# 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.

1. A subtraction $x - y$ is considered a sum $x + (-y)$.
2. Zero can have a sign, $-0$ and $0$ are distinct entities (although they compare equal when testing with ==).
3. 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 $x$.
4. 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, b, ZERO);
let (s1, c) = adder(a, b, c);
let (s2, c) = adder(a, b, c);
let (s3, c) = adder(a, b, c);
let (s4, c) = adder(a, b, c);
let (s5, c) = adder(a, b, c);
let (s6, c) = adder(a, b, c);
let (s7, _) = adder(a, b, 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);
}


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
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