Subtraction Is Functionally Complete
To be precise, IEEE-754 floating point subtraction is functionally complete. That means you can construct any binary circuit using nothing but floating point subtraction.
To see how, we must start at the bottom. I quote the IEEE 754-2019 standard, section 6.3:
6.3 The sign bit
[…] When neither the inputs nor result are NaN, […]; the sign of a sum, or of a difference $x−y$ regarded as a sum $x+(−y)$, differs from at most one of the addends’ signs; […]. These rules shall apply even when operands or results are zero or infinite.
When the sum of two operands with opposite signs (or the difference of two operands with like signs) is exactly zero, the sign of that sum (or difference) shall be $+0$ under all rounding-direction attributes except
roundTowardNegative; under that attribute, the sign of an exact zero sum (or difference) shall be $−0$.
Let’s dissect that.
- A subtraction $x - y$ is considered a sum $x + (-y)$.
- Zero can have a sign, $-0$ and $0$ are distinct entities (although they compare
equal when testing with
==). - If both of the addends have the same sign, the output must have that sign. However, for $x - y$ that means if $x$ and $y$ have different signs the output must have the sign of $x$.
- If $x$ and $y$ have the same sign, and $x - y$ is zero, the output will be
$+0$ for all rounding modes except
roundTowardNegative, then it will be $-0$.
Now since the default rounding mode in virtually every context is roundTiesToEven,
we shall assume that from now on. However, everything works analogously even for
roundTowardNegative.
A truth table
So, what does that give us when subtracting zeroes?
-0 - -0 = +0 # Same sign, must be +0.
-0 - +0 = -0 # Different signs, sign from first argument.
+0 - -0 = +0 # Different signs, sign from first argument.
+0 - +0 = +0 # Same sign, must be +0.
Interesting… What if we say that $-0$ is false and $+0$ is true?
A B | O
----+--
0 0 | 1
0 1 | 0
1 0 | 1
1 1 | 1
Our resulting truth table is equivalent to ${A \vee \neg B}$, or ${B \to A}$ (also known as the IMPLY gate, albeit with the arguments swapped). It turns out this truth table is functionally complete, which means we can make arbitrary circuits using only this gate. Technically speaking, it is only functionally complete if given access to the constant false. This is necessary to produce a NOT gate, and NOT + IMPLY is a functionally complete set. I don’t know a better term for ‘functionally complete if given access to some constant value’, however.
Subtraction circuits
Let’s build a demo in Python. First we’ll define our constants and allow us to print them nicely. Note that even though they are distinct entities, $+0$ and $-0$ compare equal in IEEE 754 floating point, so we must first extract the sign before comparing to 0 to distinguish.
import math
f_false = -0.0
f_true = 0.0
f_repr = lambda x: True if math.copysign(1, x) > 0 else False
We can now make a NOT gate by using the fact that $-0 - x$ flips the sign of zero $x$:
f_not = lambda x: f_false - x
Let’s test that:
>>> f_repr(f_not(f_false))
True
>>> f_repr(f_not(f_true))
False
Great! We can also build an OR gate by noticing that if we flip the sign of the second argument before subtracting, we always get $+0$ (true) unless both arguments are $-0$ (false):
f_or = lambda a, b: a - f_not(b)
Let’s test it out:
>>> f_repr(f_or(f_false, f_false))
False
>>> f_repr(f_or(f_true, f_false))
True
>>> f_repr(f_or(f_false, f_true))
True
>>> f_repr(f_or(f_true, f_true))
True
Now that we have OR and NOT, we can make all other gates, e.g.:
f_and = lambda a, b: f_not(f_or(f_not(a), f_not(b)))
f_xor = lambda a, b: f_or(f_and(f_not(a), b), f_and(a, f_not(b)))
>>> f_repr(f_and(f_false, f_false))
False
>>> f_repr(f_and(f_true, f_false))
False
>>> f_repr(f_and(f_false, f_true))
False
>>> f_repr(f_and(f_true, f_true))
True
>>> f_repr(f_xor(f_false, f_false))
False
>>> f_repr(f_xor(f_true, f_false))
True
>>> f_repr(f_xor(f_false, f_true))
True
>>> f_repr(f_xor(f_true, f_true))
False
Software integers
You may have heard of soft-float, software implementations of floating point
using integers. Let’s turn that on its head: an integer implementation done in
software, using only floating point ops. Let’s do it in Rust so we can look at
the final assembly output to see how horrifically slow awesome it is.
type Bit = f32;
const ZERO: Bit = -0.0;
const ONE: Bit = 0.0;
fn not(x: Bit) -> Bit { ZERO - x }
fn or(a: Bit, b: Bit) -> Bit { a - not(b) }
fn and(a: Bit, b: Bit) -> Bit { not(or(not(a), not(b))) }
fn xor(a: Bit, b: Bit) -> Bit { or(and(not(a), b), and(a, not(b))) }
fn adder(a: Bit, b: Bit, c: Bit) -> (Bit, Bit) {
let s = xor(xor(a, b), c);
let c = or(and(xor(a, b), c), and(a, b));
(s, c)
}
type SoftU8 = [Bit; 8];
pub fn softu8_add(a: SoftU8, b: SoftU8) -> SoftU8 {
let (s0, c) = adder(a[0], b[0], ZERO);
let (s1, c) = adder(a[1], b[1], c);
let (s2, c) = adder(a[2], b[2], c);
let (s3, c) = adder(a[3], b[3], c);
let (s4, c) = adder(a[4], b[4], c);
let (s5, c) = adder(a[5], b[5], c);
let (s6, c) = adder(a[6], b[6], c);
let (s7, _) = adder(a[7], b[7], c);
[s0, s1, s2, s3, s4, s5, s6, s7]
}
// Hmm? u8? What's that? Shhhh....
pub fn to_softu8(x: u8) -> SoftU8 {
std::array::from_fn(|i| if (x >> i) & 1 == 1 { ONE } else { ZERO })
}
pub fn from_softu8(x: SoftU8) -> u8 {
(0..8).filter(|i| x[*i].signum() > 0.0).map(|i| 1 << i).sum()
}
fn main() {
let a = to_softu8(23);
let b = to_softu8(19);
println!("{}", from_softu8(softu8_add(a, b)));
}
It’s horrible, but it works, it dutifully prints 42. And it only took $\approx 120$ floating point instructions to add two 8-bit integers:
example::softu8_add:
mov rax, rdi
movups xmm2, xmmword ptr [rsi]
movups xmm0, xmmword ptr [rdx]
movaps xmm3, xmm2
subps xmm3, xmm0
movaps xmm4, xmm0
subps xmm4, xmm2
movaps xmm1, xmmword ptr [rip + .LCPI0_0]
xorps xmm4, xmm1
subps xmm4, xmm3
xorps xmm3, xmm3
subss xmm3, xmm4
movaps xmm6, xmm0
xorps xmm6, xmm1
subss xmm6, xmm2
xorps xmm6, xmm1
subss xmm6, xmm3
movaps xmm3, xmm4
shufps xmm3, xmm4, 85
movaps xmm5, xmm6
subss xmm5, xmm3
xorps xmm5, xmm1
movaps xmm10, xmm6
xorps xmm10, xmm1
subss xmm10, xmm3
movaps xmm7, xmm0
shufps xmm7, xmm0, 85
xorps xmm7, xmm1
movaps xmm3, xmm2
shufps xmm3, xmm2, 85
subss xmm7, xmm3
xorps xmm7, xmm1
movaps xmm8, xmm4
unpckhpd xmm8, xmm4
movaps xmm3, xmm0
unpckhpd xmm3, xmm0
xorps xmm3, xmm1
movaps xmm9, xmm2
unpckhpd xmm9, xmm2
subss xmm3, xmm9
xorps xmm3, xmm1
xorps xmm11, xmm11
movaps xmm9, xmm4
shufps xmm9, xmm4, 255
subss xmm7, xmm10
movaps xmm10, xmm7
xorps xmm10, xmm1
subss xmm10, xmm8
subss xmm3, xmm10
unpcklps xmm7, xmm3
shufps xmm6, xmm7, 64
addps xmm11, xmm4
movlhps xmm5, xmm4
subps xmm4, xmm6
movss xmm4, xmm11
subps xmm7, xmm8
xorps xmm7, xmm1
shufps xmm5, xmm7, 66
subps xmm5, xmm4
xorps xmm3, xmm1
subss xmm3, xmm9
shufps xmm0, xmm0, 255
xorps xmm0, xmm1
shufps xmm2, xmm2, 255
subss xmm0, xmm2
xorps xmm0, xmm1
movups xmmword ptr [rdi], xmm5
movups xmm2, xmmword ptr [rdx + 16]
movaps xmm5, xmm2
xorps xmm5, xmm1
movups xmm7, xmmword ptr [rsi + 16]
subss xmm5, xmm7
xorps xmm5, xmm1
movaps xmm4, xmm2
shufps xmm4, xmm2, 85
xorps xmm4, xmm1
movaps xmm6, xmm2
movaps xmm8, xmm7
movaps xmm9, xmm7
subps xmm9, xmm2
subps xmm2, xmm7
shufps xmm7, xmm7, 85
subss xmm4, xmm7
xorps xmm4, xmm1
movhlps xmm6, xmm6
xorps xmm6, xmm1
movhlps xmm8, xmm8
subss xmm6, xmm8
xorps xmm6, xmm1
xorps xmm2, xmm1
subps xmm2, xmm9
subss xmm0, xmm3
movaps xmm3, xmm0
xorps xmm3, xmm1
subss xmm3, xmm2
subss xmm5, xmm3
unpcklps xmm0, xmm5
xorps xmm5, xmm1
movaps xmm3, xmm2
shufps xmm3, xmm2, 85
subss xmm5, xmm3
subss xmm4, xmm5
movaps xmm3, xmm4
xorps xmm3, xmm1
movaps xmm5, xmm2
unpckhpd xmm5, xmm2
subss xmm3, xmm5
subss xmm6, xmm3
unpcklps xmm4, xmm6
movlhps xmm0, xmm4
movaps xmm3, xmm2
subps xmm3, xmm0
subps xmm0, xmm2
xorps xmm0, xmm1
subps xmm0, xmm3
movups xmmword ptr [rdi + 16], xmm0
ret