Average Error: 0.0 → 0.0
Time: 13.2s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0 i\right)}\right))\]
↓
\[\Re(\left(\frac{\left(\left(e^{\left(-\left(xre + xim i\right)\right)}\right) + \left(e^{\left(xre + xim i\right)}\right)\right)}{\left(2 + 0 i\right)}\right))\]

Error

Bits error versus xre

Bits error versus xim

Derivation

  1. Initial program 0.0

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

  2. Final simplification0.0

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

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (xre xim)
  :name "exp with complex power real part (p55)"
  (re (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0))))

Details

Time bar (total: 12.0s)Debug log

sample414.0ms

Algorithm
halfpoints
Results

simplify17.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
16.0ms
(re (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))

prune53.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize105.0ms

Local error

Found 3 expressions with local error:

15.8b
(+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim))))
0.0b
(exp.c (neg.c (complex xre xim)))
0.0b
(exp.c (complex xre xim))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
1×+.c-commutative
Counts
3 → 1
Calls
3 calls:
Slowest
6.0ms
(+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim))))
0.0ms
(exp.c (neg.c (complex xre xim)))
0.0ms
(exp.c (complex xre xim))

series1.0ms

Counts
3 → 3
Calls
3 calls:
Slowest
0.0ms
(+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim))))
0.0ms
(exp.c (neg.c (complex xre xim)))
0.0ms
(exp.c (complex xre xim))

simplify28.0ms

Counts
3 → 4
Calls
3 calls:
Slowest
9.0ms
(re (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))
9.0ms
(re (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))
9.0ms
(re (/.c (+.c (exp.c (complex xre xim)) (exp.c (neg.c (complex xre xim)))) (complex 2 0)))

prune58.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0.0b

end0.0ms

sample11.4s

Algorithm
halfpoints
Results