Average Error: 0.0 → 0.0
Time: 11.9s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[(\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]

  2. Initial simplification0.0

    \[\leadsto (\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*\]

  3. Final simplification0.0

    \[\leadsto (\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*\]

Runtime

Time bar (total: 11.9s)Debug logProfile

start291.0ms

Algorithm
intervals

setup232.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize16.0ms

Local error

Found 2 expressions with local error:

0.0b
(/ (* (cos re) 0.5) (exp im))
0.0b
(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

5.0ms
(/ (* (cos re) 0.5) (exp im))
0.0ms
(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))

series166.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

111.0ms
(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))
55.0ms
(/ (* (cos re) 0.5) (exp im))

simplify943.0ms

Counts
18 → 37
Calls

18 calls. Slowest were:

628.0ms
(/ (* (* (* (cos re) 0.5) (* (cos re) 0.5)) (* (cos re) 0.5)) (* (* (exp im) (exp im)) (exp im)))
117.0ms
(- 0.5 (+ (* 0.25 (pow re 2)) (* 0.5 im)))
70.0ms
(- (+ (* 0.5 (pow im 2)) 1.0) (* 0.5 (pow re 2)))

prune354.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018357 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))