Posit numbers/decoding

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 (syntax error): {\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 (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 \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

""" Posit number, a quotient of integers, variable size and exponent length """
struct PositType3{T<:Integer}
    numbits::UInt16
    es::UInt16
    bits::T
    PositType3(nb, ne, i) = new{typeof(i)}(UInt16(nb), UInt16(ne), i)
end

""" Convert PositType3 to Rational. See also posithub.org/docs/Posits4.pdf """
function Base.Rational(p::PositType3)
    s = signbit(signed(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 or if s 1 error
    s && (pabs = (-p.bits) << 1)             # If p is negative, flip to 2's complement
    expsign = signbit(signed(pabs))          # Exponent sign from 2nd bit now MSB 
    r = expsign == 1 ? leading_ones(pabs) : leading_zeros(pabs) # r regime R size
    k = expsign ? r - 1 : -r                 # k for the exponent calculation
    pabs <<= (r + 1)                         # Shift off unwanted R bits
    pabs >>= (r + 2)                         # Shift back for E, F
    fsize = p.numbits - 1 - r - 1 - p.es     # Check how many F bits explicit
    e = fsize < 1 ? pabs : pabs >> fsize     # Get E value
    f = fsize < 1 ? 1 // 1 : 1 + (pabs & (2^fsize - 1)) // 2^fsize # Get F value
    pw = 2^p.es * k + e
    return pw >= 0 ? (-1)^s * f * big"2"^pw // 1 : (-1)^s * f // big"2"^(-pw)
end

@show Rational(PositType3(16, 3, 0b0000110111011101)) == 477 // 134217728
const tests = [
    (16, 3, 0b0000110111011101),
    (16, 3, 0b1000000000000000),
    (16, 3, 0b0000000000000000),
    (16, 1, 0b0110110010101000),
    (16, 1, 0b1001001101011000),
    (16, 2, 0b0000000000000001),
    (16, 0, 0b0111111111111111),
    (16, 6, 0b0111111111111110),
    (8, 1, 0b01000000),
    (8, 1, 0b11000000),
    (8, 1, 0b00110000),
    (8, 1, 0b00100000),
    (8, 2, 0b00000001),
    (8, 2, 0b01111111),
    (8, 7, 0b01111110),
    (32, 2, 0b00000000000000000000000000000001),
    (32, 2, 0b01111111111111111111111111111111),
    (32, 5, 0b01111111111111111111111111111110),
]

for t in tests
    r = Rational(PositType3(t...))
    println(string(t[3], base = 2, pad = t[1]), " => $r = ", float(r))
end

Output:

Rational(PositType3(16, 3, 0x0ddd)) == 477 // 134217728 = true
0000110111011101 => 477//134217728 = 3.553926944732666015625e-06
1000000000000000 => 1//0 = Inf
0000000000000000 => 0//1 = 0.0
0110110010101000 => 405//32 = 12.65625
1001001101011000 => -405//32 = -12.65625
0000000000000001 => 1//72057594037927936 = 1.387778780781445675529539585113525390625e-17
0111111111111111 => 16384//1 = 16384.0
0111111111111110 => 28638903918474961204418783933674838490721739172170652529441449702311064005352904159345284265824628375429359509218999720074396860757073376700445026041564579620512874307979212102266801261478978776245040008231745247475930553606737583615358787106474295296//1 = 2.86389039184749612044187839336748384907217391721706525294414497023110640053529e+250
01000000 => 1//1 = 1.0
11000000 => -1//1 = -1.0
00110000 => 1//2 = 0.5
00100000 => 1//4 = 0.25
00000001 => 1//16777216 = 5.9604644775390625e-08
01111111 => 16777216//1 = 1.6777216e+07
01111110 => 4562440617622195218641171605700291324893228507248559930579192517899275167208677386505912811317371399778642309573594407310688704721375437998252661319722214188251994674360264950082874192246603776//1 = 4.562440617622195218641171605700291324893228507248559930579192517899275167208677e+192
00000000000000000000000000000001 => 1//1329227995784915872903807060280344576 = 7.523163845262640050999913838222372338039459563341360137656010920181870460510254e-37
01111111111111111111111111111111 => 1329227995784915872903807060280344576//1 = 1.329227995784915872903807060280344576e+36
01111111111111111111111111111110 => 2269007733883335972287082669296112915239349672942191252221331572442536403137824056312817862695551072066953619064625508194663368599769448406663254670871573830845597595897613333042429214224697474472410882236254024057110212260250671521235807709272244389361641091086035023229622419456//1 = 2.269007733883335972287082669296112915239349672942191252221331572442536403137824e+279

Mathematica

Using code written by John Gustafson himself. Reproduced here with minor, mainly stylistic modifications (mostly to convert graphical symbols to their text equivalent). License is included as legally required by copyright laws.

(* Copyright © 2017 John L. Gustafson

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sub-license, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

This copyright and permission notice shall be included in all copies or substantial portions of the software.

THE SOFTWARE IS PROVIDED "AS IS" WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT
LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN
NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES, OR OTHER LIABILITY,
WHETHER IN AN ACTION OR CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*)

setpositenv[{n_Integer /; n >= 2, e_Integer /; e >= 0}] := (
   {nbits, es} = {n, e};
   npat = 2^nbits;
   useed = 2^2^es;
   {minpos, maxpos} = {useed^(-nbits + 2), useed^(nbits - 2)};
   qsize = Power[2, Ceiling[Log[2, (nbits - 2) 2^(es + 2) + 5]]];
   qextra = qsize - (nbits - 2) 2^(es + 2);
)
positQ[p_Integer] := 0 <= p < npat
twoscomp[sign_, p_] := Mod[If[sign > 0, p, npat - p], npat]
signbit[p_ /; positQ[p]] := IntegerDigits[p, 2, nbits][[1]]
regimebits[p_ /; positQ[p]] := 
 Module[
  {q = twoscomp[1 - signbit[p], p], bits, bit2, npower, tempbits},
  bits = IntegerDigits[q, 2, nbits];
  bit2 = bits[[2]]; (* Look for the run length after the sign bit. *)

  tempbits = Join[Drop[bits, 1], {1 - bit2}]; (* Drop the sign bit, 
  but append a complement bit as a sure-fire way to end the run. *)
  
  npower = (Position[tempbits, 1 - bit2, 1, 1])[[1]] - 1; (* 
  Find first opposite bit. *)
  
  Take[bits, {2, Min[npower + 1, nbits]}]
]
regimevalue[bits_] := If[bits[[1]] == 1, Length[bits] - 1, -Length[bits]]
exponentbits[p_ /; positQ[p]] := 
 Module[{q = twoscomp[1 - signbit[p], p], bits, startbit},
  startbit = Length[regimebits[q]] + 3;
  bits = IntegerDigits[q, 2, nbits];
  If[startbit > nbits, {}, 
   Take[bits, {startbit, Min[startbit + es - 1, nbits]}]]]
fractionbits[p_ /; positQ[p]] := 
 Module[{q = twoscomp[1 - signbit[p], p], bits, startbit},
  startbit = Length[regimebits[q]] + 3 + es;
  bits = IntegerDigits[q, 2, nbits];
  If[startbit > nbits, {}, Take[bits, {startbit, nbits}]]]
p2x[p_ /; positQ[p]] := 
 Module[{s = (-1)^signbit[p], k = regimevalue[regimebits[p]], 
   e = exponentbits[p], f = fractionbits[p]},
  e = Join[e, Table[0, es - Length[e]]]; (* 
  Pad with 0s on the right if they are clipped off. *)
  
  e = FromDigits[e, 2];
  If[f == {}, f = 1, f = 1 + FromDigits[f, 2] 2^(-Length[f])];
  Which[
   p == 0, 0,
   p == npat/2, ComplexInfinity, (* The two exception values, 
   0 and \[PlusMinus]\[Infinity] *)
   True, s useed^k*2^e*f]
]
setpositenv[{16, 3}];
p2x @ 2^^0000110111011101 // TraditionalForm

Output:

477/134217728

Phix

with javascript_semantics
function twos_compliment_2_on(string bits, integer nbits)
    for i=2 to nbits do
        bits[i] = iff(bits[i]='0'?'1':'0')
    end for
    for i=nbits to 2 by -1 do
        if bits[i]='0' then
            bits[i] = '1'
            exit
        end if
        bits[i] = '0'
    end for
    return bits
end function

function posit_decode(integer nbits, es, object bits)
    --
    -- nbits: number of bits (aka n)
    -- es: exponent scale
    -- bits: (binary) integer or string of nbits 0|1
    --
    if not string(bits) then
        string fmt = sprintf("%%0%db",nbits)
        bits = sprintf(fmt,bits)
    end if
    assert(length(bits)==nbits)
    string ibits = bits -- save for return
    integer s = bits[1]='1'
    if s then bits = twos_compliment_2_on(bits,nbits) end if
    integer r = find(xor_bits(bits[2],1),bits,3)-2,
            b2z = bits[2]='0', exponent = 0, fraction = 0
    atom fs = 1, useed = power(2,power(2,es))
    if r<0 then
        if b2z then
            if s then
                return {ibits,es,"NaR"} -- aka inf
            end if
            return {ibits,es,"zero"}
        end if
        r = nbits-1
    else
        integer estart = r+3,
               efinish = min(r+2+es,nbits)
        exponent = to_integer(bits[estart..efinish],0,2)
        fraction = to_integer(bits[efinish+1..$],0,2)
        fs = power(2,nbits-efinish)
    end if
    integer k = iff(b2z?-r:r-1)
    atom res = iff(s?-1:+1)*power(useed,k)*power(2,exponent)*(1+fraction/fs)
    return {ibits,es,res}
end function

constant tests = {{16,3,0b0000110111011101},
                  {16,3,0b1000000000000000},
                  {16,3,0b0000000000000000},
                  {16,1,0b0110110010101000},
                  {16,1,0b1001001101011000},
                  {16,2,0b0000000000000001},
                  {16,0,0b0111111111111111},
                  {16,2,0b0111111111111111},
                  {16,6,0b0111111111111110},
                  {8,1,0b01000000},
                  {8,1,0b11000000},
                  {8,1,0b00110000},
                  {8,1,0b00100000},
                  {8,2,0b00000001},
                  {8,2,0b01111111},
                  {8,7,0b01111110},
                  {32,2,0b00000000000000000000000000000001},
                  {32,2,0b01111111111111111111111111111111},
                  {32,5,0b01111111111111111111111111111110}}
for t in tests do
    printf(1,"%s (es=%d) ==> %v\n",call_func(posit_decode,t))
end for

Output:

0000110111011101 (es=3) ==> 3.553926944e-6
1000000000000000 (es=3) ==> "NaR"
0000000000000000 (es=3) ==> "zero"
0110110010101000 (es=1) ==> 12.65625
1001001101011000 (es=1) ==> -12.65625
0000000000000001 (es=2) ==> 1.38777878e-17
0111111111111111 (es=0) ==> 16384
0111111111111111 (es=2) ==> 7.205759404e+16
0111111111111110 (es=6) ==> 2.863890392e+250
01000000 (es=1) ==> 1
11000000 (es=1) ==> -1
00110000 (es=1) ==> 0.5
00100000 (es=1) ==> 0.25
00000001 (es=2) ==> 5.960464478e-8
01111111 (es=2) ==> 16777216
01111110 (es=7) ==> 4.562440618e+192
00000000000000000000000000000001 (es=2) ==> 7.523163846e-37
01111111111111111111111111111111 (es=2) ==> 1.329227996e+36
01111111111111111111111111111110 (es=5) ==> 2.269007734e+279

Raku

Translation of: Mathematica

=begin LICENSE
Copyright © 2017 John L. Gustafson

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sub-license, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

This copyright and permission notice shall be included in all copies or substantial portions of the software.

THE SOFTWARE IS PROVIDED "AS IS" WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT
LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN
NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES, OR OTHER LIABILITY,
WHETHER IN AN ACTION OR CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
=end LICENSE

constant nbits = 16;
constant es    = 3;
constant npat  = 2**nbits;
constant useed = 2**2**es;
constant minpos = useed**(-nbits + 2);
constant maxpos = useed**(+nbits - 2);
constant qsize  = 2**((nbits-2)*2**(es+2)+5).log2.ceiling;
constant qextra = qsize - (nbits - 2)*2**(es+2);

constant posit-range = 0..^2**nbits;

sub twoscomp($sign, $p) { ($sign > 0 ?? $p !! npat - $p) mod npat } 

sub sign-bit(UInt $p where posit-range) { +$p.polymod(2 xx nbits - 1).tail }
sub regime-bits(UInt $p where posit-range) {
  my $q = twoscomp(1 - sign-bit($p), $p);
  my @bits = $q.polymod(2 xx nbits - 1).reverse;
  my $bit2 = @bits[1];
  my @temp-bits = flat @bits[1..*], 1 - $bit2;
  my $npower = @temp-bits.first(1 - $bit2, :k) - 1;
  @bits[1..($npower+1)];
}
sub regime-value(@bits) { @bits.head ?? @bits.elems - 1 !! -@bits; }
sub exponent-bits(UInt $p where posit-range) {
  my $q = twoscomp(1 - sign-bit($p), $p);
  my $startbit = regime-bits($q).elems + 3;
  my @bits = $q.polymod(2 xx nbits - 1).reverse;
  @bits[$startbit-1 .. $startbit-1 + es - 1]
}
sub fraction-bits(UInt $p where posit-range) {
  my $q = twoscomp(1 - sign-bit($p), $p);
  my $startbit = regime-bits($q).elems + 3 + es;
  my @bits = $q.polymod(2 xx nbits - 1).reverse;
  @bits[$startbit-1 .. *];
}

sub p2x(UInt $p where posit-range) {
  my $s = (-1)**sign-bit($p);
  my $k = regime-value regime-bits $p;
  my @e = exponent-bits $p;
  my @f = fraction-bits $p;
  @e.push: 0 until @e == es;
  my $e = @e.reduce: 2 * * + *;
  my $f = @f == 0 ?? 1 !! 1.FatRat + @f.reduce(2 * * + *)/2**@f;
  given $p {
    when 0          { 0   }
    when npat div 2 { Inf }
    default         { $s * useed**$k * 2**$e * $f }
  }
}

dd p2x 0b0000110111011101;

Output:

FatRat.new(477, 134217728)

Wren

Library: Wren-fmt

Library: Wren-big

import "./fmt" for Conv, Fmt
import "./big" for BigRat, BigInt

var positDecode = 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) ? BigRat.zero : Conv.infinity
    }

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

var tests = [
    [3, "0000110111011101"],
    [3, "1000000000000000"],
    [3, "0000000000000000"],
    [1, "0110110010101000"],
    [1, "1001001101011000"],
    [2, "0000000000000001"],
    [0, "0111111111111111"],
    [6, "0111111111111110"],
    [1, "01000000"],
    [1, "11000000"],
    [1, "00110000"],
    [1, "00100000"],
    [2, "00000001"],
    [2, "01111111"],
    [7, "01111110"],
    [2, "00000000000000000000000000000001"],
    [2, "01111111111111111111111111111111"],
    [5, "01111111111111111111111111111110"]
]

for (test in tests) {
    var res = positDecode.call(test[1], test[0])
    var res2 = (res is BigRat) ? res.toFloat : Num.infinity
    Fmt.print("$s(es = $d) -> $s or $n", test[1], test[0], res, res2)
}

Output:

0000110111011101(es = 3) -> 477/134217728 or 3.5539269447327e-06
1000000000000000(es = 3) -> ∞ or infinity
0000000000000000(es = 3) -> 0/1 or 0
0110110010101000(es = 1) -> 405/32 or 12.65625
1001001101011000(es = 1) -> -405/32 or -12.65625
0000000000000001(es = 2) -> 1/72057594037927936 or 1.3877787807814e-17
0111111111111111(es = 0) -> 16384/1 or 16384
0111111111111110(es = 6) -> 28638903918474961204418783933674838490721739172170652529441449702311064005352904159345284265824628375429359509218999720074396860757073376700445026041564579620512874307979212102266801261478978776245040008231745247475930553606737583615358787106474295296/1 or 2.8638903918475e+250
01000000(es = 1) -> 1/1 or 1
11000000(es = 1) -> -1/1 or -1
00110000(es = 1) -> 1/2 or 0.5
00100000(es = 1) -> 1/4 or 0.25
00000001(es = 2) -> 1/16777216 or 5.9604644775391e-08
01111111(es = 2) -> 16777216/1 or 16777216
01111110(es = 7) -> 4562440617622195218641171605700291324893228507248559930579192517899275167208677386505912811317371399778642309573594407310688704721375437998252661319722214188251994674360264950082874192246603776/1 or 4.5624406176222e+192
00000000000000000000000000000001(es = 2) -> 1/1329227995784915872903807060280344576 or 7.5231638452626e-37
01111111111111111111111111111111(es = 2) -> 1329227995784915872903807060280344576/1 or 1.3292279957849e+36
01111111111111111111111111111110(es = 5) -> 2269007733883335972287082669296112915239349672942191252221331572442536403137824056312817862695551072066953619064625508194663368599769448406663254670871573830845597595897613333042429214224697474472410882236254024057110212260250671521235807709272244389361641091086035023229622419456/1 or 2.2690077338833e+279