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How approximate is Intel's floating-point reciprocal instruction?

2020/10/03

Here's a test Mu program that prints out the bits for 0.5:

fn main -> r/ebx: int {
  var two/eax: int <- copy 2
  var half/xmm0: float <- convert two
  half <- reciprocal half
  var mem: float
  copy-to mem, half
  var out/eax: int <- reinterpret mem
  print-int32-hex 0, out
  print-string 0, "\n"
  r <- copy 0
}

It gives different results when emulated and run natively:

$ cd linux
$ ./translate_debug x.mu  # debug mode = error checking
$ bootstrap/bootstrap run a.elf
0x3f000000  # correct
$ ./a.elf
0x3efff000  # wrong

I spent some time digging into this before I realized it wasn't a bug in Mu, just an artifact of the emulator not actually using the reciprocal instruction. Here's a procedure you can follow along with to convince yourself.

Start with this program (good.c):

#include<stdio.h>
int main(void) {
  int n = 2;
  float f = 1.0/n;
  printf("%f\n", f);
  return 0;
}

It works as you'd expect (compiling unoptimized to actually compute the division):

$ gcc good.c
$ ./a.out
0.5

Let's look at its Assembly:

$ gcc -S good.c

The generated good.s has a lot of stuff that doesn't interest us, surrounding these lines:

                      ; destination
movl      $2,         -8(%rbp)
cvtsi2sd  -8(%rbp),   %xmm0
movsd     .LC0(%rip), %xmm1
divsd     %xmm0,      %xmm1
movapd    %xmm1,      %xmm0

This fragment converts 2 into floating-point and then divides 1.0 (the constant .LC0) by it, leaving the result in register xmm0.

There's a way to get gcc to emit the rcpss instruction using intrinsics, but I don't know how to do it, so I'll modify the generated Assembly directly:

      movl      $2,         -8(%rbp)
<     cvtsi2sd  -8(%rbp),   %xmm0
<     movsd     .LC0(%rip), %xmm1
<     divsd     %xmm0,      %xmm1
<     movapd    %xmm1,      %xmm0
---
>     cvtsi2ss  -8(%rbp),   %xmm0
>     rcpss     %xmm0,      %xmm0
>     movss     %xmm0,      -4(%rbp)

Let's compare the result of both versions:

$ gcc good.s
$ ./a.out
0.5
$ gcc good.modified.s
$ ./a.out
0.499878

Whoa!

Reading the Intel manual more closely, it guarantees that the relative error of rcpss is less than 1.5*2^-12, and indeed 12 bits puts us squarely in the fourth decimal place.

Among the x86 instructions Mu supports, two are described in the Intel manual as "approximate": reciprocal (rcpss) and inverse-square-root (rsqrtss). Intel introduced these instructions as part of its SSE expansion in 1999. When it upgraded SSE to SSE2 (in 2000), most of its scalar[1] single-precision floating-point instructions got upgraded to double-precision — but not these two. So they seem to be an evolutionary dead-end.

[1] Thanks boulos for feedback: https://news.ycombinator.com/item?id=28501429#28507118