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Floating Point Numbers  Page 7   Wide Floating Point   Multiplication   Multiplication of WideFloat numbers consists of adding the exponents and multiplying the significands.   Multiplication of significands is like WideInteger multiplication, but with carrying in the opposite direction. Carrying will be from high multiplication to low multiplication.   Multiplication with Truncation   Multiplication of two WideFloat numbers, A and B, each having 6 digit packets, with truncation of the product is accomplished with this polynomial:

AB  =   ( A5B1 + A4B2 + A3B3 + A2B4 + A1B5 )t–6 + ( A5B0 + A4B1 + A3B2 + A2B3 + A1B4 + A0B5 )t–5 + ( A4B0 + A3B1 + A2B2 + A1B3 + A0B4 )t–4 + ( A3B0 + A2B1 + A1B2 + A0B3 )t–3 + ( A2B0 + A1B1 + A0B2 )t–2 + ( A1B0 + A0B1 )t–1 + ( A0B0 )t0   t = 230  An are digit packets of A Bn are digit packets of B

The t–6 coefficient is not used other than to create a carry into the t–5 coefficient. That is the carry from high multiplication. No other high multiplication operations are needed. The factors of the other terms of high multiplication do not need to be compared to zero.   The t–5 coefficient creates a carry into the t–4 coefficient, the t–4 coefficient creates a carry into the t–3 coefficient, etc.   If the t0 coefficient creates a carry, then that carry becomes a new digit packet, and the least significant digit packet is discarded, and the exponent is incremented.   The coefficients are two digit packets wide, and can be split into digit packets before or after carries are propagated.

 

Multiplication with Rounding   The same multiplication, to support rounding instead of truncation, is accomplished with this polynomial:

AB  =   ( A5B5 )t–10 + ( A5B4 + A4B5 )t–9 + ( A5B3 + A4B4 + A3B5 )t–8 + ( A5B2 + A4B3 + A3B4 + A2B5 )t–7 + ( A5B1 + A4B2 + A3B3 + A2B4 + A1B5 )t–6 + ( A5B0 + A4B1 + A3B2 + A2B3 + A1B4 + A0B5 )t–5 + ( A4B0 + A3B1 + A2B2 + A1B3 + A0B4 )t–4 + ( A3B0 + A2B1 + A1B2 + A0B3 )t–3 + ( A2B0 + A1B1 + A0B2 )t–2 + ( A1B0 + A0B1 )t–1 + ( A0B0 )t0

Carrying is from t–10 to t–9, from t–9 to t–8, etc.   You can do two multiplications at once with PMULUDQ. Unsigned additions of 64-bit numbers can be performed with PADDQ. The following table may be useful for organizing simd operations:

t0 t–1 t–2 t–3 t–4 t–5 t–6 t–7 t–8 t–9 t–10                A0B0             A1B0 A0B1          A2B0 A1B1 A0B2       A3B0 A2B1 A1B2 A0B3    A4B0 A3B1 A2B2 A1B3 A0B4 A5B0 A4B1 A3B2 A2B3 A1B4 A0B5 A5B1 A4B2 A3B3 A2B4 A1B5 A5B2 A4B3 A3B4 A2B5 A5B3 A4B4 A3B5 A5B4 A4B5 A5B5

If a new digit is created (if t0 has a carry-out), then increment the exponent and shift the digit packets one position so that the new digit is the only digit to the left of the radix point.

 

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