# 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:
```
$ ./translate_mu_debug x.mu # debug mode = error checking
$ ./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):
```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:
```asm
; 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:
```diff
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 single-precision floating-point
instructions got upgraded to double-precision — but not these two. So
they seem to be an evolutionary dead-end.