Posit numbers/decoding

Posit numbers/decoding is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Posit is a quantization of the real projective line proposed by John Gustafson in 2015. It is claimed to be an improvement over IEEE 754.

The purpose of this task is to write a program capable of decoding a posit number. You will use the example provided by Gustafson in his paper : 0b0000110111011101, representing a 16-bit long real number with three bits for the exponent. Once decoded, you should obtain either the fraction 477/134217728 or the floating point value 3.55393E−6.

Jeff Johnson from Facebook research, described posit numbers as such:

A more efficient representation for tapered floating points is the recent posit format by Gustafson. It has no explicit size field; the exponent is encoded using a Golomb-Rice prefix-free code, with the exponent $e$ encoded as a Golomb-Rice quotient and remainder $(q,r)$ with $q$ in unary and $r$ in binary (in posit terminology, $q$ is the regime). Remainder encoding size is defined by the exponent scale $s$, where $2^{s}$ is the Golomb-Rice divisor. Any space not used by the exponent encoding is used by the significand, which unlike IEEE 754 always has a leading 1; gradual underflow (and overflow) is handled by tapering. A posit number system is characterized by $(N,s)$, where $N$ is the word length in bits and $s$ is the exponent scale. The minimum and maximum positive finite numbers in $(N,s)$ are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_\mathrm{min} = 2^{−(N−2)2^s}} and Failed to parse (syntax error): {\displaystyle f_\mathrm{max} = 2^{(N−2)2^s}} . The number line is represented much as the projective reals, with a single point at $\pm \infty$ bounding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle −f_\mathrm{max}} and $f_{\mathrm {max} }$. $\pm \infty$ and 0 have special encodings; there is no NaN. The number system allows any choice of $N\geq 3$ and Failed to parse (syntax error): {\displaystyle 0\le s\le N − 3} .

$s$ controls the dynamic range achievable; e.g., 8-bit (8, 5)-posit $f_{\mathrm {max} }=2^{192}$ is larger than $f_{\mathrm {max} }$ in float32. (8, 0) and (8, 1) are more reasonable values to choose for 8-bit floating point representations, with $f_{\mathrm {max} }$ of 64 and 4096 accordingly. Precision is maximized in the range Failed to parse (syntax error): {\displaystyle \pm\left[2^{−(s+1)}, 2^{s+1}\right)} with Failed to parse (syntax error): {\displaystyle N − 3 − s} significand fraction bits, tapering to no fraction bits at $\pm f_{\mathrm {max} }$.

— Jeff Johnson, Rethinking floating point for deep learning, Facebook research.

Julia

struct PositType3{T<:Integer}
    numbits::UInt16
    es::UInt16
    bits::T
    PositType3(nb, ne, i) = new{typeof(i)}(UInt16(nb), UInt16(ne), i)
end

""" From posithub.org/docs/Posits4.pdf """
function Base.Rational(p::PositType3)
    s = signbit(p.bits)                      # s for S signbit, is 1 if negative
    pabs = p.bits << 1                       # shift off signbit (adds a 0 to F at LSB)
    pabs == 0 && return s ? 1 // 0 : 0 // 1  # if p is 0, return 0 if s = 0, error if s = 1
    expsign = signbit(pabs)                  # exponent sign from 2nd bit now in MSB location
    k = expsign == 1 ? leading_ones(pabs) : leading_zeros(pabs) # regime R bit count
    scaling = 2^p.es * (expsign == 0 ? -1 : 1)
    pabs <<= (k + 1)                         # shift off unwanted R bits
    pabs >>= (k + 2)                         # shift back without the extra LSB bit
    fsize = p.numbits - k - p.es - 2         # check how many F bits are actually explicit
    f = fsize > 0 ? (pabs & (2^fsize - 1)) // 2^fsize : 0 # Get F value. Can be missing -> 0
    e =
        fsize > 0 ? (pabs >> (fsize)) :      # Get E value. Can be up to p.es bits
        pabs * 2^(p.es - p.numbits - k - 2)  # implicit missing bits correction
    pw = (1 - 2s) * (scaling * k + e + s)
    return pw >= 0 ? ((1 - 3s) + f) * 2^pw // 1 : ((1 - 3s) + f) // 2^(-pw)
end

@show Rational(PositType3(16, 3, 0b0000110111011101)) == 477 // 134217728

Output:

 Rational(PositType3(16, 3, 0x0ddd)) == 477 // 134217728 = true

raku

unit role Posit[UInt $N, UInt $es];

has UInt $.UInt;
method sign { self.UInt > 2**($N - 1) ?? -1 !! +1 }

method FatRat {
  return 0   if self.UInt == 0;
  my UInt $mask = 2**($N - 1);
  return Inf if self.UInt == $mask;
  my UInt $n = self.UInt;
  my $sign = $n +& $mask ?? -1 !! +1;
  my $r = $sign;
  $n = ((2**$n - 1) +^ $n) + 1 if self.sign < 0;
  my int $count = 0;
  $mask +>= 1;
  my Bool $first-bit = ?($n +& $mask);
  repeat { $count++; $mask +>= 1;
  } while ?($n +& $mask) == $first-bit && $mask;
  my $m = $count;
  my $k = $first-bit ?? $m - 1 !! -$m;
  $r *= 2**($k*2**$es);
  return $r unless $mask > 1;
  $mask +>= 1;
  $count = 0;
  my UInt $exponent = 0;
  while $mask && $count++ < $es {
    $exponent +<= 1;
    $exponent +|= 1 if $n +& $mask;
    $mask +>= 1;
  }
  $r *= 2**$exponent;
  my $fraction = 1.FatRat;
  while $mask {
    (state $power-of-two = 1) +<= 1;
    $fraction += 1/$power-of-two if $n +& $mask;
    $mask +>= 1;
  }
  $r *= $fraction;

  return $r;
}

CHECK {
  use Test;
  # example from L<http://www.johngustafson.net/pdfs/BeatingFloatingPoint.pdf>
  is Posit[16, 3]
    .new(UInt => 0b0000110111011101)
    .FatRat, 477.FatRat/134217728;
}

Output:

ok 1 -

Wren

Library: Wren-fmt

Library: Wren-big

import "./fmt" for Conv
import "./big" for BigRat

var posit16_decode = Fn.new { |ps, maxExpSize|
    var p = ps.map { |c| c == "0" ? 0 : 1 }.toList

    // Deal with exceptional values.
    if (p[1..-1].all { |i| i == 0 }) {
        return (p[0] == 0) ? 0 : Conv.infinity
    }

    // Convert bits after sign bit to two's complement if negative.
    if (p[0] == 1) {
        for (i in 1..15) p[i] = (p[i] == 0) ? 1 : 0
        for (i in 15..1) {
            if (p[i] == 1) {
                p[i] = 0
            } else {
                p[i] = 1
                break
            }
        }
    }
    var first = p[1]
    var rs = 15  // regime size
    for (i in 2..15) {
        if (p[i] != first) {
            rs = i - 1
            break
        }
    }
    var regime = p[1..rs]
    var es = (rs == 15) ? 0 : maxExpSize.min(14-rs)  // actual exponent size
    var exponent = [0]
    if (es > 0) exponent = p[rs + 2...rs + 2 + es]
    var fs = (es == 0) ? 0 : 14 - rs - es  // function size
    var s = (p[0] == 0) ? 1 : -1  // sign
    var k = regime.all { |i| i == 0 } ? -rs : rs - 1
    var u = 2.pow(2.pow(maxExpSize))
    var e = Conv.atoi(exponent.join(""), 2)
    var f = BigRat.one
    if (fs > 0) {
        var fraction = ps[-fs..-1]
        f = Conv.atoi(fraction.join(""), 2)
        f = BigRat.one + BigRat.new(f, 2.pow(fs))
    }
    return f * BigRat.new(u, 1).pow(k) * s * 2.pow(e)
}

var ps = "0000110111011101"
System.print(posit16_decode.call(ps, 3))

Output:

477/134217728