Fraction Types

Fuzion supports fraction types with all the standard numeric operations.

Fraction Example

Here is a small example declaring two fractions and performing basic arithmetic operations on them:

calc(T type : numeric, a, b T) unit =>
{
  say("$a + $b = {a + b}")
  say("$a - $b = {a - b}")
  say("$b - $a = {b - a}")
  say("$a * $b = {a * b}")
  say("$a / $b = {a / b}")
}

calc(1 ⁄ 3, 1 ⁄ 4)

calc(24 ⁄ 63, 18 ⁄ 84)

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calc(T type : numeric, a, b T) unit =>
{
  say("$a + $b = {a + b}")
  say("$a - $b = {a - b}")
  say("$b - $a = {b - a}")
  say("$a * $b = {a * b}")
  say("$a / $b = {a / b}")
}
calc(1 ⁄ 3, 1 ⁄ 4)
calc(24 ⁄ 63, 18 ⁄ 84)
 
 
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
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What are effects?

Fractions are based on any integer type, in this case on i32.

Overflow handling

Fractions suffer from the value limitations of their underlying integer types. See this example:

f := 1 ⁄ 10000
say(f)
say(f * f)
say(f * f * f)

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f := 1 ⁄ 10000
say(f)
say(f * f)
say(f * f * f)
 
 
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
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What are effects?

Using a larger integer type solves the overflow:

f := 1.as_i64 ⁄ 10000
say(f)
say(f * f)
say(f * f * f)

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f := 1.as_i64 ⁄ 10000
say(f)
say(f * f)
say(f * f * f)
 
 
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
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What are effects?

Using int as the base type avoids overflow altogether:

for
  f := num.fraction(int(1), int(10000)), f * f
  i in (1..5)
do
  {
    say(f)
  }

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for
  f := num.fraction(int(1), int(10000)), f * f
  i in (1..5)
do
  {
    say(f)
  }
 
 
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
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What are effects?

Faulhabers formula

The Faulhaber example from rom rosettacode.org is a nice application of fraction types:

# http://rosettacode.org/wiki/Faulhaber's_formula
faulhaber_example is

  bernoulli (n i32) num.fraction i32
    pre
      n >= 0
  =>
    local_mutate : mutate.
    local_mutate.instate_self (()->
      a := local_mutate.env.new (array n+1 (i -> 1 ⁄ 1))
      for
        m in 0..n
      do
        a <- (a.get.put m (1  ⁄  (m+1)))
        for k in m..1 : -1 do
          a <- (a.get.put (k-1) ((a.get[k - 1] - a.get[k]) * (k ⁄ 1)))
      if n != 1
        a.get[0]
      else
        -a.get[0])


  binomial(n, k i32)
    pre
      n > 0
      k >= 0
    => factorial n / (factorial k * factorial n-k)


  factorial(n i32)
    =>
    for
      res := 1, res * i
      i in 1..n
    do
    else
      res


  faulhaber(p i32) =>
    yak "$p : "
    q := 1  ⁄  (p+1)
    for
      j in 0..p
      sign := 1⁄1, sign * (-1⁄1)
    do
      b := binomial(p+1, j)  ⁄  1
      coeff := q * sign / (1⁄1) * b * bernoulli j
      if coeff != (0⁄1)
        if j = 0
          if coeff = (1⁄1)
          else
            if coeff = (-1⁄1)
              yak "-"
            else
              yak "$coeff"
          else
            if coeff = (1⁄1)
              yak " + "
            else
              if coeff = (-1⁄1)
                yak " - "
              else
                if coeff > (0⁄1)
                  yak " + $coeff"
                else
                  yak " - {-coeff}"
        pwr := p + 1 - j
        if pwr > 1
          yak "n^$pwr"
        else
          yak "n"

  for i in 0..11 do
    faulhaber i
    say ""

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# http://rosettacode.org/wiki/Faulhaber's_formula
faulhaber_example is
  bernoulli (n i32) num.fraction i32
    pre
      n >= 0
  =>
    local_mutate : mutate.
    local_mutate.instate_self (()->
      a := local_mutate.env.new (array n+1 (i -> 1 ⁄ 1))
      for
        m in 0..n
      do
        a <- (a.get.put m (1  ⁄  (m+1)))
        for k in m..1 : -1 do
          a <- (a.get.put (k-1) ((a.get[k - 1] - a.get[k]) * (k ⁄ 1)))
      if n != 1
        a.get[0]
      else
        -a.get[0])
  binomial(n, k i32)
    pre
      n > 0
      k >= 0
    => factorial n / (factorial k * factorial n-k)
  factorial(n i32)
    =>
    for
      res := 1, res * i
      i in 1..n
    do
    else
      res
  faulhaber(p i32) =>
    yak "$p : "
    q := 1  ⁄  (p+1)
    for
      j in 0..p
      sign := 1⁄1, sign * (-1⁄1)
    do
      b := binomial(p+1, j)  ⁄  1
      coeff := q * sign / (1⁄1) * b * bernoulli j
      if coeff != (0⁄1)
        if j = 0
          if coeff = (1⁄1)
          else
            if coeff = (-1⁄1)
              yak "-"
            else
              yak "$coeff"
          else
            if coeff = (1⁄1)
              yak " + "
            else
              if coeff = (-1⁄1)
                yak " - "
              else
                if coeff > (0⁄1)
                  yak " + $coeff"
                else
                  yak " - {-coeff}"
        pwr := p + 1 - j
        if pwr > 1
          yak "n^$pwr"
        else
          yak "n"
  for i in 0..11 do
    faulhaber i
    say ""
 
 
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
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What are effects?

last changed: 2024-06-28

next: Type unit