Average Error: 0.0 → 0.0
Time: 36.9s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right)\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Final simplification0.0

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

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

Details

Time bar (total: 24.4s)Debug log

start288.0ms

Algorithm
intervals

setup220.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 1 expressions with local error:

0.0b
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
1 → 22
Calls

1 calls. Slowest were:

14.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

series94.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

94.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

simplify1.6s

Counts
14 → 25
Calls

14 calls. Slowest were:

505.0ms
(* 0.5 (* (cos re) (+ (exp (- im)) (exp im))))
263.0ms
(+ (* (exp (- im)) (exp (- im))) (- (* (exp im) (exp im)) (* (exp (- im)) (exp im))))
259.0ms
(* 0.5 (* (+ (exp im) (exp (* -1 im))) (cos re)))

prune314.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize19.0ms

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)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

11.0ms
(/ (* (cos re) 0.5) (exp im))
0.0ms
(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))

series159.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

99.0ms
(fma (exp im) (* (cos re) 0.5) (/ (* (cos re) 0.5) (exp im)))
60.0ms
(/ (* (cos re) 0.5) (exp im))

simplify892.0ms

Counts
18 → 37
Calls

18 calls. Slowest were:

537.0ms
(/ (* (* (* (cos re) 0.5) (* (cos re) 0.5)) (* (cos re) 0.5)) (* (* (exp im) (exp im)) (exp im)))
112.0ms
(- 0.5 (+ (* 0.25 (pow re 2)) (* 0.5 im)))
99.0ms
(- (+ (* 0.5 (pow im 2)) 1.0) (* 0.5 (pow re 2)))

prune523.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize24.0ms

Local error

Found 4 expressions with local error:

0.8b
(cbrt (* (* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im))) (/ (* (cos re) 0.5) (exp im))))
0.3b
(* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im)))
0.2b
(* (* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im))) (/ (* (cos re) 0.5) (exp im)))
0.0b
(/ (* (cos re) 0.5) (exp im))

rewrite128.0ms

Algorithm
rewrite-expression-head
Counts
4 → 133
Calls

4 calls. Slowest were:

52.0ms
(* (* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im))) (/ (* (cos re) 0.5) (exp im)))
39.0ms
(cbrt (* (* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im))) (/ (* (cos re) 0.5) (exp im))))
24.0ms
(* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im)))

series299.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

102.0ms
(cbrt (* (* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im))) (/ (* (cos re) 0.5) (exp im))))
75.0ms
(* (* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im))) (/ (* (cos re) 0.5) (exp im)))
63.0ms
(/ (* (cos re) 0.5) (exp im))
59.0ms
(* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im)))

simplify17.5s

Counts
112 → 145
Calls

112 calls. Slowest were:

985.0ms
(/ (* (* (* (cos re) 0.5) (* (cos re) 0.5)) (* (cos re) 0.5)) (* (* (exp im) (exp im)) (exp im)))
768.0ms
(* (* (* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im))) (/ (* (cos re) 0.5) (exp im))) (* (* (/ (* (cos re) 0.5) (exp im)) (/ (* (cos re) 0.5) (exp im))) (/ (* (cos re) 0.5) (exp im))))
750.0ms
(+ (- (log (* (cos re) 0.5)) im) (log (/ (* (cos re) 0.5) (exp im))))

prune2.1s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes216.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch2.0ms