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 44.2b