A refined affine approximation method of multiplication for range analysis in word-length optimization

EURASIP Journal on Advances in Signal Processing

Table 2 Comparison of analytical ranges and bits by the three methods

c.fun	True output		AATRE		AACHA		AASEE
	Range	Bits	Range	Bits	Range	Bits	Range	Bits
c₁.Y	[0,0.6931]	1	[-0.0541,0.8650]	1	[-0.0253,0.7685]	1	[-0.0068,0.7068]	1
c₂.B₀	[0,0.17]	1	[-0.13,0.17]	1	[-0.05,0.17]	1	[-0.02,0.17]	1
c₂.B₁	[0.17,0.67]	1	[-0.33,1.29]	2	[-0.05,0.98]	1	[0.10,0.92]	1
c₂.B₂	[0.17,0.67]	1	[-0.21,1.17]	2	[-0.02,0.89]	1	[0.04,0.73]	1
c₂.B₃	[-0.17,0]	1	[-0.17,0.13]	1	[-0.17,0.05]	1	[-0.17,0.02]	1
c₃.f₁	[-9453,9303]	15	[-9821,9671]	15	[-9793,9487]	15	[-9793,9487]	15
c₃.f₂	[-5.51e 4,8.79e 4]	18	[-1.75e 5,1.79e 5]	19	[-0.95e 5,1.28e 5]	18	[-1.15e 5,1.41e 5]	19
c₃.f₃	[-36,64]	8	[-256,256]	10	[-192,192]	9	[-64,64]	8
c₃.f₄	[-8,641]	11	[-1087,1121]	12	[-223,881]	11	[-335,641]	11
c₃.f₅	[0,10⁴]	15	[-10⁴,10⁴]	15	[-4800,10⁴]	15	[-4800,10⁴]	15
c₃.f₆	[0,0.94e 6]	21	[-1.07e 6,1.07e 6]	22	[-0.06e 6,1.00e 6]	21	[-0.11e 6,0.94e 6]	21
c₃.f₇	[3,1.01e 6]	21	[-1.42e 8,1.42e 8]	29	[-1.23e 8,1.13e 8]	28	[-9.87e 7,9.61e 7]	28
c₃.f₈	[-1.03,3.3e 11]	40	[-3.3e 11,3.3e 11]	40	[-2.1e 8,3.3e 11]	40	[-4.2e 10,3.3e 11]	40