Average Error: 0.0 → 0.0
Time: 18.0s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot e^{im} + \frac{0.5 \cdot \sin re}{e^{im}}\]

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

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
  2. Using strategy rm

  3. Applied distribute-rgt-in0.0

    \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)}\]

  4. Taylor expanded around -inf 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019021 
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))

Details

Time bar (total: 17.7s)Debug log

sample322.0ms

Algorithm
intervals
Results
112.0ms90×body1280valid
68.0ms85×body640valid
45.0ms267×body80valid
25.0ms46×body320valid
8.0ms17×body160valid

simplify264.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
264.0ms
(* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im)))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize41.0ms

Local error

Found 2 expressions with local error:

0.0b
(* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im)))
0.0b
(+ (exp (- 0 im)) (exp im))

rewrite39.0ms

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
26.0ms
(* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im)))
11.0ms
(+ (exp (- 0 im)) (exp im))

series128.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
98.0ms
(* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im)))
30.0ms
(+ (exp (- 0 im)) (exp im))

simplify1.7s

Counts
19 → 38
Calls
19 calls:
Slowest
358.0ms
(* 0.5 (* (sin re) (+ (exp im) (exp (* -1 im)))))
345.0ms
(* 0.5 (* (sin re) (+ (exp (- im)) (exp im))))
293.0ms
(* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im)))
206.0ms
(+ (* (exp (- 0 im)) (exp (- 0 im))) (- (* (exp im) (exp im)) (* (exp (- 0 im)) (exp im))))
176.0ms
(- (+ (* 1.0 re) (* 0.5 (* re (pow im 2)))) (* 0.16666666666666666 (pow re 3)))

prune372.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 3 expressions with local error:

0.0b
(* (exp (- 0 im)) (* 0.5 (sin re)))
0.0b
(+ (* (exp (- 0 im)) (* 0.5 (sin re))) (* (exp im) (* 0.5 (sin re))))
0.0b
(* (exp im) (* 0.5 (sin re)))

rewrite47.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-*l*
7×add-cbrt-cube
7×pow1
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×prod-exp
2×associate-*l/
2×pow-prod-down
2×exp-sum
2×associate-*r*
1×sub0-neg
1×flip-+
1×exp-diff
1×distribute-rgt-out
1×flip3-+
1×sub-neg
1×sum-log
1×+-commutative
1×exp-neg
Counts
3 → 46
Calls
3 calls:
Slowest
21.0ms
(+ (* (exp (- 0 im)) (* 0.5 (sin re))) (* (exp im) (* 0.5 (sin re))))
15.0ms
(* (exp (- 0 im)) (* 0.5 (sin re)))
10.0ms
(* (exp im) (* 0.5 (sin re)))

series267.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
138.0ms
(+ (* (exp (- 0 im)) (* 0.5 (sin re))) (* (exp im) (* 0.5 (sin re))))
70.0ms
(* (exp im) (* 0.5 (sin re)))
58.0ms
(* (exp (- 0 im)) (* 0.5 (sin re)))

simplify2.6s

Counts
28 → 55
Calls
28 calls:
Slowest
435.0ms
(* (exp (* (exp (- 0 im)) (* 0.5 (sin re)))) (exp (* (exp im) (* 0.5 (sin re)))))
307.0ms
(* (* (* (exp (- 0 im)) (exp (- 0 im))) (exp (- 0 im))) (* (* (* 0.5 (sin re)) (* 0.5 (sin re))) (* 0.5 (sin re))))
284.0ms
(+ (* 0.5 (* (sin re) (exp (* -1 im)))) (* 0.5 (* (sin re) (exp im))))
246.0ms
(* (* (* (exp im) (exp im)) (exp im)) (* (* (* 0.5 (sin re)) (* 0.5 (sin re))) (* 0.5 (sin re))))
234.0ms
(- (+ (* 0.5 re) (* 0.25 (* re (pow im 2)))) (* 0.5 (* re im)))

prune558.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize6.0ms

Local error

Found 3 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cbrt-cube
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
5×add-log-exp
5×add-exp-log
5×pow1
3×associate-*l*
3×times-frac
3×associate-/r*
1×associate-/l*
1×div-inv
1×flip-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*r*
1×div-exp
1×flip3-+
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
3 → 45
Calls
3 calls:
Slowest
10.0ms
(+ (/ (* 0.5 (sin re)) (exp im)) (* (exp im) (* 0.5 (sin re))))
5.0ms
(* (exp im) (* 0.5 (sin re)))
4.0ms
(/ (* 0.5 (sin re)) (exp im))

series214.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
105.0ms
(+ (/ (* 0.5 (sin re)) (exp im)) (* (exp im) (* 0.5 (sin re))))
57.0ms
(* (exp im) (* 0.5 (sin re)))
52.0ms
(/ (* 0.5 (sin re)) (exp im))

simplify1.6s

Counts
27 → 54
Calls
27 calls:
Slowest
540.0ms
(/ (* (* (* 0.5 (sin re)) (* 0.5 (sin re))) (* 0.5 (sin re))) (* (* (exp im) (exp im)) (exp im)))
245.0ms
(+ (* 0.5 (* re im)) (+ (* 0.5 re) (* 0.25 (* re (pow im 2)))))
242.0ms
(- (* 0.5 re) (+ (* 0.5 (* re im)) (* 0.08333333333333333 (pow re 3))))
192.0ms
(* (* (* (exp im) (exp im)) (exp im)) (* (* (* 0.5 (sin re)) (* 0.5 (sin re))) (* 0.5 (sin re))))
118.0ms
(- (+ (* 1.0 re) (* 0.5 (* re (pow im 2)))) (* 0.16666666666666666 (pow re 3)))

prune497.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes250.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample8.8s

Algorithm
intervals
Results
3.2s2895×body1280valid
2.6s2416×body640valid
1.4s8497×body80valid
616.0ms1253×body320valid
361.0ms661×body160valid