<https://arxiv.org/abs/2603.21852>
"A single two-input gate suffices for all of Boolean logic in digital >hardware. No comparable primitive has been known for continuous
mathematics: computing elementary functions such as sin, cos, sqrt, and
log has always required multiple distinct operations. Here I show that a >single binary operator, eml(x,y)=exp(x)-ln(y), together with the
constant 1, generates the standard repertoire of a scientific
calculator. This includes constants such as e, pi, and i; arithmetic >operations including addition, subtraction, multiplication, division,
and exponentiation as well as the usual transcendental and algebraic >functions."
On Tue, 14 Apr 2026 20:29:20 -0400, bitrex <user@example.net> wrote:
<https://arxiv.org/abs/2603.21852>
"A single two-input gate suffices for all of Boolean logic in digital
hardware. No comparable primitive has been known for continuous
mathematics: computing elementary functions such as sin, cos, sqrt, and
log has always required multiple distinct operations. Here I show that a
single binary operator, eml(x,y)=exp(x)-ln(y), together with the
constant 1, generates the standard repertoire of a scientific
calculator. This includes constants such as e, pi, and i; arithmetic
operations including addition, subtraction, multiplication, division,
and exponentiation as well as the usual transcendental and algebraic
functions."
I wonder how long it took the author to notice that he had a belly
button?
RL
legg <legg@nospam.magma.ca> wrote:
On Tue, 14 Apr 2026 20:29:20 -0400, bitrex <user@example.net> wrote:
<https://arxiv.org/abs/2603.21852>
"A single two-input gate suffices for all of Boolean logic in digital
hardware. No comparable primitive has been known for continuous
mathematics: computing elementary functions such as sin, cos, sqrt, and
log has always required multiple distinct operations. Here I show that a >>> single binary operator, eml(x,y)=exp(x)-ln(y), together with the
constant 1, generates the standard repertoire of a scientific
calculator. This includes constants such as e, pi, and i; arithmetic
operations including addition, subtraction, multiplication, division,
and exponentiation as well as the usual transcendental and algebraic
functions."
I wonder how long it took the author to notice that he had a belly
button?
RL
Haha yes, the Apollo Guidance Computer was pretty much entirely built out
of NOR gates 60+ years ago.
<https://arxiv.org/abs/2603.21852>
"A single two-input gate suffices for all of Boolean logic in digital >hardware. No comparable primitive has been known for continuous
mathematics: computing elementary functions such as sin, cos, sqrt, and
log has always required multiple distinct operations. Here I show that a >single binary operator, eml(x,y)=exp(x)-ln(y), together with the
constant 1, generates the standard repertoire of a scientific
calculator. This includes constants such as e, pi, and i; arithmetic >operations including addition, subtraction, multiplication, division,
and exponentiation as well as the usual transcendental and algebraic >functions."
On Tue, 14 Apr 2026 20:29:20 -0400, bitrex <user@example.net> wrote:
<https://arxiv.org/abs/2603.21852>
"A single two-input gate suffices for all of Boolean logic in digital
hardware. No comparable primitive has been known for continuous
mathematics: computing elementary functions such as sin, cos, sqrt, and
log has always required multiple distinct operations. Here I show that a
single binary operator, eml(x,y)=exp(x)-ln(y), together with the
constant 1, generates the standard repertoire of a scientific
calculator. This includes constants such as e, pi, and i; arithmetic
operations including addition, subtraction, multiplication, division,
and exponentiation as well as the usual transcendental and algebraic
functions."
I wonder how long it took the author to notice that he had a belly
button?
RL
On Tue, 14 Apr 2026 20:29:20 -0400, bitrex <user@example.net> wrote:
<https://arxiv.org/abs/2603.21852>
"A single two-input gate suffices for all of Boolean logic in digital
hardware. No comparable primitive has been known for continuous
mathematics: computing elementary functions such as sin, cos, sqrt, and
log has always required multiple distinct operations. Here I show that a
single binary operator, eml(x,y)=exp(x)-ln(y), together with the
constant 1, generates the standard repertoire of a scientific
calculator. This includes constants such as e, pi, and i; arithmetic
operations including addition, subtraction, multiplication, division,
and exponentiation as well as the usual transcendental and algebraic
functions."
That reminds me of the Whitespace programming language.
John Larkin
Highland Tech Glen Canyon Design Center
Lunatic Fringe Electronics
On 4/15/2026 1:21 PM, john larkin wrote:
On Tue, 14 Apr 2026 20:29:20 -0400, bitrex <user@example.net> wrote:
<https://arxiv.org/abs/2603.21852>
"A single two-input gate suffices for all of Boolean logic in digital
hardware. No comparable primitive has been known for continuous
mathematics: computing elementary functions such as sin, cos, sqrt, and
log has always required multiple distinct operations. Here I show that a >>> single binary operator, eml(x,y)=exp(x)-ln(y), together with the
constant 1, generates the standard repertoire of a scientific
calculator. This includes constants such as e, pi, and i; arithmetic
operations including addition, subtraction, multiplication, division,
and exponentiation as well as the usual transcendental and algebraic
functions."
That reminds me of the Whitespace programming language.
John Larkin
Highland Tech Glen Canyon Design Center
Lunatic Fringe Electronics
Yes, I believe the interesting thing is the _completeness_ of the "eml" >operator, kind of like a one-instruction set computer.
Not that NAND is boolean complete or that we can make trig functions
from exponentials, etc. which we knew already.
On Wed, 15 Apr 2026 15:41:05 -0400, bitrex <user@example.net> wrote:
On 4/15/2026 1:21 PM, john larkin wrote:
On Tue, 14 Apr 2026 20:29:20 -0400, bitrex <user@example.net> wrote:
<https://arxiv.org/abs/2603.21852>
"A single two-input gate suffices for all of Boolean logic in digital
hardware. No comparable primitive has been known for continuous
mathematics: computing elementary functions such as sin, cos, sqrt, and >>>> log has always required multiple distinct operations. Here I show that a >>>> single binary operator, eml(x,y)=exp(x)-ln(y), together with the
constant 1, generates the standard repertoire of a scientific
calculator. This includes constants such as e, pi, and i; arithmetic
operations including addition, subtraction, multiplication, division,
and exponentiation as well as the usual transcendental and algebraic
functions."
That reminds me of the Whitespace programming language.
John Larkin
Highland Tech Glen Canyon Design Center
Lunatic Fringe Electronics
Yes, I believe the interesting thing is the _completeness_ of the "eml"
operator, kind of like a one-instruction set computer.
Not that NAND is boolean complete or that we can make trig functions
from exponentials, etc. which we knew already.
That function has been suggested as the basis for a 2-button
calculator.
John Larkin
Highland Tech Glen Canyon Design Center
Lunatic Fringe Electronics
On 04/15/2026 01:08 PM, john larkin wrote:
On Wed, 15 Apr 2026 15:41:05 -0400, bitrex <user@example.net> wrote:
On 4/15/2026 1:21 PM, john larkin wrote:
On Tue, 14 Apr 2026 20:29:20 -0400, bitrex <user@example.net> wrote:
<https://arxiv.org/abs/2603.21852>
"A single two-input gate suffices for all of Boolean logic in digital >>>>> hardware. No comparable primitive has been known for continuous
mathematics: computing elementary functions such as sin, cos, sqrt,
and
log has always required multiple distinct operations. Here I show
that a
single binary operator, eml(x,y)=exp(x)-ln(y), together with the
constant 1, generates the standard repertoire of a scientific
calculator. This includes constants such as e, pi, and i; arithmetic >>>>> operations including addition, subtraction, multiplication, division, >>>>> and exponentiation as well as the usual transcendental and algebraic >>>>> functions."
That reminds me of the Whitespace programming language.
John Larkin
Highland Tech Glen Canyon Design Center
Lunatic Fringe Electronics
Yes, I believe the interesting thing is the _completeness_ of the "eml"
operator, kind of like a one-instruction set computer.
Not that NAND is boolean complete or that we can make trig functions
from exponentials, etc. which we knew already.
That function has been suggested as the basis for a 2-button
calculator.
John Larkin
Highland Tech Glen Canyon Design Center
Lunatic Fringe Electronics
The ancient Sumerians make an account for positional notation
of numbers after _increment_ what makes addition.
The ancient Egyptians make an account for ratios as fractions
of whole numbers after _partition_ what makes division.
So, it's sort of having those as two different operations
instead as multiplication after addition then inverses.
So, "long subtraction", in a sense.
The idea is that partition works on the _big_ end
while increment works on the _little_ end.
Why make a two-button calculator when all that is
is an arbitrarily long slide-rule?
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