Average Error: 0.0 → 0.0

Time: 10.2s

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

The X axis uses an exponential scale — Bits error versus `re`

Derivation

Initial program 0.0
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
Initial simplification0.0
\[\leadsto (\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*\]
Final simplification0.0
\[\leadsto (\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

Details

start252.0ms

Algorithm: intervals

setup554.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize31.0ms

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)))

rewrite11.0ms

Algorithm

rewrite-expression-head

Counts

2 → 31

Calls

2 calls. Slowest were:

10.0ms	(/ (* (cos re) 0.5) (exp im))
0.0ms	(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))

series149.0ms

Counts

2 → 6

Calls

2 calls. Slowest were:

93.0ms	(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))
56.0ms	(/ (* (cos re) 0.5) (exp im))

simplify938.0ms

Counts

18 → 37

Calls

18 calls. Slowest were:

605.0ms	(/ (* (* (* (cos re) 0.5) (* (cos re) 0.5)) (* (cos re) 0.5)) (* (* (exp im) (exp im)) (exp im)))
107.0ms	(- (+ (* 0.5 (pow im 2)) 1.0) (* 0.5 (pow re 2)))
88.0ms	(- 0.5 (+ (* 0.25 (pow re 2)) (* 0.5 im)))

prune396.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b