Average Error: 0.0 → 0.0
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\cos re \cdot \left(\frac{0.5}{e^{im}} + 0.5 \cdot e^{im}\right)\]

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. Using strategy rm

  3. Applied distribute-lft-in0.0

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

  4. Taylor expanded around -inf 0.0

    \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} \cdot \cos re\right) + 0.5 \cdot \left(e^{-1 \cdot im} \cdot \cos re\right)}\]

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

Details

Time bar (total: 1.2m)Debug log

sample255.0ms

Algorithm
intervals

simplify361.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
361.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 2 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
7×*-un-lft-identity
4×add-log-exp
4×add-exp-log
4×associate-*r*
4×add-cbrt-cube
4×pow1
3×add-cube-cbrt
3×add-sqr-sqrt
2×distribute-lft-out
2×flip-+
2×associate-*r/
2×flip3-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×distribute-lft-in
1×sum-log
1×+-commutative
1×distribute-rgt-in
Counts
2 → 32
Calls
2 calls:
Slowest
13.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))
6.0ms
(+ (exp (- im)) (exp im))

series115.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
82.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))
34.0ms
(+ (exp (- im)) (exp im))

simplify1.2s

Counts
19 → 38
Calls
19 calls:
Slowest
359.0ms
(* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))
324.0ms
(* 0.5 (* (cos re) (+ (exp (- im)) (exp im))))
150.0ms
(+ (* (exp (- im)) (exp (- im))) (- (* (exp im) (exp im)) (* (exp (- im)) (exp im))))
137.0ms
(* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (+ (exp (- im)) (exp im)) (+ (exp (- im)) (exp im))) (+ (exp (- im)) (exp im))))
127.0ms
(* 0.5 (* (+ (exp im) (exp (* -1 im))) (cos re)))

prune784.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 3 expressions with local error:

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

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cbrt-cube
7×pow1
6×associate-*r*
5×add-log-exp
5×add-cube-cbrt
5×add-exp-log
5×*-un-lft-identity
5×add-sqr-sqrt
2×cbrt-unprod
2×*-commutative
2×associate-*r/
2×prod-exp
2×pow-prod-down
2×associate-*l*
2×exp-neg
1×distribute-lft-out
1×flip-+
1×un-div-inv
1×exp-diff
1×flip3-+
1×sum-log
1×+-commutative
1×neg-sub0
Counts
3 → 45
Calls
3 calls:
Slowest
19.0ms
(+ (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp im)))
13.0ms
(* (* 0.5 (cos re)) (exp (- im)))
10.0ms
(* (* 0.5 (cos re)) (exp im))

series225.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
132.0ms
(+ (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp im)))
54.0ms
(* (* 0.5 (cos re)) (exp (- im)))
40.0ms
(* (* 0.5 (cos re)) (exp im))

simplify1.8s

Counts
27 → 54
Calls
27 calls:
Slowest
305.0ms
(* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (exp (- im)) (exp (- im))) (exp (- im))))
276.0ms
(* (exp (* (* 0.5 (cos re)) (exp (- im)))) (exp (* (* 0.5 (cos re)) (exp im))))
270.0ms
(+ (* 0.5 (* (exp im) (cos re))) (* 0.5 (* (exp (* -1 im)) (cos re))))
237.0ms
(* (* (* (* 0.5 (cos re)) (* 0.5 (cos re))) (* 0.5 (cos re))) (* (* (exp im) (exp im)) (exp im)))
200.0ms
(+ (* 0.5 (* (exp (- im)) (cos re))) (* 0.5 (* (exp im) (cos re))))

prune582.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize12.0ms

Local error

Found 3 expressions with local error:

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

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
5×add-log-exp
5×add-exp-log
5×add-cbrt-cube
5×pow1
4×associate-*l*
3×div-inv
3×associate-*r*
3×associate-/r*
2×distribute-lft-out
2×flip-+
2×associate-*l/
2×flip3-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
Counts
3 → 45
Calls
3 calls:
Slowest
12.0ms
(* (+ (/ 0.5 (exp im)) (* 0.5 (exp im))) (cos re))
5.0ms
(+ (/ 0.5 (exp im)) (* 0.5 (exp im)))
0.0ms
(/ 0.5 (exp im))

series121.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
70.0ms
(* (+ (/ 0.5 (exp im)) (* 0.5 (exp im))) (cos re))
41.0ms
(+ (/ 0.5 (exp im)) (* 0.5 (exp im)))
10.0ms
(/ 0.5 (exp im))

simplify1.1s

Counts
24 → 54
Calls
24 calls:
Slowest
391.0ms
(* (+ (pow (/ 0.5 (exp im)) 3) (pow (* 0.5 (exp im)) 3)) (cos re))
159.0ms
(* (- (* (/ 0.5 (exp im)) (/ 0.5 (exp im))) (* (* 0.5 (exp im)) (* 0.5 (exp im)))) (cos re))
115.0ms
(* (* (* (+ (/ 0.5 (exp im)) (* 0.5 (exp im))) (+ (/ 0.5 (exp im)) (* 0.5 (exp im)))) (+ (/ 0.5 (exp im)) (* 0.5 (exp im)))) (* (* (cos re) (cos re)) (cos re)))
102.0ms
(* (+ (* 0.5 (exp im)) (* 0.5 (/ 1 (exp im)))) (cos re))
84.0ms
(- (+ (* 0.25 (pow im 2)) 0.5) (* 0.5 im))

prune465.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize24.0ms

Local error

Found 4 expressions with local error:

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

rewrite277.0ms

Algorithm
rewrite-expression-head
Rules
241×associate-*r/
208×exp-neg
171×frac-times
138×pow1
104×un-div-inv
104×exp-diff
104×neg-sub0
66×pow-prod-down
63×cbrt-div
57×associate-*l/
57×add-cbrt-cube
40×add-exp-log
39×cbrt-unprod
39×prod-exp
30×pow-prod-up
15×pow-plus
6×add-log-exp
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
4×pow2
2×*-commutative
2×pow3
2×associate-*l*
2×associate-*r*
1×flip-+
1×rem-cbrt-cube
1×pow1/3
1×flip3-+
1×sum-log
1×+-commutative
1×cbrt-prod
Counts
4 → 253
Calls
4 calls:
Slowest
103.0ms
(* (* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp (- im)))) (* (* 0.5 (cos re)) (exp (- im))))
87.0ms
(cbrt (* (* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp (- im)))) (* (* 0.5 (cos re)) (exp (- im)))))
47.0ms
(+ (cbrt (* (* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp (- im)))) (* (* 0.5 (cos re)) (exp (- im))))) (* (* 0.5 (cos re)) (exp im)))
32.0ms
(* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp (- im))))

series367.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
158.0ms
(+ (cbrt (* (* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp (- im)))) (* (* 0.5 (cos re)) (exp (- im))))) (* (* 0.5 (cos re)) (exp im)))
79.0ms
(cbrt (* (* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp (- im)))) (* (* 0.5 (cos re)) (exp (- im)))))
72.0ms
(* (* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp (- im)))) (* (* 0.5 (cos re)) (exp (- im))))
58.0ms
(* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp (- im))))

simplify52.6s

Counts
302 → 265
Calls
302 calls:
Slowest
683.0ms
(* (* (* 0.5 (cos re)) (* (* 0.5 (cos re)) (exp (- im)))) (* (* 0.5 (cos re)) (exp (- im))))
673.0ms
(* (* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp (- im)))) (* (* 0.5 (cos re)) (exp (- im))))
643.0ms
(cbrt (* (* (* (* 0.5 (cos re)) (exp 0)) (* (* 0.5 (cos re)) (exp (- im)))) (* 0.5 (cos re))))
588.0ms
(* (* (* (* 0.5 (cos re)) (exp 0)) (* (* 0.5 (cos re)) (exp (- im)))) (* (* 0.5 (cos re)) (exp (- im))))
578.0ms
(* (* (* (* 0.5 (cos re)) (exp (- im))) (* (* 0.5 (cos re)) (exp 0))) (* (* 0.5 (cos re)) 1))

prune3.5s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes156.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample8.1s

Algorithm
intervals