Posit numbers/encoding
Posit numbers/encoding 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.
Encode pi as a 8-bit posit with a 2-bit exponent.
As an unsigned integer, the result should be 77.
BASIC
BASIC256
print positEncode(4 * atan(1), 8, 2)
end
function positEncode(x, nbits, es)
npat = 2 ^ nbits
useed = 2 ^ (2 ^ es)
e = 2 ^ (es - 1)
y = abs(x)
if y = 0 then return 0
if y > 1e308 then return 2 ^ (nbits - 1)
if y >= 1 then
p = 1
i = 2
while y >= useed and i < nbits
p = 2 * p + 1
y /= useed
i += 1
end while
p *= 2
i += 1
else
p = 0
i = 1
while y < 1 and i <= nbits
y *= useed
i += 1
end while
if i >= nbits then
p = 2
i = nbits + 1
else
p = 1
i += 1
end if
end if
while e > 0.5 and i <= nbits
p *= 2
if y >= 2 * e then
y /= 2 ^ e
p += 1
end if
e /= 2
i += 1
end while
y -= 1
while y > 0 and i <= nbits
y *= 2
p = 2 * p + int(y)
y -= int(y)
i += 1
end while
p *= 2 ^ (nbits + 1 - i)
i += 1
i = p and 1
p = int(p / 2)
if i <> 0 then
if y = 1 or y = 0 then
p += (p and 1)
else
p += 1
end if
end if
if x < 0 then return ((npat - p) mod npat) else return (p mod npat)
end function
Chipmunk Basic
100 cls
110 sub positencode(x)
120 if nbits = 0 then nbits = 8
130 if es = 0 then es = 2
140 npat = 2^nbits
150 useed = 2^(2^es)
160 e = 2^(es-1)
170 y = abs(x)
180 if y = 0 then positencode = 0
190 if y > 1.000000E+308 then positencode = 2^(nbits-1)
200 if y >= 1 then
210 p = 1
220 i = 2
230 while y >= useed and i < nbits
240 p = 2*p+1
250 y = y/useed
260 i = i+1
270 wend
280 p = p*2
290 i = i+1
300 else
310 p = 0
320 i = 1
330 while y < 1 and i <= nbits
340 y = y*useed
350 i = i+1
360 wend
370 if i >= nbits then
380 p = 2
390 i = nbits+1
400 else
410 p = 1
420 i = i+1
430 endif
440 endif
450 while e > 0.5 and i <= nbits
460 p = p*2
470 if y >= 2*e then
480 y = y/(2^e)
490 p = p+1
500 endif
510 e = e/2
520 i = i+1
530 wend
540 y = y-1
550 while y > 0 and i <= nbits
560 y = y*2
570 p = 2*p+int(y)
580 y = y-int(y)
590 i = i+1
600 wend
610 p = p*(2^(nbits+1-i))
620 i = i+1
630 i = p and 1
640 p = int(p/2)
650 if i <> 0 then
660 if y = 1 or y = 0 then
670 p = p+(p and 1)
680 else
690 p = p+1
700 endif
710 endif
720 if x < 0 then
730 positencode = ((npat-p) mod npat)
740 else
750 positencode = (p mod npat)
760 endif
770 end sub
780 print positencode(4*atan(1))
790 end
FreeBASIC
#define IsInf(x) Abs(x) > 1E308
Function positEncode(x As Double, nbits As Uinteger = 8, es As Uinteger = 2) As Uinteger
Dim As Double npat = 2 ^ nbits
Dim As Double useed = 2 ^ (2 ^ es)
Dim As Double e = 2 ^ (es - 1)
Dim As Double y = Abs(x)
Dim As Uinteger i, p
If y = 0 Then Return 0
If IsInf(y) Then Return 2 ^ (nbits - 1)
If y >= 1 Then
p = 1
i = 2
While y >= useed And i < nbits
p = 2 * p + 1
y /= useed
i += 1
Wend
p *= 2
i += 1
Else
p = 0
i = 1
While y < 1 And i <= nbits
y *= useed
i += 1
Wend
If i >= nbits Then
p = 2
i = nbits + 1
Else
p = 1
i += 1
End If
End If
While e > 0.5 And i <= nbits
p *= 2
If y >= 2 * e Then
y /= 2 ^ e
p += 1
End If
e /= 2
i += 1
Wend
y -= 1
While y > 0 And i <= nbits
y *= 2
p = 2 * p + Int(y)
y -= Int(y)
i += 1
Wend
p *= 2 ^ (nbits + 1 - i)
i += 1
i = p And 1
p = Int(p / 2)
If i <> 0 Then p += Iif((y = 1 Or y = 0), p And 1, 1)
Return Iif(x < 0, npat - p, p) Mod npat
End Function
Print positEncode(4 * Atn(1))
Sleep
- Output:
77
Gambas
Function positEncode(x As Float, nbits As Integer, es As Integer) As Integer
Dim npat As Integer = 2 ^ nbits
Dim useed As Float = 2 ^ (2 ^ es)
Dim e As Float = 2 ^ (es - 1)
Dim y As Float = Abs(x)
Dim i As Integer, p As Integer
If y = 0 Then Return 0
If IsInf(y) Then Return 2 ^ (nbits - 1)
If y >= 1 Then
p = 1
i = 2
While y >= useed And i < nbits
p = 2 * p + 1
y /= useed
i += 1
Wend
p *= 2
i += 1
Else
p = 0
i = 1
While y < 1 And i <= nbits
y *= useed
i += 1
Wend
If i >= nbits Then
p = 2
i = nbits + 1
Else
p = 1
i += 1
End If
End If
While e > 0.5 And i <= nbits
p *= 2
If y >= 2 * e Then
y /= 2 ^ e
p += 1
End If
e /= 2
i += 1
Wend
y -= 1
While y > 0 And i <= nbits
y *= 2
p = 2 * p + Int(y)
y -= Int(y)
i += 1
Wend
p *= 2 ^ (nbits + 1 - i)
i += 1
i = p And 1
p = Int(p / 2)
If i <> 0 Then p += IIf((y = 1 Or y = 0), p And 1, 1)
Return IIf(x < 0, npat - p, p) Mod npat
End Function
Public Sub Main()
Print positEncode(4 * Atn(1), 8, 2)
End
PureBasic
Procedure.i positEncode(x.d, nbits.i = 8, es.i = 2)
Protected npat.i, useed.d, e.d, y.d
Protected i.i, p.i
npat = Pow(2, nbits)
useed = Pow(2, Pow(2, es))
e = Pow(2, es - 1)
y = Abs(x)
If y = 0
ProcedureReturn 0
EndIf
If y > 1E308
ProcedureReturn Pow(2, nbits - 1)
EndIf
If y >= 1
p = 1
i = 2
While y >= useed And i < nbits
p = 2 * p + 1
y / useed
i + 1
Wend
p * 2
i + 1
Else
p = 0
i = 1
While y < 1 And i <= nbits
y * useed
i + 1
Wend
If i >= nbits
p = 2
i = nbits + 1
Else
p = 1
i + 1
EndIf
EndIf
While e > 0.5 And i <= nbits
p * 2
If y >= 2 * e
y / Pow(2, e)
p + 1
EndIf
e / 2
i + 1
Wend
y - 1
While y > 0 And i <= nbits
y * 2
p = 2 * p + Int(y)
y - Int(y)
i + 1
Wend
p * Pow(2, nbits + 1 - i)
i + 1
i = p & 1
p = Int(p / 2)
If i <> 0
If y = 1 Or y = 0
p + (p & 1)
Else
p + 1
EndIf
EndIf
If x < 0
ProcedureReturn (npat - p) % npat
Else
ProcedureReturn p % npat
EndIf
EndProcedure
OpenConsole()
PrintN(Str(positEncode(4 * ATan(1))))
PrintN(#CRLF$ + "Press ENTER to exit"): Input()
CloseConsole()
QBasic
The QB64 solution works without any changes.
QB64
Print positEncode(4 * Atn(1), 8, 2)
Function positEncode (x, nbits, es)
Dim npat As Double, useed As Double
Dim e As Double, y As Double
Dim i As Integer, p As Integer
npat = 2 ^ nbits
useed = 2 ^ (2 ^ es)
e = 2 ^ (es - 1)
y = Abs(x)
If y = 0 Then positEncode = 0: Exit Function
If y > 1E+38 Then positEncode = 2 ^ (nbits - 1): Exit Function
If y >= 1 Then
p = 1
i = 2
While y >= useed And i < nbits
p = 2 * p + 1
y = y / useed
i = i + 1
Wend
p = p * 2
i = i + 1
Else
p = 0
i = 1
While y < 1 And i <= nbits
y = y * useed
i = i + 1
Wend
If i >= nbits Then
p = 2
i = nbits + 1
Else
p = 1
i = i + 1
End If
End If
While e > 0.5 And i <= nbits
p = p * 2
If y >= 2 * e Then
y = y / (2 ^ e)
p = p + 1
End If
e = e / 2
i = i + 1
Wend
y = y - 1
While y > 0 And i <= nbits
y = y * 2
p = 2 * p + Int(y)
y = y - Int(y)
i = i + 1
Wend
p = p * (2 ^ (nbits + 1 - i))
i = i + 1
i = p And 1
p = Int(p / 2)
If i <> 0 Then
If y = 1 Or y = 0 Then
p = p + (p And 1)
Else
p = p + 1
End If
End If
If x < 0 Then positEncode = ((npat - p) Mod npat) Else positEncode = (p Mod npat)
End Function
True BASIC
FUNCTION positencode(x, nbits, es)
LET npat = 2^nbits
LET useed = 2^(2^es)
LET e = 2^(es-1)
LET y = ABS(x)
IF y = 0 THEN
LET positencode = 0
EXIT FUNCTION
END IF
IF y > 1e308 THEN
LET positencode = 2^(nbits-1)
EXIT FUNCTION
END IF
IF y >= 1 THEN
LET p = 1
LET i = 2
DO WHILE y >= useed AND i < nbits
LET p = 2*p+1
LET y = y/useed
LET i = i+1
LOOP
LET p = p*2
LET i = i+1
ELSE
LET p = 0
LET i = 1
DO WHILE y < 1 AND i <= nbits
LET y = y*useed
LET i = i+1
LOOP
IF i >= nbits THEN
LET p = 2
LET i = nbits+1
ELSE
LET p = 1
LET i = i+1
END IF
END IF
DO WHILE e > 0.5 AND i <= nbits
LET p = p*2
IF y >= 2*e THEN
LET y = y/(2^e)
LET p = p+1
END IF
LET e = e/2
LET i = i+1
LOOP
LET y = y-1
DO WHILE y > 0 AND i <= nbits
LET y = y*2
LET p = 2*p+INT(y)
LET y = y-INT(y)
LET i = i+1
LOOP
LET p = p*(2^(nbits+1-i))
LET i = i+1
!LET i = p AND 1
IF REMAINDER(p, 2) = 1 THEN
LET i = 1
ELSE
LET i = 0
END IF
LET p = INT(p/2)
IF i <> 0 THEN
IF y = 1 OR y = 0 THEN
!LET p = p + (p AND 1)
IF REMAINDER(p, 2) = 1 THEN
LET p = p + 1
END IF
ELSE
LET p = p+1
END IF
END IF
IF x < 0 THEN LET positencode = REMAINDER((npat-p),npat) ELSE LET positencode = REMAINDER(p,npat)
END FUNCTION
PRINT positencode(4*ATN(1), 8, 2)
END
Yabasic
print posit_encode(4 * atan(1))
end
sub posit_encode(x, nbits, es)
if not nbits nbits = 8
if not es es = 2
npat = 2 ^ nbits
useed = 2 ^ (2 ^ es)
e = 2 ^ (es - 1)
y = abs(x)
if y = 0 return 0
if y > 1e308 return 2 ^ (nbits - 1)
if y >= 1 then
p = 1
i = 2
while y >= useed and i < nbits
p = 2 * p + 1
y = y / useed
i = i + 1
wend
p = p * 2
i = i + 1
else
p = 0
i = 1
while y < 1 and i <= nbits
y = y * useed
i = i + 1
wend
if i >= nbits then
p = 2
i = nbits + 1
else
p = 1
i = i + 1
fi
fi
while e > 0.5 and i <= nbits
p = p * 2
if y >= 2 * e then
y = y / (2 ^ e)
p = p + 1
fi
e = e / 2
i = i + 1
wend
y = y - 1
while y > 0 and i <= nbits
y = y * 2
p = 2 * p + int(y)
y = y - int(y)
i = i + 1
wend
p = p * (2 ^ (nbits + 1 - i))
i = i + 1
i = p and 1
p = int(p / 2)
if i <> 0 if y = 1 or y = 0 then p = p + (p and 1) else p = p + 1 : fi
if x < 0 then return mod((npat - p), npat) else return mod(p, npat) : fi
end sub
JavaScript
/* 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 .
*/
const nbits = 8
const es = 2
const npat = 1 << nbits
const useed = 1 << (1 << es)
function x2p(x) {
"use strict";
let i, p,
e = 1 << (es - 1),
y = Math.abs(x);
if (y == 0) return 0
if (y == Math.Infinity) return 1 << (nbits - 1)
if (y >= 1) {
p = 1
i = 2
while (y >= useed && i < nbits) {
p = 2 * p + 1
y = y / useed
i = i + 1
}
p = 2 * p
i = i + 1
} else {
p = 0
i = 1
while (y < 1 && i <= nbits) {
y = y * useed
i = i + 1
}
if (i >= nbits) {
p = 2
i = nbits + 1
} else {
p = 1
i = i + 1
}
}
while (e > 0.5 && i <= nbits) {
p = 2 * p
if (y >= 2 * e) {
y = y / (1 << e)
p = p + 1
}
e = e / 2
i = i + 1
}
y = y - 1
while (y > 0 && i <= nbits) {
y = 2 * y
p = 2 * p + Math.floor(y)
y = y - Math.floor(y)
i = i + 1
}
p = p * (1 << (nbits + 1 - i))
i = i + 1
i = p & 1
p = Math.floor((p/2))
if (i != 0) {
if (y == 1 || y == 0) {
p = p + (p & 1)
} else {
p = p + 1
}
}
return (x < 0 ? npat - p : p) % npat
}
console.log(x2p(Math.PI));
- Output:
77
Julia
""" Posit floating point numbers """
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, then F value next line
f = fsize < 1 ? 1 // 1 : big"1" + (pabs & (2^fsize - 1)) // big"2"^fsize
pw = 2^p.es * k + e # pw multiplier, power of 2 exponent
return pw >= 0 ? (-1)^s * f * big"2"^pw // 1 : (-1)^s * f // big"2"^(-pw)
end
""" Get bits representation of a posit of size numbits and from a real number """
function positbits(x::Real, numbits, es)
tindex = Int(round(log2(numbits / 8))) + 1 # choice of output type
1 <= tindex <= 5 || error("Cannot create posit of bit size $numbits")
T = [UInt8, UInt16, UInt32, UInt64, UInt128][tindex]
x == 0 && return zero(T) # bits for 0 if 0, Inf if Inf, etc
x in [-Inf, Inf, NaN] && return typemax(T) - typemax((signed(typemax(T))))
s = x < 0 # sign bit, 1 if negative
xabs = abs(x) # work with abs(x)
useed = 2^es # the useed
pw = Int(floor(log2(xabs))) # xabs = 1.bits.. * 2^pw
k, e = divrem(pw, useed) # from pw = 2^p.es * k + e
if e < 0
k, e = k - 1, e + useed # e must be unsigned
end
r = k < 0 ? -k : k + 1 # r is number of R repetitions
rbits = pw >= 0 ? (2^(r+1)-1) ⊻ 1 : 01 # bit pattern of R portion
fsize = numbits - 1 - r - 1 - es # size of F portion
f = round((xabs / (2^pw) - 1) * 2^fsize) # f (mantissa - 1 as binary digits)
pabs = T(f) | T(e << fsize) | T(BigInt(rbits) << (fsize + es)) # rbits | e | f
return s ? -pabs : pabs # S and two's complement if negative
end
""" Construct various bit sizes of Posit """
posit8(x, es = 2) = PositType3(8, 2, positbits(x, 8, es))
posit16(x, es = 2) = PositType3(16, 2, positbits(x, 16, es))
posit32(x, es = 2) = PositType3(32, 2, positbits(x, 32, es))
posit64(x, es = 2) = PositType3(64, 2, positbits(x, 64, es))
const tests = [0, Inf, 1, -1, π, -π, 10π, -10π]
for t in tests, posit in (posit8, posit16, posit32, posit64)
p = posit(t)
i = signed(p.bits)
ending = BigFloat(Rational(p))
err = Float64(abs(t - ending))
println("\n$t to $(p.numbits)-bit posit is $p.")
println("This posit reinterpreted as integer is $i.")
println("This posit as float is $ending,\n with error $err.")
end
- Output:
0.0 to 8-bit posit is PositType3{UInt8}(0x0008, 0x0002, 0x00).
This posit reinterpreted as integer is 0.
This posit as float is 0.0,
with error 0.0.
0.0 to 16-bit posit is PositType3{UInt16}(0x0010, 0x0002, 0x0000).
This posit reinterpreted as integer is 0.
This posit as float is 0.0,
with error 0.0.
0.0 to 32-bit posit is PositType3{UInt32}(0x0020, 0x0002, 0x00000000).
This posit reinterpreted as integer is 0.
This posit as float is 0.0,
with error 0.0.
0.0 to 64-bit posit is PositType3{UInt64}(0x0040, 0x0002, 0x0000000000000000).
This posit reinterpreted as integer is 0.
This posit as float is 0.0,
with error 0.0.
Inf to 8-bit posit is PositType3{UInt8}(0x0008, 0x0002, 0x80).
This posit reinterpreted as integer is -128.
This posit as float is Inf,
with error NaN.
Inf to 16-bit posit is PositType3{UInt16}(0x0010, 0x0002, 0x8000).
This posit reinterpreted as integer is -32768.
This posit as float is Inf,
with error NaN.
Inf to 32-bit posit is PositType3{UInt32}(0x0020, 0x0002, 0x80000000).
This posit reinterpreted as integer is -2147483648.
This posit as float is Inf,
with error NaN.
Inf to 64-bit posit is PositType3{UInt64}(0x0040, 0x0002, 0x8000000000000000).
This posit reinterpreted as integer is -9223372036854775808.
This posit as float is Inf,
with error NaN.
1.0 to 8-bit posit is PositType3{UInt8}(0x0008, 0x0002, 0x40).
This posit reinterpreted as integer is 64.
This posit as float is 1.0,
with error 0.0.
1.0 to 16-bit posit is PositType3{UInt16}(0x0010, 0x0002, 0x4000).
This posit reinterpreted as integer is 16384.
This posit as float is 1.0,
with error 0.0.
1.0 to 32-bit posit is PositType3{UInt32}(0x0020, 0x0002, 0x40000000).
This posit reinterpreted as integer is 1073741824.
This posit as float is 1.0,
with error 0.0.
1.0 to 64-bit posit is PositType3{UInt64}(0x0040, 0x0002, 0x4000000000000000).
This posit reinterpreted as integer is 4611686018427387904.
This posit as float is 1.0,
with error 0.0.
-1.0 to 8-bit posit is PositType3{UInt8}(0x0008, 0x0002, 0xc0).
This posit reinterpreted as integer is -64.
This posit as float is -1.0,
with error 0.0.
-1.0 to 16-bit posit is PositType3{UInt16}(0x0010, 0x0002, 0xc000).
This posit reinterpreted as integer is -16384.
This posit as float is -1.0,
with error 0.0.
-1.0 to 32-bit posit is PositType3{UInt32}(0x0020, 0x0002, 0xc0000000).
This posit reinterpreted as integer is -1073741824.
This posit as float is -1.0,
with error 0.0.
-1.0 to 64-bit posit is PositType3{UInt64}(0x0040, 0x0002, 0xc000000000000000).
This posit reinterpreted as integer is -4611686018427387904.
This posit as float is -1.0,
with error 0.0.
3.141592653589793 to 8-bit posit is PositType3{UInt8}(0x0008, 0x0002, 0x4d).
This posit reinterpreted as integer is 77.
This posit as float is 3.25,
with error 0.10840734641020688.
3.141592653589793 to 16-bit posit is PositType3{UInt16}(0x0010, 0x0002, 0x4c91).
This posit reinterpreted as integer is 19601.
This posit as float is 3.1416015625,
with error 8.908910206884002e-6.
3.141592653589793 to 32-bit posit is PositType3{UInt32}(0x0020, 0x0002, 0x4c90fdaa).
This posit reinterpreted as integer is 1284570538.
This posit as float is 3.1415926516056060791015625,
with error 1.984187036896401e-9.
3.141592653589793 to 64-bit posit is PositType3{UInt64}(0x0040, 0x0002, 0x4c90fdaa22168c00).
This posit reinterpreted as integer is 5517188450687028224.
This posit as float is 3.141592653589793115997963468544185161590576171875,
with error 0.0.
-3.141592653589793 to 8-bit posit is PositType3{UInt8}(0x0008, 0x0002, 0xb3).
This posit reinterpreted as integer is -77.
This posit as float is -3.25,
with error 0.10840734641020688.
-3.141592653589793 to 16-bit posit is PositType3{UInt16}(0x0010, 0x0002, 0xb36f).
This posit reinterpreted as integer is -19601.
This posit as float is -3.1416015625,
with error 8.908910206884002e-6.
-3.141592653589793 to 32-bit posit is PositType3{UInt32}(0x0020, 0x0002, 0xb36f0256).
This posit reinterpreted as integer is -1284570538.
This posit as float is -3.1415926516056060791015625,
with error 1.984187036896401e-9.
-3.141592653589793 to 64-bit posit is PositType3{UInt64}(0x0040, 0x0002, 0xb36f0255dde97400).
This posit reinterpreted as integer is -5517188450687028224.
This posit as float is -3.141592653589793115997963468544185161590576171875,
with error 0.0.
31.41592653589793 to 8-bit posit is PositType3{UInt8}(0x0008, 0x0002, 0x64).
This posit reinterpreted as integer is 100.
This posit as float is 32.0,
with error 0.5840734641020688.
31.41592653589793 to 16-bit posit is PositType3{UInt16}(0x0010, 0x0002, 0x63db).
This posit reinterpreted as integer is 25563.
This posit as float is 31.421875,
with error 0.00594846410206884.
31.41592653589793 to 32-bit posit is PositType3{UInt32}(0x0020, 0x0002, 0x63da9e8a).
This posit reinterpreted as integer is 1675271818.
This posit as float is 31.415926456451416015625,
with error 7.944651514435463e-8.
31.41592653589793 to 64-bit posit is PositType3{UInt64}(0x0040, 0x0002, 0x63da9e8a554e1780).
This posit reinterpreted as integer is 7195237671651645312.
This posit as float is 31.41592653589793115997963468544185161590576171875,
with error 0.0.
-31.41592653589793 to 8-bit posit is PositType3{UInt8}(0x0008, 0x0002, 0x9c).
This posit reinterpreted as integer is -100.
This posit as float is -32.0,
with error 0.5840734641020688.
-31.41592653589793 to 16-bit posit is PositType3{UInt16}(0x0010, 0x0002, 0x9c25).
This posit reinterpreted as integer is -25563.
This posit as float is -31.421875,
with error 0.00594846410206884.
-31.41592653589793 to 32-bit posit is PositType3{UInt32}(0x0020, 0x0002, 0x9c256176).
This posit reinterpreted as integer is -1675271818.
This posit as float is -31.415926456451416015625,
with error 7.944651514435463e-8.
-31.41592653589793 to 64-bit posit is PositType3{UInt64}(0x0040, 0x0002, 0x9c256175aab1e880).
This posit reinterpreted as integer is -7195237671651645312.
This posit as float is -31.41592653589793115997963468544185161590576171875,
with error 0.0.
Mathematica
John Gustafson's code.
(*
* 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);
)
positableQ[x_] := (Abs[x] == ∞ || x ∈ Reals)
x2p[x_ /; positableQ[x]] := Module[
{i, p, e = 2^(es - 1), y = Abs[x]},
Which[
(* First, take care of the two exception values: *)
y == 0, 0, (* all 0 bits s *)
y == ∞, BitShiftLeft[1, nbits - 1], (* 1 followed by all 0 bits *)
True,
If[
y >= 1, (* Northeast quadrant: *)
p = 1; i = 2; (* Shift in 1s from the right and scale down. *)
While[y >= useed &&
i < nbits, {p, y, i} = {2 p + 1, y/useed, i + 1}];
p = 2 p; i++,
(* Else, southeast quadrant: *)
p = 0; i = 1; (* Shift in 0s from the right and scale up. *)
While[y < 1 && i <= nbits, {y, i} = {y useed, i + 1}];
If[i >= nbits, p = 2; i = nbits + 1, p = 1; i++]
];(* Extract exponent bits: *)
While[e > 1/2 && i <= nbits, p = 2 p;
If[y >= 2^e, y /= 2^e; p++]; e /= 2; i++];
y--; (* Fraction bits; subtract the hidden bit *)
While[y > 0 && i <= nbits, y = 2 y;
p = 2 p + ⌊y⌋;
y -= ⌊y⌋; i++];
p *= 2^(nbits + 1 - i); i++;(* Round to nearest; tie goes to even *)
i = BitAnd[p, 1]; p = ⌊p/2⌋;
p = Which[
i == 0, p, (* closer to lower value *)
y == 1 || y == 0,
p + BitAnd[p, 1], (* tie goes to nearest even *)
True,
p + 1 (* closer to upper value *)];
Mod[If[x < 0, npat - p, p], npat (* Simulate 2's complement *)]
]
]
setpositenv[{8,2}];
x2p @ Pi
- Output:
77
Perl
# 20240928 Perl programming solution
use strict;
use warnings;
use POSIX qw(INFINITY);
my ($npat, $useed) = (1 << (my $nbits = 8), 1 << (1 << (my $es = 2)));
sub x2p {
my $x = shift;
my ($e, $y, $i, $p) = (1 << ($es - 1), abs($x));
return 0 if ($y == 0);
return (1 << ($nbits - 1)) if ($y == INFINITY);
if ($y >= 1) {
($p, $i) = (1, 2);
while ($y >= $useed && $i < $nbits) {
$p += $p + 1;
$y /= $useed;
$i++;
}
$p += $p;
$i++;
} else {
($p, $i) = (0, 1);
while ($y < 1 && $i <= $nbits) {
$y *= $useed;
$i++;
}
if ($i >= $nbits) {
($p, $i) = (2, $nbits + 1);
} else {
$p = 1;
$i++;
}
}
while ($e > 0.5 && $i <= $nbits) {
$p = 2 * $p;
if ($y >= 2 * $e) {
$y /= (1 << $e);
$p++;
}
$e /= 2;
$i++;
}
$y -= 1;
while ($y > 0 && $i <= $nbits) {
$y *= 2;
$p = 2 * $p + int($y);
$y -= int($y);
$i++;
}
$p *= (1 << ($nbits + 1 - $i));
$i++;
my $i_tmp = $p & 1;
$p >>= 1;
if ($i_tmp != 0) { ($y == 1 || $y == 0) ? $p += ($p & 1) : $p++ }
return ($x < 0 ? $npat - $p : $p) % $npat;
}
print x2p(3.14159265358979), "\n";
You may Attempt This Online!
Phix
with javascript_semantics function posit_encode(atom x, integer nbits=8, es=2) atom npat = power(2,nbits), useed = power(2,power(2,es)), e = power(2,es-1), y = abs(x), i, p if y == 0 then return 0 end if if is_inf(y) then return power(2,nbits-1) end if if y>=1 then p = 1 i = 2 while y>=useed and i<nbits do p = 2 * p + 1 y /= useed i += 1 end while p *= 2 i += 1 else p = 0 i = 1 while y<1 and i<=nbits do y *= useed i += 1 end while if i>=nbits then p = 2 i = nbits + 1 else p = 1 i += 1 end if end if while e>0.5 and i<=nbits do p *= 2 if y>=2*e then y /= power(2,e) p += 1 end if e /= 2 i = i + 1 end while y -= 1 while y>0 and i<=nbits do y *= 2 p = 2 * p + floor(y) y -= floor(y) i += 1 end while p = p * power(2,nbits+1-i) i += 1 i = and_bits(p,1) p = floor(p/2) if i!=0 then if y=1 or y=0 then p += and_bits(p,1) else p += 1 end if end if return remainder(iff(x<0 ? npat-p : p),npat) end function ?posit_encode(PI)
- Output:
77
Can also re-encode all the outputs from the decode task perfectly.
Raku
=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
# L<https://posithub.org/docs/Posits4.pdf>
constant nbits = 8;
constant es = 2;
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 = ^npat;
subset posit of UInt where posit-range;
sub x2p(Real $x --> posit) {
# first, take care of the two exceptional values
return 0 if $x == 0;
return npat div 2 if $x == Inf;
# working variables
my ($i, $p, $e, $y) = $, $, 2**(es - 1), $x.abs;
if $y ≥ 1 { # north-east quadrant
($p, $i) = 1, 2;
# Shift in 1s from the right and scale down.
($p, $y, $i) = 2*$p+1, $y/useed, $i+1 while $y ≥ useed && $i ≤ nbits;
$p *= 2; $i++;
} else { # south-east quadrant
($p, $i) = 0, 1;
# Shift in 0 s from the right and scale up.
($y, $i) = $y*useed, $i+1 while $y < 1 && $i ≤ nbits;
if $i ≥ nbits {
$p = 2; $i = nbits + 1;
} else { $p = 1; $i++; }
}
# Extract exponent bits:
while $e > 1/2 and $i ≤ nbits {
$p *= 2;
if $y ≥ 2**$e { $y /= 2**$e; $p++ }
$e /= 2;
$i++;
}
$y--;
# Fraction bits; substract the hidden bit
while $y > 0 and $i ≤ nbits {
$y *= 2;
$p = 2*$p + $y.floor;
$y -= $y.floor;
$i++
}
$p *= 2**(nbits+1-$i);
$i++;
# Round to nearest; tie goes to even
$i = $p +& 1;
$p = ($p/2).floor;
$p = $i == 0 ?? $p !!
$y == 0|1 ?? $p + $p+&1 !!
$p + 1;
# Simulate 2's complement
($x < 0 ?? npat - $p !! $p) mod npat;
}
say x2p pi;
- Output:
77
You may Attempt This Online!
Wren
/* See original Mathematica example for copyright notice and comments. */
var nbits = 8
var es = 2
var npat = 1 << nbits
var useed = 1 << (1 << es)
var x2p = Fn.new { |x|
var i
var p
var e = 1 << (es - 1)
var y = x.abs
if (y == 0) return 0
if (y.isInfinity) return 1 << (nbits - 1)
if (y >= 1) {
p = 1
i = 2
while (y >= useed && i < nbits) {
p = 2 * p + 1
y = y / useed
i = i + 1
}
p = 2 * p
i = i + 1
} else {
p = 0
i = 1
while (y < 1 && i <= nbits) {
y = y * useed
i = i + 1
}
if (i >= nbits) {
p = 2
i = nbits + 1
} else {
p = 1
i = i + 1
}
}
while (e > 0.5 && i <= nbits) {
p = 2 * p
if (y >= 2 * e) {
y = y / (1 << e)
p = p + 1
}
e = e / 2
i = i + 1
}
y = y - 1
while (y > 0 && i <= nbits) {
y = 2 * y
p = 2 * p + y.floor
y = y - y.floor
i = i + 1
}
p = p * (1 << (nbits + 1 - i))
i = i + 1
i = p & 1
p = (p/2).floor
if (i != 0) {
if (y == 1 || y == 0) {
p = p + (p & 1)
} else {
p = p + 1
}
}
return (x < 0 ? npat - p : p) % npat
}
System.print(x2p.call(Num.pi))
- Output:
77