Average Error: 0.0 → 0.0
Time: 13.9s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{\left(\left(e^{\left(xre + xim i\right)}\right) + \left(e^{\left(-\left(xre + xim i\right)\right)}\right)\right)}{\left(2 + 0 i\right)}\right))\]
↓
\[\Re(\left(\frac{\left(\left(e^{\left(\left(-xre\right) + \left(-xim\right) i\right)}\right) + \left(e^{\left(xre + xim i\right)}\right)\right)}{\left(2 + 0 i\right)}\right))\]

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. Initial simplification0.0

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

  3. Final simplification0.0

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

Runtime

Time bar (total: 13.9s)Debug logProfile

start640.0ms

Algorithm
halfpoints

setup29.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize106.0ms

Local error

Found 3 expressions with local error:

17.2b
(+.c (exp.c (complex (- xre) (- xim))) (exp.c (complex xre xim)))
0.0b
(exp.c (complex (- xre) (- xim)))
0.0b
(exp.c (complex xre xim))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
3 → 1
Calls

3 calls. Slowest were:

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

series1.0ms

Counts
3 → 3
Calls

3 calls. Slowest were:

0.0ms
(+.c (exp.c (complex (- xre) (- xim))) (exp.c (complex xre xim)))
0.0ms
(exp.c (complex (- xre) (- xim)))
0.0ms
(exp.c (complex xre xim))

simplify31.0ms

Counts
3 → 4
Calls

3 calls. Slowest were:

11.0ms
(re (/.c (+.c (exp.c (complex (- xre) (- xim))) (exp.c (complex xre xim))) (complex 2 0)))
10.0ms
(re (/.c (+.c (exp.c (complex (- xre) (- xim))) (exp.c (complex xre xim))) (complex 2 0)))
9.0ms
(re (/.c (+.c (exp.c (complex (- xre) (- xim))) (exp.c (complex xre xim))) (complex 2 0)))

prune85.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0.0b

herbie shell --seed 2018357 +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))))