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I'm trying to figure out just how expensive floating point fundamentally is, at the hardware level. For example, how many more transistors does a 32-bit multiplier cost in floating point compared to integer.

To be specific:

  • A 32-bit floating point multiplier, versus a 32-bit integer multiplier.
  • Both have a throughput of one clock cycle.
  • The FP does not need IEEE semantics; it can make the simplifications typical of GPUs e.g. no exceptions, rounding mode is not configurable, denormals flush to zero.
  • The integer multiplier only produces 32 bits of result and throws away the rest.
  • If it matters, say the target clock speed is 50 MHz and the implementation technology is CMOS.
  • I'm just considering the arithmetic hardware itself, not other issues like control logic, register renaming etc.

Roughly how much more expensive is the floating point circuit? For example, twice as many transistors?

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  • \$\begingroup\$ there are various tradeoffs to make in an actual implementation. you could fire up your favorite hdl package and synthesize two units with your choices for those tradeoffs and compare. \$\endgroup\$
    – PlasmaHH
    Commented Apr 16, 2018 at 11:50
  • \$\begingroup\$ @PlasmaHH What sort of tradeoffs exist, aside from the ones I mentioned? \$\endgroup\$
    – rwallace
    Commented Apr 16, 2018 at 11:53
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    \$\begingroup\$ real implementations have lots of complex things like register renaming, ways to access the data, superscalar execution. Then there is speed, you say you want one clock cycle, but its easier on 5MHz than on 5GHz. On the former you can go some lazy routes and have internally basically loops running, but on 5GHz you would have a really fine tuned design. And then there are probably fine prints like carry bits or how the 64bit result in the integer case is stored and whatnot that may or may not influence whether you get it into the time constrain of a cycle of the desired clock or not \$\endgroup\$
    – PlasmaHH
    Commented Apr 16, 2018 at 11:58
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    \$\begingroup\$ Have you ever seen a block diagram of a floating-point multiplier? It has an integer multiplier at its core, to handle the multiplication of the mantissas. It also has shifters before and after to handle denormalization and renormalization. Then there's the separate path that deals with the exponents. Some of these blocks are \$O(n^2)\$ and some of them are \$O(n)\$, but they all have different scale factors for the number of transistors. That's a lot of variables, and I don't think it's really possible to distill it down to a single generic factor. \$\endgroup\$
    – Dave Tweed
    Commented Apr 16, 2018 at 12:31
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    \$\begingroup\$ You say, "throughput of one clock cycle", which implies that you want a result every cycle. This is not difficult at all. The key difference is latency -- integer multiplication can take just one cycle, but floating-point will typically require 3-4 cycles (pipelined) because of the pre- and post-processing required on the numbers. \$\endgroup\$
    – Dave Tweed
    Commented Apr 16, 2018 at 12:54

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I'm writing this with no experience what so ever, so take this answer with a grain of salt. That said...

A typical 32-bit float has 23 bits of fraction. Multiplying two of these only requires a 23x23 multiplier, keeping the upper 23 bits. The exponents are then added. Adds are cheap.

Your 32-bit integer has 32 bits, so you need a 32x32 bit multiplier, keeping the upper or lower 32 bits.

So: Your floating point multiplier ought to be cheaper.

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  • \$\begingroup\$ That is a consideration, certainly. On the other hand, floating point operands need to be shifted to bring them into alignment, and a barrel shifter has, like a multiplier, quadratic cost, so that's a counterweight to start with. \$\endgroup\$
    – rwallace
    Commented Apr 16, 2018 at 12:29
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    \$\begingroup\$ @pipe, Actually, they are 24-bit mantissas. The 24th bit is not stored explicitly, because its value is implied by the exponent -- 1 for normalized numbers, and 0 for denormalized numbers. \$\endgroup\$
    – Dave Tweed
    Commented Apr 16, 2018 at 12:36
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    \$\begingroup\$ @rwallace: You're thinking of floating-point addition/subtraction. Multiplication does not require that kind of alignment. \$\endgroup\$
    – Dave Tweed
    Commented Apr 16, 2018 at 12:38
  • \$\begingroup\$ @DaveTweed Good point. I stand corrected. \$\endgroup\$
    – rwallace
    Commented Apr 16, 2018 at 12:40
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    \$\begingroup\$ Floating point multiplication should be cheaper, add and subtract require barrel shifters to get single cycle operation. \$\endgroup\$
    – Spehro 'speff' Pefhany
    Commented Apr 16, 2018 at 15:12

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