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We convert double precision float values between integer coordinates which provides 32 bits8 of precision which is more than sufficient.
While PKDGRAV3 does use mixed precision float code, the measured 10 Pflops compares quite well with the 17.59 measured LINPACK performance.
In the GPU implementation, all the involved arrays are single precision (float 32 bits) as most GPUs are relatively not efficient with 64 bits operations.
In order to encode a non-zero real number as a single precision float, the first step is to write as a normalized binary number: :\begin{align} x &= \pm 1.b_1b_2b_3\ldots \times 2^{e_x}\\ &= \pm 2^{e_x} (1 + m_x) \end{align} where the exponent is an integer,, and is the binary representation of the "significand".
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Three plaintext implementations are considered: (1) 16-bit fixed point (XP); (2) single precision floating point (FP1); (3) double precision floating point (FP2).
and are equivalent to a single precision and a double precision floating point implementations, respectively.
Data types can be: signed/unsigned 8, 16, 32, 64-bits or single precision floating point.
The MicroBlaze is also configured to use single precision floating-point unit for the software modules.
The alternative transforms used in the experiments have been implemented using double precision floating point.
Using a pair of floating point numbers double precision accuracy is approximated through single precision floating point operations.
A single precision floating point multiplier is designed with these two VM structures.
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