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On Jun 8, 2018, at 5:47 AM, KHMan <keinhong@gmail.com> wrote: Operation / Op_Count / Diff_Count ================================= A+B 1,000,000,000 0 A-B 1,000,000,000 0 A*B 1,000,000,000 243,584 A/B 1,000,000,000 244,107 For A*B and A/B, this is roughly 1 in 4000. All different results differ by 1 ULP. Nothing more than 1 ULP was seen. I found another blog doing the same test, error also about 1:4000 Just some math behind the observation: Double rounding occurs when both rounded in the same direction. Difference between 53-bits and 64-bits are 11-bits. --> chances of same direction rounding is about 50% --> chances of double rounding = 50% / 2^11 = 1 / 4096 With just 1 op, of course the error is at most 1 ULP Operation : * Here are the hex representations of the significands: fpu8087 = 7e65b9c2a810e binary128 = 7E65B9C2A810E5EAB7A33195161D fpu64bit = 7e65b9c2a810f Using this comparison, it appears that the binary128 result is nearer to fpu8087 than to fpu64bit. We can also verify this by getting a more accurate decimal version of the 64-bit binary representation (done by manual fiddling, so not definitive): You do realize we have to work in 53-bits for comparison. Getting more precise result with more bits does not show anything. My guess is you set (n1, n2) as decimal string -> __float128. Had you use the actual hexfloats, you should get this: n1 * n2 = 0x1.84daa65360288p-3 * 0x1.f77f9cd91528bp-1 = 0x1.7e65b9c2a810e 800eb4c1577ec p-3 --> value should round-up, to 0x1.7e65b9c2a810fp-3 --> 53-bits rounding is doing the best it can with 2 doubles. |