Average Error: 0 → 0
Time: 15.9s
Precision: 64
Internal Precision: 1088
\[\Im(\left(\left(\left(\left(\left(\left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) + \left(\left(\left(\left(\left(-2\right) + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(\left(5 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(4 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(7 + 0 i\right)\right))\]
↓
\[\Im(\left(\left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) + \left(6 + \frac{\sqrt{3} \cdot -2}{2} i\right)\right)\right) + \left(5 + \frac{\sqrt{3} \cdot 4}{2} i\right)\right))\]

Error

Derivation

  1. Initial program 0

    \[\Im(\left(\left(\left(\left(\left(\left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) + \left(\left(\left(\left(\left(-2\right) + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(\left(5 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(4 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(7 + 0 i\right)\right))\]

  2. Simplified0

    \[\leadsto \color{blue}{\Im(\left(\left(5 + \frac{4 \cdot \sqrt{3}}{2} i\right) + \left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\left(6 + \frac{\sqrt{3} \cdot -2}{2} i\right) + \left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right)\right)\right))}\]

  3. Final simplification0

    \[\leadsto \Im(\left(\left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) + \left(6 + \frac{\sqrt{3} \cdot -2}{2} i\right)\right)\right) + \left(5 + \frac{\sqrt{3} \cdot 4}{2} i\right)\right))\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore ()
  :name "3.9.2 imaginary part (p56)"
  (im (+.c (+.c (+.c (+.c (*.c (*.c (*.c (complex (/ (- 1) 2) (/ (sqrt 3) 2)) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (*.c (*.c (*.c (complex (- 2) 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (*.c (*.c (complex 5 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (*.c (complex 4 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (complex 7 0))))

Details

Time bar (total: 14.0s)Debug log

sample818.0ms

Algorithm
halfpoints
Results

simplify682.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
682.0ms
(im (+.c (+.c (+.c (+.c (*.c (*.c (*.c (complex (/ (- 1) 2) (/ (sqrt 3) 2)) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (*.c (*.c (*.c (complex (- 2) 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (*.c (*.c (complex 5 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (*.c (complex 4 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (complex 7 0)))

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize83.0ms

Local error

Found 2 expressions with local error:

1.6b
(*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2)))
1.6b
(*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2)))

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
2×complex-mul-def
2×*.c-commutative
Counts
2 → 4
Calls
2 calls:
Slowest
1.0ms
(*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2)))
1.0ms
(*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2)))

series0.0ms

Counts
2 → 2
Calls
2 calls:
Slowest
0.0ms
(*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2)))
0.0ms
(*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2)))

simplify662.0ms

Counts
6 → 6
Calls
6 calls:
Slowest
217.0ms
(im (+.c (complex 5 (/ (* 4 (sqrt 3)) 2)) (*.c (*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2))) (+.c (complex 6 (/ (* (sqrt 3) -2) 2)) (*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2)))))))
194.0ms
(im (+.c (complex 5 (/ (* 4 (sqrt 3)) 2)) (*.c (*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2))) (+.c (complex 6 (/ (* (sqrt 3) -2) 2)) (*.c (complex -1/2 (/ (sqrt 3) 2)) (complex -1/2 (/ (sqrt 3) 2)))))))
90.0ms
(- (* -1/2 -1/2) (* (/ (sqrt 3) 2) (/ (sqrt 3) 2)))
74.0ms
(- (* -1/2 -1/2) (* (/ (sqrt 3) 2) (/ (sqrt 3) 2)))
47.0ms
(+ (* -1/2 (/ (sqrt 3) 2)) (* (/ (sqrt 3) 2) -1/2))

prune69.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample11.7s

Algorithm
halfpoints
Results