Fuzion • Tutorial • Floating-Point Types

Floating-Point Types

Introduction

Digital computers are great, but when it comes to computing numbers with arbitrary precision there is a problem. There is no way of representing every possible number as this would mean that a number might take up an infinite amount of space. Thankfully, clever engineers invented floating-point values which offer a reasonable trade-off between range and precision.

History of floating-point

According to wikipedia the invention of floating-point values dates back to as early as 1914 when Leonardo Torres y Quevedo proposed an early form of floating-point values for a calculator design. The Z1 built by Konrad Zuse in 1938 was the first freely programmable computer and it already had full support for 24-bit floating-point values! The Z3 completed in 1941 even had representations for +/-∞.
Because there are many subtleties in floating-point numbers without one obvious way of doing things, by the 1980s there where many incompatible floating-point implementations. This was the source of many problems and lead to the standardization of floating-points in IEEE 754

(Binary) floating-point numbers - a refresher

The way computers store (binary) floating-point numbers is by breaking them up into the sign, n digits of the fraction and an exponent which is stored in m bits. The floating-point number thus needs 1 sing bit + n fraction bits + m exponent bits of space. Special values that floating-point numbers can represent are +/-zero, +/-∞ and NaN (not a number)

In IEEE754, every operation (+-*/...) is always allowed for floating-points even e.g. dividing by zero. The result of dividing by zero can be NaN or ∞. A NaN is always unequal to any other number including itself. If numbers get too big to be represented by the floating-point number, the result will be ∞.

Because of the nature how floating-point numbers are stored in computers, the density of floating-point numbers decreases the further you are from zero. The result of this is that within the range of a floating-point the relative error will always be less than a certain epsilon.

32-bit floats

If the type of a float is not inferable it is assumed to be 64-bit. That is why we have to call f32 with 4.0 in below example.

four := f32 4.0;
π := f32.π;
say("square root of 4 is {four.sqrt}");
say("sinus of π/2 = {(π/2).sin}");
say("π = $π");
say("1/0 = {(f32 1.0)/0}");
say("0/0 = {(f32 0.0)/0}");

1
2
3
4
5
6
7
8
four := f32 4.0;
π := f32.π;
say("square root of 4 is {four.sqrt}");
say("sinus of π/2 = {(π/2).sin}");
say("π = $π");
say("1/0 = {(f32 1.0)/0}");
say("0/0 = {(f32 0.0)/0}");
 
 
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

What are effects?

64-bit floats

π := f64.π;
say("square root of 4 is {4.0.sqrt}");
say("sinus of π/2 = {(π/2).sin}");
say("π = $π");
say("1/0 = {1.0/0}");
say("0/0 = {0.0/0}");

1
2
3
4
5
6
7
π := f64.π;
say("square root of 4 is {4.0.sqrt}");
say("sinus of π/2 = {(π/2).sin}");
say("π = $π");
say("1/0 = {1.0/0}");
say("0/0 = {0.0/0}");
 
 
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

What are effects?

16-bit floats

Type f16 is currently not supported.

128-bit floats

Type f128 is currently not supported.

Mandelbrot example

mandelbrotexample is
  is_in_mandelbrot_set(c num.complex f64, max_escape_iterations i32, z num.complex f64) =>
    max_escape_iterations = 0 || z.abs² <= 4 && is_in_mandelbrot_set c max_escape_iterations-1 z*z+c

  steps(start, step f64) => start : steps start+step step

  mandelbrot_image(y_start, y_step, x_start, x_step f64, height, width i32) =>
    for _ in 1..height; y in steps y_start y_step do
      for _ in 1..width; x in steps x_start x_step do
        if is_in_mandelbrot_set (num.complex x y) 50 (num.complex 0.0 0.0)
          yak "#"
        else
          yak " "
      say ""

  mandelbrot_image 1 -0.05 -2 0.0315 40 80

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
mandelbrotexample is
  is_in_mandelbrot_set(c num.complex f64, max_escape_iterations i32, z num.complex f64) =>
    max_escape_iterations = 0 || z.abs² <= 4 && is_in_mandelbrot_set c max_escape_iterations-1 z*z+c
  steps(start, step f64) => start : steps start+step step
  mandelbrot_image(y_start, y_step, x_start, x_step f64, height, width i32) =>
    for _ in 1..height; y in steps y_start y_step do
      for _ in 1..width; x in steps x_start x_step do
        if is_in_mandelbrot_set (num.complex x y) 50 (num.complex 0.0 0.0)
          yak "#"
        else
          yak " "
      say ""
  mandelbrot_image 1 -0.05 -2 0.0315 40 80
 
 
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

What are effects?

last changed: 2024-06-28

next: Numeric Overflows