MATL, 10 bytes

Xr*+Jr4*^*

Inputs are r, then s. Output is a complex number.

Try it online! Or see the plot for 1000 points at MATL Online! (it takes 10‒15 seconds).

How it works

Uses method 2 described in the challenge.

Xr % Push random number with standard Gaussian distribution
* % Implicit input: r. Multiply
+ % Implicit input: s. Add
J % Push imaginary unit
r % Push random number with stantard uniform distribution
4 % Push 4
* % Multiply
^ % Power
* % Multiply. Implicit output

MATL, 10 bytes

Xr*+Jr4*^*

Inputs are r, then s. Output is a complex number.

Try it online! Or see the plot for 1000 points at MATL Online! (it takes 10‒15 seconds).

How it works

Uses method 2 described in the challenge.

Xr % Push random number with standard Gaussian distribution
* % Implicit input: r. Multiply
+ % Implicit input: s. Add
J % Push imaginary unit
r % Push random number with stantard uniform distribution
4 % Push 4
* % Multiply
^ % Power
* % Multiply. Implicit output

MATL, 10 bytes

Xr*+Jr4*^*

Inputs are r, then s. Output is a complex number.

Try it online! Or see the plot for 1000 points at MATL Online! (it takes 10‒15 seconds).

Uses method 2 described in the challenge.

MATL, 10 bytes

Xr*+Jr4*^*

Inputs are r, then s. Output is a complex number.

Try it online! Or see the plot for 1000 points at MATL Online! (it takes 10‒15 seconds).

How it works

Uses method 2 described in the challenge.

Xr % Push random number with standard Gaussian distribution
* % Implicit input: r. Multiply
+ % Implicit input: s. Add
J % Push imaginary unit
r % Push random number with stantard uniform distribution
4 % Push 4
* % Multiply
^ % Power
* % Multiply. Implicit output