# Elliptic curve arithmetic

Elliptic curve arithmetic
You are encouraged to solve this task according to the task description, using any language you may know.

Elliptic curves   are sometimes used in   cryptography   as a way to perform   digital signatures.

The purpose of this task is to implement a simplified (without modular arithmetic) version of the elliptic curve arithmetic which is required by the   elliptic curve DSA   protocol.

In a nutshell, an elliptic curve is a bi-dimensional curve defined by the following relation between the x and y coordinates of any point on the curve:

${\displaystyle y^2 = x^3 + a x + b}$

a and b are arbitrary parameters that define the specific curve which is used.

For this particular task, we'll use the following parameters:

a=0,   b=7

The most interesting thing about elliptic curves is the fact that it is possible to define a   group   structure on it.

To do so we define an   internal composition   rule with an additive notation +,   such that for any three distinct points P, Q and R on the curve, whenever these points are aligned, we have:

P + Q + R = 0

Here   0   (zero)   is the infinity point,   for which the x and y values are not defined.   It's basically the same kind of point which defines the horizon in   projective geometry.

We'll also assume here that this infinity point is unique and defines the   neutral element   of the addition.

This was not the definition of the addition, but only its desired property.   For a more accurate definition, we proceed as such:

Given any three aligned points P, Q and R,   we define the sum   S = P + Q   as the point (possibly the infinity point) such that   S, R   and the infinity point are aligned.

Considering the symmetry of the curve around the x-axis, it's easy to convince oneself that two points S and R can be aligned with the infinity point if and only if S and R are symmetric of one another towards the x-axis   (because in that case there is no other candidate than the infinity point to complete the alignment triplet).

S is thus defined as the symmetric of R towards the x axis.

The task consists in defining the addition which, for any two points of the curve, returns the sum of these two points.   You will pick two random points on the curve, compute their sum and show that the symmetric of the sum is aligned with the two initial points.

You will use the a and b parameters of secp256k1, i.e. respectively zero and seven.

Hint:   You might need to define a "doubling" function, that returns P+P for any given point P.

Extra credit:   define the full elliptic curve arithmetic (still not modular, though) by defining a "multiply" function that returns,
for any point P and integer n,   the point P + P + ... + P     (n times).

## 11l

Translation of: Python
```T Point
:b = 7

Float x, y

F (x = Float.infinity, y = Float.infinity)
.x = x
.y = y

F.const copy()
R Point(.x, .y)

F.const is_zero()
R .x > 1e20 | .x < -1e20

F neg()
R Point(.x, -.y)

F dbl()
I .is_zero()
R .copy()
I .y == 0
R Point()
V l = (3 * .x * .x) / (2 * .y)
V x = l * l - 2 * .x
R Point(x, l * (.x - x) - .y)

I .x == q.x & .y == q.y
R .dbl()
I .is_zero()
R q.copy()
I q.is_zero()
R .copy()
I q.x - .x == 0
R Point()
V l = (q.y - .y) / (q.x - .x)
V x = l * l - .x - q.x
R Point(x, l * (.x - x) - .y)

F mul(n)
V p = .copy()
V r = Point()
V i = 1
L i <= n
I i [&] n
p = p.dbl()
i <<= 1
R r

F String()
R ‘(#.3, #.3)’.format(.x, .y)

F show(s, p)
print(s‘ ’(I p.is_zero() {‘Zero’} E p))

F from_y(y)
V n = y * y - Point.:b
V x = I n >= 0 {n ^ (1. / 3)} E -((-n) ^ (1. / 3))
R Point(x, y)

V a = from_y(1)
V b = from_y(2)
show(‘a =’, a)
show(‘b =’, b)
show(‘c = a + b =’, c)
V d = c.neg()
show(‘d = -c =’, d)
show(‘a * 12345 =’, a.mul(12345))```
Output:
```a = (-1.817, 1.000)
b = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c = (10.375, 33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)
```

## C

```#include <stdio.h>
#include <math.h>

#define C 7
typedef struct { double x, y; } pt;

pt zero(void) { return (pt){ INFINITY, INFINITY }; }

// should be INFINITY, but numeric precision is very much in the way
int is_zero(pt p) { return p.x > 1e20 || p.x < -1e20; }

pt neg(pt p) { return (pt){ p.x, -p.y }; }

pt dbl(pt p) {
if (is_zero(p)) return p;

pt r;
double L = (3 * p.x * p.x) / (2 * p.y);
r.x = L * L - 2 * p.x;
r.y = L * (p.x - r.x) - p.y;
return r;
}

pt add(pt p, pt q) {
if (p.x == q.x && p.y == q.y) return dbl(p);
if (is_zero(p)) return q;
if (is_zero(q)) return p;

pt r;
double L = (q.y - p.y) / (q.x - p.x);
r.x = L * L - p.x - q.x;
r.y = L * (p.x - r.x) - p.y;
return r;
}

pt mul(pt p, int n) {
int i;
pt r = zero();

for (i = 1; i <= n; i <<= 1) {
if (i & n) r = add(r, p);
p = dbl(p);
}
return r;
}

void show(const char *s, pt p) {
printf("%s", s);
printf(is_zero(p) ? "Zero\n" : "(%.3f, %.3f)\n", p.x, p.y);
}

pt from_y(double y) {
pt r;
r.x = pow(y * y - C, 1.0/3);
r.y = y;
return r;
}

int main(void) {
pt a, b, c, d;

a = from_y(1);
b = from_y(2);

show("a = ", a);
show("b = ", b);
show("c = a + b = ", c = add(a, b));
show("d = -c = ", d = neg(c));
show("c + d = ", add(c, d));
show("a * 12345 = ", mul(a, 12345));

return 0;
}
```
Output:
```a = (-1.817, 1.000)
b = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c = (10.375, 33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)
```

## C++

Uses C++11 or later

```#include <cmath>
#include <iostream>

using namespace std;

// define a type for the points on the elliptic curve that behaves
// like a built in type.
class EllipticPoint
{
double m_x, m_y;
static constexpr double ZeroThreshold = 1e20;
static constexpr double B = 7; // the 'b' in y^2 = x^3 + a * x + b
//  'a' is 0

void Double() noexcept
{
if(IsZero())
{
// doubling zero is still zero
return;
}
// based on the property of the curve, the line going through the
// current point and the negative doubled point is tangent to the
// curve at the current point.  wikipedia has a nice diagram.
if(m_y == 0)
{
// at this point the tangent to the curve is vertical.
// this point doubled is 0
*this = EllipticPoint();
}
else
{
double L = (3 * m_x * m_x) / (2 * m_y);
double newX = L * L -  2 * m_x;
m_y = L * (m_x - newX) - m_y;
m_x = newX;
}
}

public:
friend std::ostream& operator<<(std::ostream&, const EllipticPoint&);

// Create a point that is initialized to Zero
constexpr EllipticPoint() noexcept : m_x(0), m_y(ZeroThreshold * 1.01) {}

// Create a point based on the yCoordiante.  For a curve with a = 0 and b = 7
// there is only one x for each y
explicit EllipticPoint(double yCoordinate) noexcept
{
m_y = yCoordinate;
m_x = cbrt(m_y * m_y - B);
}

// Check if the point is 0
bool IsZero() const noexcept
{
// when the elliptic point is at 0, y = +/- infinity
bool isNotZero =  abs(m_y) < ZeroThreshold;
return !isNotZero;
}

// make a negative version of the point  (p = -q)
EllipticPoint operator-() const noexcept
{
EllipticPoint negPt;
negPt.m_x = m_x;
negPt.m_y = -m_y;

return negPt;
}

// add a point to this one  ( p+=q )
EllipticPoint& operator+=(const EllipticPoint& rhs) noexcept
{
if(IsZero())
{
*this = rhs;
}
else if (rhs.IsZero())
{
// since rhs is zero this point does not need to be
// modified
}
else
{
double L = (rhs.m_y - m_y) / (rhs.m_x - m_x);
if(isfinite(L))
{
double newX = L * L - m_x - rhs.m_x;
m_y = L * (m_x - newX) - m_y;
m_x = newX;
}
else
{
if(signbit(m_y) != signbit(rhs.m_y))
{
// in this case rhs == -lhs, the result should be 0
*this = EllipticPoint();
}
else
{
// in this case rhs == lhs.
Double();
}
}
}

return *this;
}

// subtract a point from this one  (p -= q)
EllipticPoint& operator-=(const EllipticPoint& rhs) noexcept
{
*this+= -rhs;
return *this;
}

// multiply the point by an integer (p *= 3)
EllipticPoint& operator*=(int rhs) noexcept
{
EllipticPoint r;
EllipticPoint p = *this;

if(rhs < 0)
{
// change p * -rhs to -p * rhs
rhs = -rhs;
p = -p;
}

for (int i = 1; i <= rhs; i <<= 1)
{
if (i & rhs) r += p;
p.Double();
}

*this = r;
return *this;
}
};

// add points  (p + q)
inline EllipticPoint operator+(EllipticPoint lhs, const EllipticPoint& rhs) noexcept
{
lhs += rhs;
return lhs;
}

// subtract points  (p - q)
inline EllipticPoint operator-(EllipticPoint lhs, const EllipticPoint& rhs) noexcept
{
lhs += -rhs;
return lhs;
}

// multiply by an integer  (p * 3)
inline EllipticPoint operator*(EllipticPoint lhs, const int rhs) noexcept
{
lhs *= rhs;
return lhs;
}

// multiply by an integer (3 * p)
inline EllipticPoint operator*(const int lhs, EllipticPoint rhs) noexcept
{
rhs *= lhs;
return rhs;
}

// print the point
ostream& operator<<(ostream& os, const EllipticPoint& pt)
{
if(pt.IsZero()) cout << "(Zero)\n";
else cout << "(" << pt.m_x << ", " << pt.m_y << ")\n";
return os;
}

int main(void) {
const EllipticPoint a(1), b(2);
cout << "a = " << a;
cout << "b = " << b;
const EllipticPoint c = a + b;
cout << "c = a + b = "       << c;
cout << "a + b - c = "       << a + b - c;
cout << "a + b - (b + a) = " << a + b - (b + a) << "\n";

cout << "a + a + a + a + a - 5 * a = "         << a + a + a + a + a - 5 * a;
cout << "a * 12345 = "                         << a * 12345;
cout << "a * -12345 = "                        << a * -12345;
cout << "a * 12345 + a * -12345 = "            << a * 12345 + a * -12345;
cout << "a * 12345 - (a * 12000 + a * 345) = " << a * 12345 - (a * 12000 + a * 345);
cout << "a * 12345 - (a * 12001 + a * 345) = " << a * 12345 - (a * 12000 + a * 344) << "\n";

const EllipticPoint zero;
EllipticPoint g;
cout << "g = zero = "      << g;
cout << "g += a = "        << (g+=a);
cout << "g += zero = "     << (g+=zero);
cout << "g += b = "        << (g+=b);
cout << "b + b - b * 2 = " << (b + b - b * 2) << "\n";

EllipticPoint special(0);  // the point where the curve crosses the x-axis
cout << "special = "      << special; // this has the minimum possible value for x
cout << "special *= 2 = " << (special*=2); // doubling it gives zero

return 0;
}
```
Output:
```a = (-1.81712, 1)
b = (-1.44225, 2)
c = a + b = (10.3754, -33.5245)
a + b - c = (Zero)
a + b - (b + a) = (Zero)

a + a + a + a + a - 5 * a = (Zero)
a * 12345 = (10.7586, 35.3874)
a * -12345 = (10.7586, -35.3874)
a * 12345 + a * -12345 = (Zero)
a * 12345 - (a * 12000 + a * 345) = (Zero)
a * 12345 - (a * 12001 + a * 345) = (-1.81712, 1)

g = zero = (Zero)
g += a = (-1.81712, 1)
g += zero = (-1.81712, 1)
g += b = (10.3754, -33.5245)
b + b - b * 2 = (Zero)

special = (-1.91293, 0)
special *= 2 = (Zero)
```

## D

Translation of: Go
```import std.stdio, std.math, std.string;

enum bCoeff = 7;

struct Pt {
double x, y;

@property static Pt zero() pure nothrow @nogc @safe {
return Pt(double.infinity, double.infinity);
}

@property bool isZero() const pure nothrow @nogc @safe {
return x > 1e20 || x < -1e20;
}

@property static Pt fromY(in double y) nothrow /*pure*/ @nogc @safe {
return Pt(cbrt(y ^^ 2 - bCoeff), y);
}

@property Pt dbl() const pure nothrow @nogc @safe {
if (this.isZero)
return this;
immutable L = (3 * x * x) / (2 * y);
immutable x2 = L ^^ 2  - 2 * x;
return Pt(x2, L * (x - x2) - y);
}

string toString() const pure /*nothrow*/ @safe {
if (this.isZero)
return "Zero";
else
return format("(%.3f, %.3f)", this.tupleof);
}

Pt opUnary(string op)() const pure nothrow @nogc @safe
if (op == "-") {
return Pt(this.x, -this.y);
}

Pt opBinary(string op)(in Pt q) const pure nothrow @nogc @safe
if (op == "+") {
if (this.x == q.x && this.y == q.y)
return this.dbl;
if (this.isZero)
return q;
if (q.isZero)
return this;
immutable L = (q.y - this.y) / (q.x - this.x);
immutable x = L ^^ 2 - this.x - q.x;
return Pt(x, L * (this.x - x) - this.y);
}

Pt opBinary(string op)(in uint n) const pure nothrow @nogc @safe
if (op == "*") {
auto r = Pt.zero;
Pt p = this;
for (uint i = 1; i <= n; i <<= 1) {
if ((i & n) != 0)
r = r + p;
p = p.dbl;
}
return r;
}
}

void main() @safe {
immutable a = Pt.fromY(1);
immutable b = Pt.fromY(2);
writeln("a = ", a);
writeln("b = ", b);
immutable c = a + b;
writeln("c = a + b = ", c);
immutable d = -c;
writeln("d = -c = ", d);
writeln("c + d = ", c + d);
writeln("a + b + d = ", a + b + d);
writeln("a * 12345 = ", a * 12345);
}
```
Output:
```a = (-1.817, 1.000)
b = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c = (10.375, 33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)```

## EchoLisp

### Arithmetic

```(require 'struct)
(decimals 4)
(string-delimiter "")
(struct pt (x y))

(define-syntax-id _.x (struct-get _ #:pt.x))
(define-syntax-id _.y (struct-get _ #:pt.y))

(define (E-zero) (pt Infinity Infinity))
(define (E-zero? p) (=  (abs p.x) Infinity))
(define (E-neg p) (pt  p.x  (- p.y)))

;; magic formulae from "C"
;; p + p
(define (E-dbl p)
(if (E-zero? p) p
(let* (
[L (// (* 3 p.x p.x) (* 2 p.y))]
[rx (- (* L L) (* 2 p.x))]
[ry (- (* L (- p.x rx)) p.y)]
)
(pt rx ry))))

;; p + q
(cond
[ (and (= p.x p.x) (= p.y q.y)) (E-dbl p)]
[ (E-zero? p) q ]
[ (E-zero? q) p ]
[ else
(let* (
[L (// (- q.y p.y) (- q.x p.x))]
[rx (- (* L L) p.x q.x)] ;; match
[ry (- (* L (- p.x rx)) p.y)]
)
(pt rx ry))]))

;; (E-add* a b c ...)

;; p * n
(define (E-mul p n (r (E-zero)) (i 1))
(while (<= i n)
(when (!zero? (bitwise-and i n))  (set! r (E-add r p)))
(set! p (E-dbl p))
(set! i (* i 2)))
r)

;; make points from x or y
(define (Ey.pt y  (c 7))
(pt (expt (- (* y y) c) 1/3 ) y))
(define (Ex.pt x  (c 7))
(pt x (sqrt (+ ( * x x x ) c))))

;; Check floating point precision
;; P * n is not always P+P+P+P....P

(define (E-ckmul a n )
(define e a)
(for ((i (in-range 1 n))) (set! e (E-add a e)))
(printf "%d additions a+(a+(a+...)))  → %a" n e)
(printf "multiplication a x %d        → %a" n (E-mul a n)))
```
Output:
```(define P (Ey.pt 1))
(define Q (Ey.pt 2))
→ #<pt> (10.3754 -33.5245)
(E-zero? (E-add* P Q (E-neg R)))
→ #t
(E-mul P 12345)
→ #<pt> (10.7586 35.3874)

;; check floating point precision
(E-ckmul P 10) ;; OK
10 additions a+(a+(a+...))) → #<pt> (0.3797 -2.6561)
multiplication a x 10       → #<pt> (0.3797 -2.6561)

(E-ckmul P 12345) ;; KO
12345 additions a+(a+(a+...))) → #<pt> (-1.3065 2.4333)
multiplication a x 12345 → #<pt> (10.7586 35.3874)
```

### Plotting

Result at http://www.echolalie.org/echolisp/help.html#plot-xy
```(define (E-plot (r 3))
(define (Ellie x y) (- (* y y) (* x x x) 7))
(define P (Ey.pt 0))
(define Q (Ex.pt 0))

(plot-clear)
(plot-xy Ellie -10 -10) ;; curve
(plot-axis 0 0 "red")
(plot-circle P.x P.y r) ;; points
(plot-circle Q.x Q.y r)
(plot-circle R.x R.y r)
(plot-circle R.x (- R.y) r)
(plot-segment P.x P.y R.x (- R.y))
(plot-segment R.x R.y R.x (- R.y))
)
```

## Go

Translation of: C
```package main

import (
"fmt"
"math"
)

const bCoeff = 7

type pt struct{ x, y float64 }

func zero() pt {
return pt{math.Inf(1), math.Inf(1)}
}

func is_zero(p pt) bool {
return p.x > 1e20 || p.x < -1e20
}

func neg(p pt) pt {
return pt{p.x, -p.y}
}

func dbl(p pt) pt {
if is_zero(p) {
return p
}
L := (3 * p.x * p.x) / (2 * p.y)
x := L*L - 2*p.x
return pt{
x: x,
y: L*(p.x-x) - p.y,
}
}

func add(p, q pt) pt {
if p.x == q.x && p.y == q.y {
return dbl(p)
}
if is_zero(p) {
return q
}
if is_zero(q) {
return p
}
L := (q.y - p.y) / (q.x - p.x)
x := L*L - p.x - q.x
return pt{
x: x,
y: L*(p.x-x) - p.y,
}
}

func mul(p pt, n int) pt {
r := zero()
for i := 1; i <= n; i <<= 1 {
if i&n != 0 {
}
p = dbl(p)
}
return r
}

func show(s string, p pt) {
fmt.Printf("%s", s)
if is_zero(p) {
fmt.Println("Zero")
} else {
fmt.Printf("(%.3f, %.3f)\n", p.x, p.y)
}
}

func from_y(y float64) pt {
return pt{
x: math.Cbrt(y*y - bCoeff),
y: y,
}
}

func main() {
a := from_y(1)
b := from_y(2)
show("a = ", a)
show("b = ", b)
show("c = a + b = ", c)
d := neg(c)
show("d = -c = ", d)
show("c + d = ", add(c, d))
show("a * 12345 = ", mul(a, 12345))
}
```
Output:
```a = (-1.817, 1.000)
b = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c = (10.375, 33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)```

First, some useful imports:

```import Data.Monoid
import Test.QuickCheck (quickCheck)
```

The datatype for a point on an elliptic curve:

```import Data.Monoid

data Elliptic = Elliptic Double Double | Zero
deriving Show

instance Eq Elliptic where
p == q = dist p q < 1e-14
where
dist Zero Zero = 0
dist Zero p = 1/0
dist p Zero = 1/0
dist (Elliptic x1 y1) (Elliptic x2 y2) = (x2-x1)^2 + (y2-y1)^2

inv Zero = Zero
inv (Elliptic x y) = Elliptic x (-y)
```

Points on elliptic curve form a monoid:

```instance Monoid Elliptic where
mempty = Zero

mappend Zero p = p
mappend p Zero = p
mappend p@(Elliptic x1 y1) q@(Elliptic x2 y2)
| p == inv q = Zero
| p == q     = mkElliptic \$ 3*x1^2/(2*y1)
| otherwise  = mkElliptic \$ (y2 - y1)/(x2 - x1)
where
mkElliptic l = let x = l^2 - x1 - x2
y = l*(x1 - x) - y1
in Elliptic x y
```

Examples given in other solutions:

```ellipticX b y = Elliptic (qroot (y^2 - b)) y
where qroot x = signum x * abs x ** (1/3)
```
```λ> let a = ellipticX 7 1
λ> let b = ellipticX 7 2
λ> a
Elliptic (-1.8171205928321397) 1.0
λ> b
Elliptic (-1.4422495703074083) 2.0
λ> let c = a <> b
λ> c
Elliptic 10.375375389201409 (-33.524509096269696)
λ> let d = inv c
λ> c <> d
Zero
λ> a <> b <> d
Zero```

Extra credit: multiplication.

1. direct monoidal solution:

```mult :: Int -> Elliptic -> Elliptic
mult n = mconcat . replicate n
```

2. efficient recursive solution:

```n `mult` p
| n == 0 = Zero
| n == 1 = p
| n == 2 = p <> p
| n < 0  = inv ((-n) `mult` p)
| even n = 2 `mult` ((n `div` 2) `mult` p)
| odd n  = p <> (n -1) `mult` p
```
```λ> 12345 `mult` a
Elliptic 10.758570529320476 35.387434774280486```

### Testing

We use QuickCheck to test general properties of points on arbitrary elliptic curve.

```-- for given a, b and x returns a point on the positive branch of elliptic curve (if point exists)
elliptic a b Nothing  = Just Zero
elliptic a b (Just x) =
do let y2 = x**3 + a*x + b
guard (y2 > 0)
return \$ Elliptic x (sqrt y2)

addition a b x1 x2 =
let p = elliptic a b
s = p x1 <> p x2
in (s /= Nothing) ==> (s <> (inv <\$> s) == Just Zero)

associativity a b x1 x2 x3 =
let p = elliptic a b
in (p x1 <> p x2) <> p x3 == p x1 <> (p x2 <> p x3)

commutativity a b x1 x2 =
let p = elliptic a b
in p x1 <> p x2 == p x2 <> p x1
```
```λ> quickCheck addition
+++ OK, passed 100 tests.
λ> quickCheck associativity
+++ OK, passed 100 tests.
λ> quickCheck commutativity
+++ OK, passed 100 tests.```

## J

Follows the C contribution.

```zero=: _j_

isZero=: 1e20 < |@{.@+.

neg=: +

'p_x p_y'=. +. p=. y
if. isZero p do. p return. end.
L=. 1.5 * p_x*p_x % p_y
r=. (L*L) - 2*p_x
r j. (L * p_x-r) - p_y
)

'p_x p_y'=. +. p=. x
'q_x q_y'=. +. q=. y
if. x=y do. dbl x return. end.
if. isZero x do. y return. end.
if. isZero y do. x return. end.
L=. %~/ +. q-p
r=. (L*L) - p_x + q_x
r j. (L * p_x-r) - p_y
)

a=. zero
for_bit.|.#:y do.
if. bit do.
end.
x=. dbl x
end.
a
)

NB. C is 7
from=: j.~ [:(* * 3 |@%: ]) _7 0 1 p. ]

if. isZero y do. 'Zero' else.
'a b'=. ":each +.y
'(',a,', ', b,')'
end.
)

a=. from 1
b=. from 2

echo 'a         = ', show a
echo 'b         = ', show b
echo 'c = a + b = ', show c =. a add b
echo 'd = -c    = ', show d =. neg c
echo 'c + d     = ', show c add d
echo 'a + b + d = ', show add/ a, b, d
echo 'a * 12345 = ', show a mul 12345
)
```
Output:
```   task ''
a         = (_1.81712, 1)
b         = (_1.44225, 2)
c = a + b = (10.3754, _33.5245)
d = -c    = (10.3754, 33.5245)
c + d     = Zero
a + b + d = Zero
a * 12345 = (10.7586, 35.3874)
```

## Java

Translation of: D
```import static java.lang.Math.*;
import java.util.Locale;

public class Test {

public static void main(String[] args) {
Pt a = Pt.fromY(1);
Pt b = Pt.fromY(2);
System.out.printf("a = %s%n", a);
System.out.printf("b = %s%n", b);
Pt c = a.plus(b);
System.out.printf("c = a + b = %s%n", c);
Pt d = c.neg();
System.out.printf("d = -c = %s%n", d);
System.out.printf("c + d = %s%n", c.plus(d));
System.out.printf("a + b + d = %s%n", a.plus(b).plus(d));
System.out.printf("a * 12345 = %s%n", a.mult(12345));
}
}

class Pt {
final static int bCoeff = 7;

double x, y;

Pt(double x, double y) {
this.x = x;
this.y = y;
}

static Pt zero() {
return new Pt(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY);
}

boolean isZero() {
return this.x > 1e20 || this.x < -1e20;
}

static Pt fromY(double y) {
return new Pt(cbrt(pow(y, 2) - bCoeff), y);
}

Pt dbl() {
if (isZero())
return this;
double L = (3 * this.x * this.x) / (2 * this.y);
double x2 = pow(L, 2) - 2 * this.x;
return new Pt(x2, L * (this.x - x2) - this.y);
}

Pt neg() {
return new Pt(this.x, -this.y);
}

Pt plus(Pt q) {
if (this.x == q.x && this.y == q.y)
return dbl();

if (isZero())
return q;

if (q.isZero())
return this;

double L = (q.y - this.y) / (q.x - this.x);
double xx = pow(L, 2) - this.x - q.x;
return new Pt(xx, L * (this.x - xx) - this.y);
}

Pt mult(int n) {
Pt r = Pt.zero();
Pt p = this;
for (int i = 1; i <= n; i <<= 1) {
if ((i & n) != 0)
r = r.plus(p);
p = p.dbl();
}
return r;
}

@Override
public String toString() {
if (isZero())
return "Zero";
return String.format(Locale.US, "(%.3f,%.3f)", this.x, this.y);
}
}
```
```a = (-1.817,1.000)
b = (-1.442,2.000)
c = a + b = (10.375,-33.525)
d = -c = (10.375,33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759,35.387)```

## Julia

Works with: Julia version 0.6
Translation of: Python
```struct Point{T<:AbstractFloat}
x::T
y::T
end
Point{T}() where T<:AbstractFloat = Point{T}(Inf, Inf)
Point() = Point{Float64}()

Base.show(io::IO, p::Point{T}) where T = iszero(p) ? print(io, "Zero{\$T}") : @printf(io, "{%s}(%.3f, %.3f)", T, p.x, p.y)
Base.copy(p::Point) = Point(p.x, p.y)
Base.iszero(p::Point{T}) where T = p.x in (-Inf, Inf)
Base.:-(p::Point) = Point(p.x, -p.y)

function dbl(p::Point{T}) where T
iszero(p) && return p

L = 3p.x ^ 2 / 2p.y
x = L ^ 2 - 2p.x
y = L * (p.x - x) - p.y
return Point{T}(x, y)
end
Base.:(==)(a::Point{T}, C::Point{T}) where T = a.x == C.x && a.y == C.y

function Base.:+(p::Point{T}, q::Point{T}) where T
p == q && return dbl(p)
iszero(p) && return q
iszero(q) && return p

L = (q.y - p.y) / (q.x - p.x)
x = L ^ 2 - p.x - q.x
y = L * (p.x - x) - p.y
return Point{T}(x, y)
end
function Base.:*(p::Point, n::Integer)
r = Point()
i = 1
while i ≤ n
if i & n != 0 r += p end
p = dbl(p)
i <<= 1
end
return r
end
Base.:*(n::Integer, p::Point) = p * n

const C = 7
function Point(y::AbstractFloat)
n = y ^ 2 - C
x = n ≥ 0 ? n ^ (1 / 3) : -((-n) ^ (1 / 3))
return Point{typeof(y)}(x, y)
end

a = Point(1.0)
b = Point(2.0)
@show a b
@show c = a + b
@show d = -c
@show c + d
@show a + b + d
@show 12345a
```
Output:
```a = {Float64}(-1.817, 1.000)
b = {Float64}(-1.442, 2.000)
c = a + b = {Float64}(10.375, -33.525)
d = -c = {Float64}(10.375, 33.525)
c + d = Zero{Float64}
a + b + d = Zero{Float64}
12345a = {Float64}(10.759, 35.387)```

### Julia 1.x compatible version

Adds assertion checks for points to be on the curve.

```using Printf
import Base.in

struct EllipticCurve{T <: AbstractFloat}
a::T
b::T
EllipticCurve(a::T, b::T) where T <: AbstractFloat = new{T}(a, b)
end

in(c::EllipticCurve, x, y) = (x == Inf || y == Inf || y^2 ≈ x^3 + c.a * x + c.b)

const Curve07 = EllipticCurve(0.0, 7.0)

struct EPoint{T <: AbstractFloat}
x::T
y::T
curve::EllipticCurve
end
EPoint{T}() where T <: AbstractFloat = EPoint{T}(Inf, Inf, Curve07)
EPoint() = EPoint{Float64}()
function EPoint(x, y, c::EllipticCurve=Curve07)
@assert in(c, x, y)
EPoint(x, y, c)
end

Base.show(io::IO, p::EPoint{T}) where T = iszero(p) ? print(io, "Zero{\$T}") : @printf(io, "{%s}(%.3f, %.3f)", T, p.x, p.y)
Base.copy(p::EPoint) = EPoint(p.x, p.y, p.curve)
Base.iszero(p::EPoint{T}) where T = p.x in (-Inf, Inf, p.curve)
Base.:-(p::EPoint) = EPoint(p.x, -p.y, p.curve)

function dbl(p::EPoint{T}) where T
iszero(p) && return p

L = 3p.x ^ 2 / 2p.y
x = L ^ 2 - 2p.x
y = L * (p.x - x) - p.y
return EPoint{T}(x, y, p.curve)
end

function Base.:(==)(a::EPoint{T}, C::EPoint{T}) where T
@assert a.curve == C.curve
return (iszero(a) && iszero(C)) || (a.x == C.x && a.y == C.y)
end

function Base.:+(p::EPoint{T}, q::EPoint{T}) where T
@assert p.curve == q.curve
p == q && return dbl(p)
iszero(p) && return q
iszero(q) && return p

L = (q.y - p.y) / (q.x - p.x)
x = L ^ 2 - p.x - q.x
y = L * (p.x - x) - p.y
return EPoint{T}(x, y, p.curve)
end

function Base.:*(p::EPoint, n::Integer)
r = EPoint()
i = 1
while i ≤ n
if i & n != 0 r += p end
p = dbl(p)
i <<= 1
end
return r
end
Base.:*(n::Integer, p::EPoint) = p * n

const C = 7.0

function EPoint(y::AbstractFloat)
n = y ^ 2 - C
x = n ≥ 0 ? n ^ (1 / 3) : -((-n) ^ (1 / 3))
return EPoint{typeof(y)}(x, y, Curve07)
end

a = EPoint(1.0)
b = EPoint(2.0)
@show a b
@show c = a + b
@show d = -c
@show c + d
@show a + b + d
@show 12345a
```

Output: Same as the original, Julia 0.6 version code.

## Kotlin

Translation of: C
```// version 1.1.4

const val C = 7

class Pt(val x: Double, val y: Double) {
val zero get() = Pt(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY)

val isZero get() = x > 1e20 || x < -1e20

fun dbl(): Pt {
if (isZero) return this
val l = 3.0 * x * x / (2.0 * y)
val t = l * l - 2.0 * x
return Pt(t, l * (x - t) - y)
}

operator fun unaryMinus() = Pt(x, -y)

operator fun plus(other: Pt): Pt {
if (x == other.x && y == other.y) return dbl()
if (isZero) return other
if (other.isZero) return this
val l = (other.y - y) / (other.x - x)
val t = l * l - x - other.x
return Pt(t, l * (x - t) - y)
}

operator fun times(n: Int): Pt {
var r: Pt = zero
var p = this
var i = 1
while (i <= n) {
if ((i and n) != 0) r += p
p = p.dbl()
i = i shl 1
}
return r
}

override fun toString() =
if (isZero) "Zero" else "(\${"%.3f".format(x)}, \${"%.3f".format(y)})"
}

fun Double.toPt() = Pt(Math.cbrt(this * this - C), this)

fun main(args: Array<String>) {
val a = 1.0.toPt()
val b = 2.0.toPt()
val c = a + b
val d = -c
println("a         = \$a")
println("b         = \$b")
println("c = a + b = \$c")
println("d = -c    = \$d")
println("c + d     = \${c + d}")
println("a + b + d = \${a + b + d}")
println("a * 12345 = \${a * 12345}")
}
```
Output:
```a         = (-1.817, 1.000)
b         = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c    = (10.375, 33.525)
c + d     = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)
```

## Nim

Translation of: C
```import math, strformat

const B = 7

type Point = tuple[x, y: float]

#---------------------------------------------------------------------------------------------------

template zero(): Point =
(Inf, Inf)

#---------------------------------------------------------------------------------------------------

func isZero(pt: Point): bool {.inline.} =
pt.x > 1e20 or pt.x < -1e20

#---------------------------------------------------------------------------------------------------

func `-`(pt: Point): Point {.inline.} =
(pt.x, -pt.y)

#---------------------------------------------------------------------------------------------------

func double(pt: Point): Point =

if pt.isZero: return pt

let t = (3 * pt.x * pt.x) / (2 * pt.y)
result.x = t * t - 2 * pt.x
result.y = t * (pt.x - result.x) - pt.y

#---------------------------------------------------------------------------------------------------

func `+`(pt1, pt2: Point): Point =

if pt1.x == pt2.x and pt1.y == pt2.y: return double(pt1)
if pt1.isZero: return pt2
if pt2.isZero: return pt1

let t = (pt2.y - pt1.y) / (pt2.x - pt1.x)
result.x = t * t - pt1.x - pt2.x
result.y = t * (pt1.x - result.x) - pt1.y

#---------------------------------------------------------------------------------------------------

func `*`(pt: Point; n: int): Point =

result = zero()
var pt = pt
var i = 1

while i <= n:
if (i and n) != 0:
result = result + pt
pt = double(pt)
i = i shl 1

#---------------------------------------------------------------------------------------------------

func `\$`(pt: Point): string =
if pt.isZero: "Zero" else: fmt"({pt.x:.3f}, {pt.y:.3f})"

#---------------------------------------------------------------------------------------------------

func fromY(y: float): Point {.inline.} =
(cbrt(y * y - B), y)

#———————————————————————————————————————————————————————————————————————————————————————————————————

when isMainModule:
let a = fromY(1)
let b = fromY(2)

echo "a = ", a
echo "b = ", b
let c = a + b
echo "c = a + b = ", c
let d = -c
echo "d = -c = ", d
echo "c + d = ", c + d
echo "a + b + d = ", a + b + d
echo "a * 12345 = ", a * 12345
```
Output:
```a = (-1.817, 1.000)
b = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c = (10.375, 33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.388)```

## OCaml

Original version by User:Vanyamil. Some float-precision issues but overall works.

```(* Task : Elliptic_curve_arithmetic *)

(*
Using the secp256k1 elliptic curve (a=0, b=7),
define the addition operation on points on the curve.
Extra credit:   define the full elliptic curve arithmetic
(still not modular, though) by defining a "multiply" function.
*)

(*** Helpers ***)

type ec_point = Point of float * float | Inf

type ec_curve = { a : float; b : float }

(* By default, cube root doesn't work for negative bases *)
let cube_root : float -> float =
let third = 1. /. 3. in
let f x =
if x > 0.
then x ** third
else ~-. (~-. x ** third)
in
f

(* Finds the left-most x on this curve *)
let ec_minx ({a; b} : ec_curve) : float =
let factor = ~-. b *. 0.5 in
let discr = (factor ** 2.) +. (a ** 3. /. 27.) in
if discr <= 0.
then failwith "Not a simple curve"
else
let root = sqrt discr in
cube_root (factor +. root) +. cube_root (factor -. root)

(* Negates the point by negating y coord *)
let ec_neg : ec_point -> ec_point = function
| Inf -> Inf
| Point (x, y) -> Point (x, ~-. y)

(*** Actual task at hand ***)

(* Generates a random point in the vicinity of x=0 *)
let ec_random ({a; b} as c : ec_curve) : ec_point =
let minx = ec_minx c in
let x = Random.float (~-. minx *. 2.) +. minx in
let rhs = x ** 3. +. a *. x +. b in
Point (x, sqrt rhs)

(*  Verifies that the point is on curve.
Due to rounding errors, sometimes these calculations aren't perfect.
*)
let on_curve ?(debug : bool = false) ({a; b} : ec_curve) : ec_point -> bool = function
| Inf -> true
| Point (x, y) ->
let lhs = y *. y in
let rhs = x ** 3. +. a *. x +. b in
let delta = abs_float (lhs -. rhs) in
(
if debug then Printf.printf "Delta = %.8f" delta;
delta < 0.000001
)

(* Doubles a point on the curve (adds a point to itself) *)
let ec_double ({a; b} as c : ec_curve) : ec_point -> ec_point = function
| Inf -> Inf
| Point (x, y) as p ->
if not (on_curve c p)
then failwith "Point not on this curve."
else if y = 0.
then Inf
else
let s = (3. *. x *. x +. a) /. (2. *. y) in
let x' = s *. s -. 2. *. x in
let y' = y +. s *. (x' -. x) in
Point (x', -. y')

(* Adds any two points on the curve *)
let ec_add ({a; b} as c : ec_curve) (p : ec_point) (q : ec_point) : ec_point =
match p, q with
| Inf, x | x, Inf -> x
| Point (px, py), Point (qx, qy) ->
if not (on_curve c p) || not (on_curve c q)
then failwith "Point not on this curve."
else if abs_float (px -. qx) < 0.000001 then
begin
if abs_float (py +. qy) < 0.000001
then Inf
else
(* py must equal qy here, otherwise something goes real bad *)
ec_double c p |> ec_neg
end
else
let s = (py -. qy) /. (px -. qx) in
let rx = s *. s -. px -. qx in
let ry = py +. s *. (rx -. px) in
Point (rx, -. ry)

(* Extra credit : multiplies a point by a scalar *)
let ec_mul ({a; b} as c : ec_curve) (p : ec_point) (n : int) : ec_point =
let rec helper n curPow acc =
if n = 0 then acc
else
let doubled = ec_double c curPow in
if n mod 2 = 0
then helper (n / 2) doubled acc
else helper (n / 2) doubled (ec_add c acc curPow)
in
helper n p Inf

(*** Output ***)

let string_of_point : ec_point -> string = function
| Inf -> "Zero"
| Point (x, y) -> Printf.sprintf "(%.4f, %.4f)" x y

let print_output () =
let c = { a = 0.; b = 7. } in
let p = ec_random c in
let q = ec_random c in
let r = ec_add c p q in
let t = ec_neg r in
Printf.printf "p = %s\n" (string_of_point p);
Printf.printf "q = %s\n" (string_of_point q);
Printf.printf "r = p + q = %s\n" (string_of_point r);
Printf.printf "t = -r = %s\n" (string_of_point t);
Printf.printf "r + t = %s\n" (ec_add c r t |> string_of_point);
Printf.printf "p + (q + t) = %s\n" (ec_add c q t |> ec_add c p |> string_of_point);
Printf.printf "p * 12345 = %s\n" (ec_mul c p 12345 |> string_of_point)

let _ =
print_output ();
print_output ()
```
Output:
```p = (-0.0489, 2.6457)
q = (-0.0236, 2.6457)
r = p + q = (0.0726, -2.6458)
t = -r = (0.0726, 2.6458)
r + t = Zero
p + (q + t) = Zero
p * 12345 = (-1.7905, -1.1224)

p = (1.2715, 3.0092)
q = (1.6370, 3.3745)
r = p + q = (-1.9103, 0.1696)
t = -r = (-1.9103, -0.1696)
r + t = Zero
p + (q + t) = Zero
p * 12345 = (-0.3948, -2.6341)
```

## PARI/GP

The examples were borrowed from C, though the coding is built-in for GP and so not ported.

```e=ellinit([0,7]);
a=[-6^(1/3),1]
b=[-3^(1/3),2]
d=ellneg(e,c)
ellmul(e,a,12345)```
Output:
```%1 = [-1.8171205928321396588912117563272605024, 1]
%2 = [-1.4422495703074083823216383107801095884, 2]
%3 = [10.375375389201411959219947350723254093, -33.524509096269714974732957666465317961]
%4 = [10.375375389201411959219947350723254093, 33.524509096269714974732957666465317961]
%5 = [0]
%6 = [0]
%7 = [10.758570529079026647817660298097473136, 35.387434773095871032744887640370612568]```

## Perl

Translation of: C
```package EC;
{
our (\$A, \$B) = (0, 7);
package EC::Point;
sub new { my \$class = shift; bless [ @_ ], \$class }
sub zero { bless [], shift }
sub x { shift->[0] }; sub y { shift->[1] };
sub double {
my \$self = shift;
return \$self unless @\$self;
my \$L = (3 * \$self->x**2) / (2*\$self->y);
my \$x = \$L**2 - 2*\$self->x;
bless [ \$x, \$L * (\$self->x - \$x) - \$self->y ], ref \$self;
}
'==' => sub { my (\$p, \$q) = @_; \$p->x == \$q->x and \$p->y == \$q->y },
'+' => sub {
my (\$p, \$q) = @_;
return \$p->double if \$p == \$q;
return \$p unless @\$q;
return \$q unless @\$p;
my \$slope = (\$q->y - \$p->y) / (\$q->x - \$p->x);
my \$x = \$slope**2 - \$p->x - \$q->x;
bless [ \$x, \$slope * (\$p->x - \$x)  - \$p->y ], ref \$p;
},
q{""} => sub {
my \$self = shift;
return @\$self
? sprintf "EC-point at x=%f, y=%f", @\$self
: 'EC point at infinite';
}
}

package Test;
my \$p = +EC::Point->new(-(\$EC::B - 1)**(1/3), 1);
my \$q = +EC::Point->new(-(\$EC::B - 4)**(1/3), 2);
my \$s = \$p + \$q, "\n";
print "\$_\n" for \$p, \$q, \$s;
print "check alignment... ";
print abs((\$q->x - \$p->x)*(-\$s->y - \$p->y) - (\$q->y - \$p->y)*(\$s->x - \$p->x)) < 0.001
? "ok" : "wrong";
```
Output:
```EC-point at x=-1.817121, y=1.000000
EC-point at x=-1.442250, y=2.000000
EC-point at x=10.375375, y=-33.524509
check alignment... ok
```

## Phix

Translation of: C
```with javascript_semantics
constant X=1, Y=2, bCoeff=7, INF = 1e300*1e300

type point(object pt)
return sequence(pt) and length(pt)=2 and atom(pt[X]) and atom(pt[Y])
end type

function zero()
point pt = {INF, INF}
return pt
end function

function is_zero(point p)
return p[X]>1e20 or p[X]<-1e20
end function

function neg(point p)
p = {p[X], -p[Y]}
return p
end function

function dbl(point p)
point r = p
if not is_zero(p) then
atom L = (3*p[X]*p[X])/(2*p[Y])
atom x = L*L-2*p[X]
r = {x, L*(p[X]-x)-p[Y]}
end if
return r
end function

if p==q then return dbl(p) end if
if is_zero(p) then return q end if
if is_zero(q) then return p end if
atom L = (q[Y]-p[Y])/(q[X]-p[X])
atom x = L*L-p[X]-q[X]
point r = {x, L*(p[X]-x)-p[Y]}
return r
end function

function mul(point p, integer n)
point r = zero()
integer i = 1
while i<=n do
if and_bits(i, n) then r = add(r, p) end if
p = dbl(p)
i = i*2
end while
return r
end function

procedure show(string s, point p)
puts(1, s&iff(is_zero(p)?"Zero\n":sprintf("(%.3f, %.3f)\n", p)))
end procedure

function cbrt(atom c)
return iff(c>=0?power(c,1/3):-power(-c,1/3))
end function

function from_y(atom y)
point r = {cbrt(y*y-bCoeff), y}
return r
end function

point a, b, c, d

a = from_y(1)
b = from_y(2)
d = neg(c)

show("a = ", a)
show("b = ", b)
show("c = a + b = ", c)
show("d = -c = ", d)
show("c + d = ", add(c, d))
show("a * 12345 = ", mul(a, 12345))
```
Output:
```a = (-1.817, 1.000)
b = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c = (10.375, 33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)
```

## PicoLisp

```(scl 16)
(setq *B 7)
(de from_y (Y)
(let
(A (* 1.0 (- (* Y Y) *B))
B (pow (abs A) (*/ 1.0 1.0 3.0)) )
(list
(if (gt0 A) B (- B))
(* Y 1.0) ) ) )
(de prn (P)
(if (is_zero P)
"Zero"
(pack
(round (car P) 3)
" "
(round (cadr P) 3) ) ) )
(de neg (P)
(list (car P) (*/ -1.0 (cadr P) 1.0)) )
(de is_zero (P)
(or
(=T (car P))
(> (length (car P)) 20) ) )
(de dbl (P)
(if (is_zero P)
P
(let
(Y
(*/
1.0
(*/ 3.0 (car P) (car P) (** 1.0 2))
(*/ 2.0 (cadr P) 1.0) )
X
(-
(*/ Y Y 1.0)
(*/ 2.0 (car P) 1.0) ) )
(list
X
(-
(*/ Y (- (car P) X) 1.0)
(cadr P) ) ) ) ) )
(cond
((= A B) (dbl A))
((is_zero A) B)
((is_zero B) A)
(T
(let Z (- (car B) (car A))
(if (=0 Z)
(list T T)
(let
X
(- (*/ Y Y 1.0) (car A) (car B)) )
(list
X
(-
(*/ Y (- (car A) X) 1.0)
(cadr A) ) ) ) ) ) ) ) )
(de mul (P N)
(let R (list T T)
(for (I 1 (>= N I) (* I 2))
(when (gt0 (& I N))
(setq R (add R P)) )
(setq P (dbl P)) )
R ) )
(setq
A (from_y 1)
B (from_y 2) )
(prinl "A: " (prn A))
(prinl "B: " (prn B))
(prinl "C: " (prn C))
(setq D (neg C))
(prinl "D: " (prn D))
(prinl "D + C: " (prn (add C D)))
(prinl "A + B + D: " (prn (add A (add B D))))
(prinl "A * 12345: " (prn (mul A 12345)))```
Output:
```A: -1.817 1.000
B: -1.442 2.000
C: 10.375 -33.525
D: 10.375 33.525
D + C: Zero
A + B + D: Zero
A * 12345: 10.759 35.387
```

## Python

Translation of: C
```#!/usr/bin/env python3

class Point:
b = 7
def __init__(self, x=float('inf'), y=float('inf')):
self.x = x
self.y = y

def copy(self):
return Point(self.x, self.y)

def is_zero(self):
return self.x > 1e20 or self.x < -1e20

def neg(self):
return Point(self.x, -self.y)

def dbl(self):
if self.is_zero():
return self.copy()
try:
L = (3 * self.x * self.x) / (2 * self.y)
except ZeroDivisionError:
return Point()
x = L * L - 2 * self.x
return Point(x, L * (self.x - x) - self.y)

if self.x == q.x and self.y == q.y:
return self.dbl()
if self.is_zero():
return q.copy()
if q.is_zero():
return self.copy()
try:
L = (q.y - self.y) / (q.x - self.x)
except ZeroDivisionError:
return Point()
x = L * L - self.x - q.x
return Point(x, L * (self.x - x) - self.y)

def mul(self, n):
p = self.copy()
r = Point()
i = 1
while i <= n:
if i&n:
p = p.dbl()
i <<= 1
return r

def __str__(self):
return "({:.3f}, {:.3f})".format(self.x, self.y)

def show(s, p):
print(s, "Zero" if p.is_zero() else p)

def from_y(y):
n = y * y - Point.b
x = n**(1./3) if n>=0 else -((-n)**(1./3))
return Point(x, y)

# demonstrate
a = from_y(1)
b = from_y(2)
show("a =", a)
show("b =", b)
show("c = a + b =", c)
d = c.neg()
show("d = -c =", d)
show("a * 12345 =", a.mul(12345))
```
Output:
```a = (-1.817, 1.000)
b = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c = (10.375, 33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)
```

## Racket

```#lang racket
(define a 0) (define b 7)
(define (ε? x) (<= (abs x) 1e-14))
(define (== p q) (for/and ([pi p] [qi q]) (ε? (- pi qi))))
(define zero #(0 0))
(define (zero? p) (== p zero))
(define (neg p) (match-define (vector x y) p) (vector x (- y)))
(define (⊕ p q)
(cond [(== q (neg p)) zero]
[else
(match-define (vector px py) p)
(match-define (vector qx qy) q)
(define (done λ px py qx)
(define x (- (* λ λ) px qx))
(vector x (- (+ (* λ (- x px)) py))))
(cond [(and (== p q) (ε? py)) zero]
[(or (== p q) (ε? (- px qx)))
(done (/ (+ (* 3 px px) a) (* 2 py)) px py qx)]
[(done (/ (- py qy) (- px qx)) px py qx)])]))
(define (⊗ p n)
(cond [(= n 0)       zero]
[(= n 1)       p]
[(= n 2)       (⊕ p p)]
[(negative? n) (neg (⊗ p (- n)))]
[(even? n)     (⊗ (⊗ p (/ n 2)) 2)]
[(odd? n)      (⊕ p (⊗ p (- n 1)))]))
```

Test:

```(define (root3 x) (* (sgn x) (expt (abs x) 1/3)))
(define (y->point y) (vector (root3 (- (* y y) b)) y))
(define p (y->point 1))
(define q (y->point 2))
(displayln (~a "p = " p))
(displayln (~a "q = " q))
(displayln (~a "p+q = " (⊕ p q)))
(displayln (~a "-(p+q) = " (neg (⊕ p q))))
(displayln (~a "(p+q)+(-(p+q)) = " (⊕ (⊕ p q) (neg (⊕ p q)))))
(displayln (~a "p+(q+(-(p+q))) = 0 " (zero? (⊕ p (⊕ q (neg (⊕ p q)))))))
(displayln (~a "p*12345 " (⊗ p 12345)))
```

Output:

```p = #(-1.8171205928321397 1)
q = #(-1.4422495703074083 2)
p+q = #(10.375375389201409 -33.524509096269696)
-(p+q) = #(10.375375389201409 33.524509096269696)
(p+q)+(-(p+q)) = #(0 0)
p+(q+(-(p+q))) = 0 #t
p*12345 #(10.758570529320806 35.387434774282106)
```

## Raku

(formerly Perl 6)

```unit module EC;
our (\$A, \$B) = (0, 7);

role Horizon { method gist { 'EC Point at horizon' } }
class Point {
has (\$.x, \$.y);
multi method new(
\$x, \$y where \$y**2 ~~ \$x**3 + \$A*\$x + \$B
) { self.bless(:\$x, :\$y) }
multi method new(Horizon \$) { self.bless but Horizon }
method gist { "EC Point at x=\$.x, y=\$.y" }
}

multi prefix:<->(Point \$p) { Point.new: x => \$p.x, y => -\$p.y }
multi prefix:<->(Horizon \$) { Horizon }
multi infix:<->(Point \$a, Point \$b) { \$a + -\$b }

multi infix:<+>(Horizon \$, Point \$p) { \$p }
multi infix:<+>(Point \$p, Horizon) { \$p }

multi infix:<*>(Point \$u, Int \$n) { \$n * \$u }
multi infix:<*>(Int \$n, Horizon) { Horizon }
multi infix:<*>(0, Point) { Horizon }
multi infix:<*>(1, Point \$p) { \$p }
multi infix:<*>(2, Point \$p) {
my \$l = (3*\$p.x**2 + \$A) / (2 *\$p.y);
my \$y = \$l*(\$p.x - my \$x = \$l**2 - 2*\$p.x) - \$p.y;
\$p.bless(:\$x, :\$y);
}
multi infix:<*>(Int \$n where \$n > 2, Point \$p) {
2 * (\$n div 2 * \$p) + \$n % 2 * \$p;
}

multi infix:<+>(Point \$p, Point \$q) {
if \$p.x ~~ \$q.x {
return \$p.y ~~ \$q.y ?? 2 * \$p !! Horizon;
}
else {
my \$slope = (\$q.y - \$p.y) / (\$q.x - \$p.x);
my \$y = \$slope*(\$p.x - my \$x = \$slope**2 - \$p.x - \$q.x) - \$p.y;
return \$p.new(:\$x, :\$y);
}
}

say my \$p = Point.new: x => \$_, y => sqrt(abs(\$_**3 + \$A*\$_ + \$B)) given 1;
say my \$q = Point.new: x => \$_, y => sqrt(abs(\$_**3 + \$A*\$_ + \$B)) given 2;
say my \$s = \$p + \$q;

say "checking alignment:  ", abs (\$p.x - \$q.x)*(-\$s.y - \$q.y) - (\$p.y - \$q.y)*(\$s.x - \$q.x);
```
Output:
```EC Point at x=1, y=2.8284271247461903
EC Point at x=2, y=3.872983346207417
EC Point at x=-1.9089023002066448, y=0.21008487055753378
checking alignment:  0```

## REXX

REXX doesn't have any higher math functions, so a cube root   (cbrt)   function was included here as well as a
general purpose root   (and accompanying rootG, and rootI)   functions.

Also, some code was added to have the output better aligned   (for instance, negative and positive numbers).

```/*REXX program defines (for any 2 points on the curve), returns the sum of the 2 points.*/
numeric digits 100                               /*try to ensure a min. of accuracy loss*/
a= func(1)             ;            say  '    a =              '          show(a)
b= func(2)             ;            say  '    b =              '          show(b)
c= add(a, b)           ;            say  '    c = (a+b)       ='          show(c)
d= neg(c)              ;            say  '    d = -c          ='          show(d)
e= add(c, d)           ;            say  '    e = (c+d)       ='          show(e)
g= add(a, add(b, d))   ;            say  '    g = (a+b+d)     ='          show(g)
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
cbrt:  procedure; parse arg x;                                            return root(x,3)
conv:  procedure; arg z; if isZ(z)  then return 'zero'; return left('',z>=0)format(z,,5)/1
root:  procedure; parse arg x,y;  if x=0 | y=1  then return x/1;   d=5;   return rootI()/1
rootG: parse value format(x,2,1,,0)  'E0'  with  ? 'E' _ .;    return (?/y'E'_ %y) + (x>1)
func:  procedure; parse arg y,k;  if k=='' then k=7;           return cbrt(y**2-k) y
inf:                                                           return '1e' || (digits()%2)
isZ:   procedure; parse arg px . ;                             return abs(px) >= inf()
neg:   procedure; parse arg px py;                             return px         (-py)
show:  procedure; parse arg x  y ;                             return conv(x)  conv(y)
zero:                                                          return inf()    inf()
/*──────────────────────────────────────────────────────────────────────────────────────*/
add:   procedure; parse arg px py, qx qy;     if px=qx & py=qy  then return dbl(px py)
if isZ(px  py)  then return qx qy;     if isZ(qx qy)     then return     px py
z= qx - px;       if z=0  then do;     \$= inf();         rx= inf();         end
else do;     \$= (qy-py) / z;   rx= \$*\$ - px - qx; end
ry= \$ * (px-rx) - py;                                         return  rx  ry
/*──────────────────────────────────────────────────────────────────────────────────────*/
dbl:   procedure;  parse arg px py;      if isZ(px py)  then return px py;        z= py+py
if z=0  then \$= inf()
else \$= (3*px*py) / (py+py)
rx= \$*\$ - px*px;      ry= \$ * (px-rx) - py;       return rx ry
/*──────────────────────────────────────────────────────────────────────────────────────*/
rootI: ox=x;  oy=y;  x=abs(x);  y=abs(y);  a=digits()+5;  numeric form;  g=rootG();  m=y-1
do  until  d==a;    d=min(d+d,a);               numeric digits d;          o=0
do  until o=g;    o=g;   g=format((m*g**y+x)/y/g**m,,d-2);  end  /*until o=g*/
end  /*until d==a*/;       _=g*sign(ox);   if oy<0  then _=1/_;         return _
```
output:
```    a =               -1.81712  1
b =               -1.44225  2
c = (a+b)       =  10.37538 -33.52451
d = -c          =  10.37538  33.52451
e = (c+d)       = zero zero
g = (a+b+d)     = zero zero
```

## Sage

Examples from C, using the built-in Elliptic curves library.

```Ellie = EllipticCurve(RR,[0,7]) # RR = field of real numbers

# a point (x,y) on Ellie, given y
def point ( y) :
x = var('x')
x = (y^2 - 7 - x^3).roots(x,ring=RR,multiplicities = False)[0]
P = Ellie([x,y])
return P

print(Ellie)
P = point(1)
print('P',P)
Q = point(2)
print('Q',Q)
S = P+Q
print('S = P + Q',S)
print('P+Q-S', P+Q-S)
print('P*12345' ,P*12345)
```
Output:
```Elliptic Curve defined by y^2 = x^3 + 7.00000000000000 over Real Field
with 53 bits of precision

P (-1.81712059283214 : 1.00000000000000 : 1.00000000000000)
Q (-1.44224957030741 : 2.00000000000000 : 1.00000000000000)
S = P + Q (10.3753753892014 : -33.5245090962697 : 1.00000000000000)
P+Q-S (0.000000000000000 : 1.00000000000000 : 0.000000000000000) ## Zero
P*12345 (10.7585721817304 : 35.3874428812067 : 1.00000000000000)
```

## Sidef

Translation of: Raku
```module EC {

var A = 0
var B = 7

class Horizon {
method to_s {
"EC Point at horizon"
}

method *(_) {
self
}

method -(_) {
self
}
}

class Point(Number x, Number y) {
method to_s {
"EC Point at x=#{x}, y=#{y}"
}

method neg {
Point(x, -y)
}

method -(Point p) {
self + -p
}

method +(Point p) {

if (x == p.x) {
return (y == p.y ? self*2 : Horizon())
}
else {
var slope = (p.y - y)/(p.x - x)
var x2 = (slope**2 - x - p.x)
var y2 = (slope * (x - x2) - y)
Point(x2, y2)
}
}

method +(Horizon _) {
self
}

method *((0)) {
Horizon()
}

method *((1)) {
self
}

method *((2)) {
var l = (3 * x**2 + A)/(2 * y)
var x2 = (l**2 - 2*x)
var y2 = (l * (x - x2) - y)
Point(x2, y2)
}

method *(Number n) {
2*(self * (n>>1)) + self*(n % 2)
}
}

class Horizon {
method +(Point p) {
p
}
}

class Number {
method +(Point p) {
p + self
}
method *(Point p) {
p * self
}
method *(Horizon h) {
h
}
method -(Point p) {
-p + self
}
}
}

say var p = with(1) {|v| EC::Point(v, sqrt(abs(1 - v**3 - EC::A*v - EC::B))) }
say var q = with(2) {|v| EC::Point(v, sqrt(abs(1 - v**3 - EC::A*v - EC::B))) }
say var s = (p + q)

say ("checking alignment:  ", abs((p.x - q.x)*(-s.y - q.y) - (p.y - q.y)*(s.x - q.x)) < 1e-20)
```
Output:
```EC Point at x=1, y=2.64575131106459059050161575363926042571025918308
EC Point at x=2, y=3.74165738677394138558374873231654930175601980778
EC Point at x=-1.79898987322333068322364213893577309997540625528, y=0.421678696849803028974882458314430376814790014487
checking alignment:  true
```

## Tcl

Translation of: C
```set C 7
set zero {x inf y inf}
proc tcl::mathfunc::cuberoot n {
# General power operator doesn't like negative, but its defined for root3
expr {\$n>=0 ? \$n**(1./3) : -((-\$n)**(1./3))}
}
proc iszero p {
dict with p {}
return [expr {\$x > 1e20 || \$x<-1e20}]
}
proc negate p {
dict set p y [expr {-[dict get \$p y]}]
}
proc double p {
if {[iszero \$p]} {return \$p}
dict with p {}
set L [expr {(3.0 * \$x**2) / (2.0 * \$y)}]
set rx [expr {\$L**2 - 2.0 * \$x}]
set ry [expr {\$L * (\$x - \$rx) - \$y}]
return [dict create x \$rx y \$ry]
}
if {[dict get \$p x]==[dict get \$q x] && [dict get \$p y]==[dict get \$q y]} {
return [double \$p]
}
if {[iszero \$p]} {return \$q}
if {[iszero \$q]} {return \$p}

dict with p {}
set L [expr {([dict get \$q y]-\$y) / ([dict get \$q x]-\$x)}]
dict set r x [expr {\$L**2 - \$x - [dict get \$q x]}]
dict set r y [expr {\$L * (\$x - [dict get \$r x]) - \$y}]
return \$r
}
proc multiply {p n} {
set r \$::zero
for {set i 1} {\$i <= \$n} {incr i \$i} {
if {\$i & int(\$n)} {
}
set p [double \$p]
}
return \$r
}
```

Demonstrating:

```proc show {s p} {
if {[iszero \$p]} {
puts "\${s}Zero"
} else {
dict with p {}
puts [format "%s(%.3f, %.3f)" \$s \$x \$y]
}
}
proc fromY y {
global C
dict set r x [expr {cuberoot(\$y**2 - \$C)}]
dict set r y [expr {double(\$y)}]
}

set a [fromY 1]
set b [fromY 2]
show "a = " \$a
show "b = " \$b
show "c = a + b = " [set c [add \$a \$b]]
show "d = -c = " [set d [negate \$c]]
show "c + d = " [add \$c \$d]
show "a + b + d = " [add \$a [add \$b \$d]]
show "a * 12345 = " [multiply \$a 12345]
```
Output:
```a = (-1.817, 1.000)
b = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c = (10.375, 33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)
```

## V (Vlang)

Translation of: go
```import math

const b_coeff = 7

struct Pt {
x f64
y f64
}

fn zero() Pt {
return Pt{math.inf(1), math.inf(1)}
}

fn is_zero(p Pt) bool {
return p.x > 1e20 || p.x < -1e20
}

fn neg(p Pt) Pt {
return Pt{p.x, -p.y}
}

fn dbl(p Pt) Pt {
if is_zero(p) {
return p
}
l := (3 * p.x * p.x) / (2 * p.y)
x := l*l - 2*p.x
return Pt{
x: x,
y: l*(p.x-x) - p.y,
}
}

fn add(p Pt, q Pt) Pt {
if p.x == q.x && p.y == q.y {
return dbl(p)
}
if is_zero(p) {
return q
}
if is_zero(q) {
return p
}
l := (q.y - p.y) / (q.x - p.x)
x := l*l - p.x - q.x
return Pt{
x: x,
y: l*(p.x-x) - p.y,
}
}

fn mul(mut p Pt, n int) Pt {
mut r := zero()
for i := 1; i <= n; i <<= 1 {
if i&n != 0 {
}
p = dbl(p)
}
return r
}

fn show(s string, p Pt) {
print("\$s")
if is_zero(p) {
println("Zero")
} else {
println("(\${p.x:.3f}, \${p.y:.3f})")
}
}

fn from_y(y f64) Pt {
return Pt{
x: math.cbrt(y*y - b_coeff),
y: y,
}
}

fn main() {
mut a := from_y(1)
b := from_y(2)
show("a = ", a)
show("b = ", b)
show("c = a + b = ", c)
d := neg(c)
show("d = -c = ", d)
show("c + d = ", add(c, d))
show("a * 12345 = ", mul(mut a, 12345))
}```
Output:
```a = (-1.817, 1.000)
b = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c = (10.375, 33.525)
c + d = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)
```

## Wren

Translation of: C
Library: Wren-fmt
```import "/fmt" for Fmt

var C = 7

class Pt {
static zero { Pt.new(1/0, 1/0) }

construct new(x, y) {
_x = x
_y = y
}

x { _x }
y { _y }

static fromNum(n) { Pt.new((n*n - C).cbrt, n) }

isZero { x > 1e20 || x < -1e20 }

double {
if (isZero) return this
var l = 3 * x * x / (2 * y)
var t = l*l - 2*x
return Pt.new(t, l*(x - t) - y)
}

- { Pt.new(x, -y) }

+(other) {
if (other.type != Pt) Fiber.abort("Argument must be a Pt object.")
if (x == other.x && y == other.y) return double
if (isZero) return other
if (other.isZero) return this
var l = (other.y - y) / (other.x - x)
var t = l*l - x - other.x
return Pt.new(t, l*(x-t) - y)
}

*(n) {
if (n.type != Num || !n.isInteger) {
Fiber.abort("Argument must be an integer.")
}
var r = Pt.zero
var p = this
var i = 1
while (i <= n) {
if ((i & n) != 0) r = r + p
p = p.double
i = i << 1
}
return r
}

toString { isZero ? "Zero" : Fmt.swrite("(\$0.3f, \$0.3f)", x, y) }
}

var a = Pt.fromNum(1)
var b = Pt.fromNum(2)
var c = a + b
var d = -c
System.print("a         = %(a)")
System.print("b         = %(b)")
System.print("c = a + b = %(c)")
System.print("d = -c    = %(d)")
System.print("c + d     = %(c + d)")
System.print("a + b + d = %(a + b + d)")
System.print("a * 12345 = %(a * 12345)")
```
Output:
```a         = (-1.817, 1.000)
b         = (-1.442, 2.000)
c = a + b = (10.375, -33.525)
d = -c    = (10.375, 33.525)
c + d     = Zero
a + b + d = Zero
a * 12345 = (10.759, 35.387)
```

## zkl

Translation of: C
```const C=7, INFINITY=(0.0).inf;

fcn zero{ T(INFINITY, INFINITY) }

// should be INFINITY, but numeric precision is very much in the way
fcn is_zero(p){ x,_:=p; (x < -1e20 or x > 1e20) }

fcn neg(p){ return(p[0], -p[1]) }
fcn dbl(p){
if(is_zero(p)) return(p);

px,py := p;
L:=(3.0 * px * px) / (2.0 * py);
rx,ry := L * L - 2.0 * px, L * (px - rx) - py;
return(rx,ry);
}

px,py := p;
qx,qy := q;
if(px == qx and py == qy) return(dbl(p));
if(is_zero(p)) return(q);
if(is_zero(q)) return(p);

L := (qy - py) / (qx - px);
rx,ry := L * L - px - qx, L * (px - rx) - py;
return(rx,ry);
}

fcn mul(p,n){
r := zero();

i:=1; while(i <= n){
p = dbl(p);
i*=2;
}
r
}```
```fcn show(str,p)
{ println(str, is_zero(p) and "Zero" or "(%.3f, %.3f)".fmt(p.xplode())) }

fcn from_y(y){
y3:=y * y - C;  // cube root of -6 --> -1.817
return(y3.abs().pow(1.0/3) * y3.sign, y)
}

a,b := from_y(1.0), from_y(2.0);
show("a = ", a);
show("b = ", b);
show("c = a + b = ", c := add(a, b));
show("d = -c = ",    d := neg(c));
show("c + d = ",     add(c, d));
```a = (-1.817, 1.000)