++fn

Float

A mold for the floating-point arithmetic using the base of two, the formula is (-1)^s * a * 2^e.

Produces either a float (%f), an infinity of other sign (%i), or not-a-number (%n). s refers to sign, the flags & or |; e to exponent, a signed decimal; and a to the significand, an unsigned integer.

Source

++  fn
  $%  [%f s=? e=@s a=@u]
      [%i s=?]
      [%n ~]
  ==

Examples

> *fn
[%n ~]

> (add:fl [%f & --33 2] [%f | --23 2])
[%f s=%.y e=-79 a=10.374.452.512.267.829.421.849.019.032.797.184]

> (add:fl [%i &] [%i &])
[%i s=%.y]

> (add:fl [%n ~] [%i |])
[%n ~]

> (sun:fl 961.193.554.848.514.048.973.893.027.381.506.219.443)
[%f s=%.y e=--17 a=7.333.324.850.834.000.007.430.214.137.126.970]

++dn

Decimal float

A mold for the floating-point arithmetic using the base of 10; the formula is (-1)^s * a *10^e.

Valid values are a float (%d), an infinity (%i), or a not-a-number (%n). s refers to sign, the flags & or |; e to exponent, a signed decimal; and a to the significand, an unsigned integer.

++  dn
  $%  [%d s=? e=@s a=@u]
      [%i s=?]
      [%n ~]
  ==

Examples

> `dn`[%d & --0 17.163.091.968]
[%d s=%.y e=--0 a=17.163.091.968]

> `dn`[%i s=%.y]
[%i s=%.y]

> `dn`[%n ~]
[%n ~]

++rn

Parsed decimal float

A mold for the floating-point arithmetic using the base of 10; the formula is (-1)^s * a *10^e.

Produces either a parsed float (%d), infinity of either sign (%i), or not-a-number (%n).

Source

++  rn
  $%  [%d a=? b=[c=@ [d=@ e=@] f=? i=@]]
      [%i a=?]
      [%n ~]
  ==

Examples

> `rn`[%d | [2 [3 4] | 17.163]]
[%d a=%.n b=[c=2 [d=3 e=4] f=%.n i=17.163]]

++fl

Arbitrary-precision floating-point

Container arm for floating-point arithmetic functions.

• Precision (p): number of bits in the significand; must be at least 2. Default is 113.
• Minimum exponent (v): minimum value of e. Default is -16.494.
• Width (w): Max value of e minus min value of e. 0 is for fixed-point. Default is 32.765.
• Rounding mode (r): Possible modes are nearest (%n), up (%u), down (%d), to zero (%z), and away from zero (%a). Default value is %n.
• Behavior (d): return denormals (%d), flush denormals to zero (%z), infinite exponent range (%i). Default value is %d.

Source

++  fl
  =/  [[p=@u v=@s w=@u] r=$?(%n %u %d %z %a) d=$?(%d %f %i)]
    [[113 -16.494 32.765] %n %d]
  =>
    ~%  %cofl  +>  ~
    |%

^rou:fl

Round

Rounds a to a the nearest float that can be represented with a 113-bit significand. There is no term to sign the significand, meaning that a positive sign will always be produced.

Accepts

a is a cell of a signed integer and an unsigned integer.

Produces

An fn.

Source

++  rou
  |=  [a=[e=@s a=@u]]  ^-  fn  (rau a &)

Examples

> =a 10.161.487.211.429.486.882.397.572.894.294.017.777
> (^rou:fl [--12 a])
[%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]

> (^rou:fl [--12 (add a 1)])
[%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]

> (^rou:fl [--12 (add a 300)])
[%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]

> (^rou:fl [--12 (add a 1.000)])
[%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.003]

++rau:fl

Various roundings

Rounds a based on what the state of of r in the core contained in fl. t is a sticky bit that represents a value less than ULP(a) = 2^(e.a) when passed to lug:fl.

Accepts

a is a cell of a signed integer and an unsigned integer.

t is a flag.

Produces

An fn.

Source

++  rau
  |=  [a=[e=@s a=@u] t=?]  ^-  fn
  ?-  r
    %z  (lug %fl a t)  %d  (lug %fl a t)
    %a  (lug %ce a t)  %u  (lug %ce a t)
    %n  (lug %ne a t)
  ==

Examples

> (rau:fl [-18 342.602.577] &)
[%f s=%.y e=-102 a=6.626.897.619.228.945.634.459.505.846.648.832]

Discussion

See lug:fl for possible rounding operations.

^add:fl

Add

Produces the sum of a and b. e is used to choose between an exact result (any-sized significand) or a rounded result (113-bit significand).

There is no term to sign the significands, so a positive sign will always be produced.

Accepts

a is an fn.

b is an fn

e is a flag.

Produces

An fn.

Source

++  add
  |=  [a=[e=@s a=@u] b=[e=@s a=@u] e=?]  ^-  fn
  =+  q=(dif:si e.a e.b)
  |-  ?.  (syn:si q)  $(b a, a b, q +(q))
  ?:  e
    [%f & e.b (^add (lsh [0 (abs:si q)] a.a) a.b)]
  =+  [ma=(met 0 a.a) mb=(met 0 a.b)]
  =+  ^=  w  %+  dif:si  e.a  %-  sun:si
    ?:  (gth prc ma)  (^sub prc ma)  0
  =+  ^=  x  %+  sum:si  e.b  (sun:si mb)
  ?:  =((cmp:si w x) --1)
    ?-  r
      %z  (lug %fl a &)  %d  (lug %fl a &)
      %a  (lug %lg a &)  %u  (lug %lg a &)
      %n  (lug %na a &)
    ==
  (rou [e.b (^add (lsh [0 (abs:si q)] a.a) a.b)])

Examples

> (^add:fl [--33 2.718] [--23 11] %.y)
[%f s=%.y e=--23 a=2.783.243]

> (^add:fl [--33 2.718] [--23 11] %.n)
[%f s=%.y e=-68 a=6.890.975.897.521.519.304.902.126.405.156.864]

^sub:fl

Subtract

Produces the difference of a minus b. e is used to choose between an exact result (any-sized significand) or a rounded result (113-bit significand).

Accepts

a is a cell of a signed integer and an unsigned integer.

b is a cell of a signed integer and an unsigned integer.

e is a flag.

Produces

An fn.

Source

++  sub
  |=  [a=[e=@s a=@u] b=[e=@s a=@u] e=?]  ^-  fn
  =+  q=(dif:si e.a e.b)
  |-  ?.  (syn:si q)
    (fli $(b a, a b, q +(q), r swr))
  =+  [ma=(met 0 a.a) mb=(met 0 a.b)]
  =+  ^=  w  %+  dif:si  e.a  %-  sun:si
    ?:  (gth prc ma)  (^sub prc ma)  0
  =+  ^=  x  %+  sum:si  e.b  (sun:si +(mb))
  ?:  &(!e =((cmp:si w x) --1))
    ?-  r
      %z  (lug %sm a &)  %d  (lug %sm a &)
      %a  (lug %ce a &)  %u  (lug %ce a &)
      %n  (lug %nt a &)
    ==
  =+  j=(lsh [0 (abs:si q)] a.a)
  |-  ?.  (gte j a.b)
    (fli $(a.b j, j a.b, r swr))
  =+  i=(^sub j a.b)
  ?~  i  [%f & zer]
  ?:  e  [%f & e.b i]  (rou [e.b i])

Examples

> (^sub:fl [--33 2.718] [--23 11] %.y)
[%f s=%.y e=--23 a=2.783.221]

> (^sub:fl [--33 2.718] [--63 11] %.y)
[%f s=%.n e=--33 a=11.811.157.346]

^mul:fl

Multiply

Produces the product of a multiplied by b. There is no term to sign the significands, so a positive sign will always be produced.

Accepts

a is a cell of a signed integer and an unsigned integer.

b is a cell of a signed integer and an unsigned integer.

Produces

An fn.

Source

++  mul
  |=  [a=[e=@s a=@u] b=[e=@s a=@u]]  ^-  fn
  (rou (sum:si e.a e.b) (^mul a.a a.b))

Examples

> (^mul:fl [--3 2.718] [--23 11])
[%f s=%.y e=-72 a=9.475.054.411.405.900.661.487.108.108.582.912]

^div:fl

Divide

Produces the quotient of a divided by b. There is no term to sign the significands, so a positive sign will always be produced.

Accepts

a is a cell of a signed integer and an unsigned integer.

b is a cell of a signed integer and an unsigned integer.

Produces

An fn.

Source

++  div
  |=  [a=[e=@s a=@u] b=[e=@s a=@u]]  ^-  fn
  =+  [ma=(met 0 a.a) mb=(met 0 a.b)]
  =+  v=(dif:si (sun:si ma) (sun:si +((^add mb prc))))
  =.  a  ?:  (syn:si v)  a
  a(e (sum:si v e.a), a (lsh [0 (abs:si v)] a.a))
  =+  [j=(dif:si e.a e.b) q=(dvr a.a a.b)]
  (rau [j p.q] =(q.q 0))

Examples

> (^div:fl [--13 2.718] [--23 11])
[%f s=%.y e=-115 a=10.023.198.055.040.952.765.870.659.817.343.907]

^sqt:fl

Square root

Produces the square root of a.

Accepts

a is a cell of a signed integer and an unsigned integer.

Produces

An fn.

Source

++  sqt
  |=  [a=[e=@s a=@u]]  ^-  fn
  =.  a
    =+  [w=(met 0 a.a) x=(^mul +(prc) 2)]
    =+  ?:((^lth w x) (^sub x w) 0)
    =+  ?:  =((dis - 1) (dis (abs:si e.a) 1))  -
      (^add - 1)
    a(e (dif:si e.a (sun:si -)), a (lsh [0 -] a.a))
  =+  [y=(^sqt a.a) z=(fra:si e.a --2)]
  (rau [z p.y] =(q.y 0))

Examples

> (^sqt:fl [-18 342.602.577])
[%f s=%.y e=-107 a=5.865.903.143.604.945.574.132.671.852.050.553]

^lth:fl

Less than

Tests if a is less than b.

Accepts

a is a cell of a signed integer and an unsigned integer.

b is a cell of a signed integer and an unsigned integer.

Produces

An flag.

Source

++  lth
  |=  [a=[e=@s a=@u] b=[e=@s a=@u]]  ^-  ?
  ?:  =(e.a e.b)  (^lth a.a a.b)
  =+  c=(cmp:si (ibl a) (ibl b))
  ?:  =(c -1)  &  ?:  =(c --1)  |
  ?:  =((cmp:si e.a e.b) -1)
    (^lth (rsh [0 (abs:si (dif:si e.a e.b))] a.a) a.b)
  (^lth (lsh [0 (abs:si (dif:si e.a e.b))] a.a) a.b)

Examples

> (^lth:fl [-4 684] [--0 35])
%.n

> (^lth:fl [-4 684] [--0 90])
%.y

^equ:fl

Equals

Tests if a is equal to b.

Accepts

a is a cell of a signed integer and an unsigned integer.

b is a cell of a signed integer and an unsigned integer.

Produces

A flag.

Source

++  equ
  |=  [a=[e=@s a=@u] b=[e=@s a=@u]]  ^-  ?
  ?.  =((ibl a) (ibl b))  |
  ?:  =((cmp:si e.a e.b) -1)
    =((lsh [0 (abs:si (dif:si e.a e.b))] a.b) a.a)
  =((lsh [0 (abs:si (dif:si e.a e.b))] a.a) a.b)

Examples

> (^equ:fl [-4 480] [-0 50])
%.n

> (^equ:fl [-4 480] [-0 30])
%.y

++ibl:fl

Integer binary logarithm

Produces the lowest power to which the number 2 must be raised to obtain a or greater.

Accepts

a is a cell of a signed integer and an unsigned integer.

Produces

A signed integer.

Source

++  ibl
  |=  [a=[e=@s a=@u]]  ^-  @s
  (sum:si (sun:si (dec (met 0 a.a))) e.a)

Examples

> (ibl:fl [-18 342.602.577])
--10

++uni:fl

Change representation to odd

Produces another representation of the floating point a where the significand is odd. Every floating-point number has a unique representation of this kind. If the significand of a is already odd, nothing changes.

Accepts

a is a cell of a signed integer and an unsigned integer.

Produces

A cell of a signed integer and an unsigned integer.

Source

++  uni
  |=  [a=[e=@s a=@u]]
  |-  ?:  =((end 0 a.a) 1)  a
  $(a.a (rsh 0 a.a), e.a (sum:si e.a --1))

Examples

> (uni:fl [-8 342.602.578])
[e=-7 a=171.301.289]

> (uni:fl [-8 342.602.577])
[e=-8 a=342.602.577]

++xpd:fl

Expand

Produces the fully precise form of a, or the denormalized form of a.

Accepts

a is a cell of a signed integer and an unsigned integer.

Produces

A cell of a signed integer and an unsigned integer.

Sources

++  xpd
  |=  [a=[e=@s a=@u]]
  =+  ma=(met 0 a.a)
  ?:  (gte ma prc)  a
  =+  ?:  =(den %i)  (^sub prc ma)
      =+  ^=  q
        =+  w=(dif:si e.a emn)
        ?:  (syn:si w)  (abs:si w)  0
      (min q (^sub prc ma))
  a(e (dif:si e.a (sun:si -)), a (lsh [0 -] a.a))

Examples

> (xpd:fl [--3 12])
[e=-106 a=7.788.445.287.802.241.442.795.744.493.830.144]

> (xpd:fl [-8 342.602.577])
[e=-92 a=6.626.897.619.228.945.634.459.505.846.648.832]

> (xpd:fl [-92 6.626.897.619.228.945.634.459.505.846.648.832])
[e=-92 a=6.626.897.619.228.945.634.459.505.846.648.832]

++lug:fl

Central rounding mechanism

Performs various rounding operations on a. An operation is chosen based on the value selected for t. s is a sticky bit that represents a value less than ULP(a) = 2^(e.a)

Possible rounding operations:

• Floor (%fl)
• Ceiling (%ce)
• Smaller (%sm)
• Larger (%lg)
• Nearest (%ne) -- Rounds ties away from 0 if the number is even, rounds toward 0 if the number is odd.

Accepts

t is one of the following: %fl, %ce, %sm, %lg, %ne, %na, or %nt.

a is a cell of a signed integer and an unsigned integer.

s is a flag.

Produces

An fn.

Source

++  lug
  ~/  %lug
  |=  [t=$?(%fl %ce %sm %lg %ne %na %nt) a=[e=@s a=@u] s=?]  ^-  fn
  ?<  =(a.a 0)
  =-
    ?.  =(den %f)  -                                ::  flush denormals
    ?.  ?=([%f *] -)  -
    ?:  =((met 0 ->+>) prc)  -  [%f & zer]
  ::
  =+  m=(met 0 a.a)
  ?>  |(s (gth m prc))                              ::  require precision
  =+  ^=  q  %+  max
      ?:  (gth m prc)  (^sub m prc)  0              ::  reduce precision
    %-  abs:si  ?:  =(den %i)  --0                  ::  enforce min. exp
    ?:  =((cmp:si e.a emn) -1)  (dif:si emn e.a)  --0
  =^  b  a  :-  (end [0 q] a.a)
    a(e (sum:si e.a (sun:si q)), a (rsh [0 q] a.a))
  ::
  ?~  a.a
    ?<  =(den %i)
    ?-  t
      %fl  [%f & zer]
      %sm  [%f & zer]
      %ce  [%f & spd]
      %lg  [%f & spd]
      %ne  ?:  s  [%f & ?:((lte b (bex (dec q))) zer spd)]
           [%f & ?:((^lth b (bex (dec q))) zer spd)]
      %nt  ?:  s  [%f & ?:((lte b (bex (dec q))) zer spd)]
           [%f & ?:((^lth b (bex (dec q))) zer spd)]
      %na  [%f & ?:((^lth b (bex (dec q))) zer spd)]
    ==
  ::
  =.  a  (xpd a)
  ::
  =.  a
    ?-  t
      %fl  a
      %lg  a(a +(a.a))
      %sm  ?.  &(=(b 0) s)  a
           ?:  &(=(e.a emn) !=(den %i))  a(a (dec a.a))
           =+  y=(dec (^mul a.a 2))
           ?.  (lte (met 0 y) prc)  a(a (dec a.a))
           [(dif:si e.a --1) y]
      %ce  ?:  &(=(b 0) s)  a  a(a +(a.a))
      %ne  ?~  b  a
           =+  y=(bex (dec q))
           ?:  &(=(b y) s)                          ::  round halfs to even
             ?~  (dis a.a 1)  a  a(a +(a.a))
           ?:  (^lth b y)  a  a(a +(a.a))
      %na  ?~  b  a
           =+  y=(bex (dec q))
           ?:  (^lth b y)  a  a(a +(a.a))
      %nt  ?~  b  a
           =+  y=(bex (dec q))
           ?:  =(b y)  ?:  s  a  a(a +(a.a))
           ?:  (^lth b y)  a  a(a +(a.a))
    ==
  ::
  =.  a  ?.  =((met 0 a.a) +(prc))  a
    a(a (rsh 0 a.a), e (sum:si e.a --1))
  ?~  a.a  [%f & zer]
  ::
  ?:  =(den %i)  [%f & a]
  ?:  =((cmp:si emx e.a) -1)  [%i &]  [%f & a]      ::  enforce max. exp

^drg:fl

Get printable decimal

Produces the decimal form of a using the Dragon4 algorithm. Guarantees accurate results for rounded floats.

Accepts

a is a cell of a signed integer and an unsigned integer.

Produces

A cell of a signed integer and an unsigned integer.

Source

++  drg                                             ::  dragon4; get
  ~/  %drg                                          ::  printable decimal;
  |=  [a=[e=@s a=@u]]  ^-  [@s @u]                  ::  guaranteed accurate
  ?<  =(a.a 0)                                      ::  for rounded floats
  =.  a  (xpd a)
  =+  r=(lsh [0 ?:((syn:si e.a) (abs:si e.a) 0)] a.a)
  =+  s=(lsh [0 ?.((syn:si e.a) (abs:si e.a) 0)] 1)
  =+  mn=(lsh [0 ?:((syn:si e.a) (abs:si e.a) 0)] 1)
  =+  mp=mn
  =>  ?.
        ?&  =(a.a (bex (dec prc)))                  ::  if next smallest
            |(!=(e.a emn) =(den %i))                ::  float is half ULP,
        ==                                          ::  tighten lower bound
      .
    %=  .
      mp  (lsh 0 mp)
      r  (lsh 0 r)
      s  (lsh 0 s)
    ==
  =+  [k=--0 q=(^div (^add s 9) 10)]
  |-  ?:  (^lth r q)
    %=  $
      k  (dif:si k --1)
      r  (^mul r 10)
      mn  (^mul mn 10)
      mp  (^mul mp 10)
    ==
  |-  ?:  (gte (^add (^mul r 2) mp) (^mul s 2))
    $(s (^mul s 10), k (sum:si k --1))
  =+  [u=0 o=0]
  |-                                                ::  r/s+o = a*10^-k
  =+  v=(dvr (^mul r 10) s)
  =>  %=  .
      k  (dif:si k --1)
      u  p.v
      r  q.v
      mn  (^mul mn 10)
      mp  (^mul mp 10)
    ==
  =+  l=(^lth (^mul r 2) mn)                        ::  in lower bound
  =+  ^=  h                                         ::  in upper bound
    ?|  (^lth (^mul s 2) mp)
        (gth (^mul r 2) (^sub (^mul s 2) mp))
    ==
  ?:  &(!l !h)
    $(o (^add (^mul o 10) u))
  =+  q=&(h |(!l (gth (^mul r 2) s)))
  =.  o  (^add (^mul o 10) ?:(q +(u) u))
  [k o]

Examples

> (sun:fl 218.116)
[%f s=%.y e=-95 a=8.640.464.947.480.640.129.276.716.135.743.488]
> (^drg:fl [e=-95 a=8.640.464.947.480.640.129.276.716.135.743.488])
[--0 218.116]

> (sun:fl 102.057.673.128.349)
[%f s=%.y e=-66 a=7.530.527.107.827.833.883.675.587.233.447.936]
> (^drg:fl [e=-66 a=7.530.527.107.827.833.883.675.587.233.447.936])
[--0 102.057.673.128.349]

^toj:fl

Round to integer

Rounds float a to the nearest decimal float with an exponent of 0.

Accepts

a is a cell of a signed integer and an unsigned integer.

Produces

An fn.

Source

++  toj
  |=  [a=[e=@s a=@u]]  ^-  fn
  ?.  =((cmp:si e.a --0) -1)  [%f & a]
  =+  x=(abs:si e.a)
  =+  y=(rsh [0 x] a.a)
  ?:  |(=(r %d) =(r %z))  [%f & --0 y]
  =+  z=(end [0 x] a.a)
  ?:  |(=(r %u) =(r %a))  [%f & --0 ?~(z y +(y))]
  =+  i=(bex (dec x))
  ?:  &(=(z i) =((dis y 1) 0))  [%f & --0 y]
  ?:  (^lth z i)  [%f & --0 y]  [%f & --0 +(y)]

Examples

> (^toj:fl [-11 7.530.107.827.833.587])
[%f s=%.y e=--0 a=3.676.810.462.809]

> (^toj:fl [-11 7.530.107.827.833.589])
[%f s=%.y e=--0 a=3.676.810.462.809]

++ned:fl

Require float

Produces a if a is a is of floating-point representation. If a is another case of fn, such as infinity or not-a-number, a crash is produced.

Accepts

a is an fn.

Produces

A cell of a signed integer and an unsigned integer.

Source

++  ned
  |=  [a=fn]  ^-  [%f s=? e=@s a=@u]
  ?:  ?=([%f *] a)  a
  ~_  leaf+"need-float"
  !!

Examples

> (ned:fl [%f s=%.y e=-11 a=7.530.107.827.833.587])
[%f s=%.y e=-11 a=7.530.107.827.833.587]

> (ned:fl [%n ~])
! need-float
! exit

> (ned:fl [%i |])
! need-float
! exit

++shf:fl

Shift power

Multiplies a by 2 to the b power without rounding. This results in shifting the exponent term by b.

Accepts

a is an fn.

b is a signed integer.

Produces

An fn.

Source

++  shf
  |=  [a=fn b=@s]
  ?:  |(?=([%n *] a) ?=([%i *] a))  a
  a(e (sum:si e.a b))

Examples

> (shf:fl [[%f & -2 7] --2])
[%f s=%.y e=--0 a=7]

> (shf:fl [[%f & -2 7] -2])
[%f s=%.y e=-4 a=7]

> (shf:fl [%f & -11 7.530.107.827.833.587] --5)
[%f s=%.y e=-6 a=7.530.107.827.833.587]

++fli:fl

Flip sign

Produces a with its signed changed from positive to negative, or vice versa.

Accepts

a is an fn.

Produces

An fn.

Source

++  fli
  |=  [a=fn]  ^-  fn
  ?-(-.a %f a(s !s.a), %i a(s !s.a), %n a)

Examples

> (fli:fl [%f %.y -2 7])
[%f s=%.n e=-2 a=7]

> (fli:fl [%f %.n --2 30.617])
[%f s=%.y e=--2 a=30.617]

> (fli:fl [%f | --2 30.617])
[%f s=%.y e=--2 a=30.617]

++swr:fl

Switch rounding

Switches the rounding mode of r:fl.

Source

++  swr  ?+(r r %d %u, %u %d)

Examples

> r:fl
%n
> swr:fl
%n

> =new-fl fl                              ::  new fl core with changed state
> =new-fl new-fl(r %u)
> swr:new-fl
%d

++prc:fl

Force precision of 2 or greater

Produces p, the core's precision, if p is greater than or equal to 2. Otherwise, a crash is produced.

Source

++  prc  ?>((gth p 1) p)

Examples

> prc:fl
113

> =new-fl fl
> =new-fl new-fl(p 1)
> prc:new-fl
! exit

> =new-fl new-fl(p 2)
> prc:new-fl
2

++den:fl

Behavior

Produces d:fl. Denormalizes if d:fl is %d. Flushes denormals to zero if d:fl is %f.

• Denormalizes if d:fl is %d.
• Flushes denormals to zero if d:fl is %f.
• Infinite exponent range if %d is %i.

The default value of d is %d.

Source

++  den  d

Examples

> den:fl
%d

> =new-fl fl
> =new-fl new-fl(d %f)
> den:new-fl
%f

++emn:fl

Minimum exponent

Produces v:fl, the minimum exponent. The default minimum exponent is -16.494.

Source

++  emn  v

Examples

> emn:fl
-16.494

++emx:fl

Maximum exponent

Returns the maximum exponent of fl. The default maximum exponent is --16.271.

Source

++  emx  (sum:si emn (sun:si w))

Examples

> emx:fl
--16.271

> `@u`emx:fl
32.542

++spd:fl

Smallest denormal

Produces the smallest possible denormalized float.

Source

++  spd  [e=emn a=1]

Examples

> spd:fl
[e=-16.494 a=1]

++spn:fl

Smallest normal

Produces the smallest representable normal float.

Source

++  spn  [e=emn a=(bex (dec prc))]

Examples

> spn:fl
[e=-16.494 a=5.192.296.858.534.827.628.530.496.329.220.096]

++lfn:fl

Largest normal

Produces the largest representable normal float.

Source

++  lfn  [e=emx a=(fil 0 prc 1)]

Examples

> lfn:fl
[e=--16.271 a=10.384.593.717.069.655.257.060.992.658.440.191]

++lfe:fl

Maximum

Produces the sum of emx:fl plus prc:fl.

Source

++  lfe  (sum:si emx (sun:si prc))

Examples

> lfe:fl
--16.384

++zer:fl

Zero

Produces zero represented as a float.

Source

++  zer  [e=--0 a=0]

Examples

> zer:fl
[e=--0 a=0]

++rou:fl

Round

Rounds a. The way in which a is rounded depends on the value of r:fl.

Accepts

a is an fn.

Produes

An fn.

Source

++  rou
  |=  [a=fn]  ^-  fn
  ?.  ?=([%f *] a)  a
  ?~  a.a  [%f s.a zer]
  ?:  s.a  (^rou +>.a)
  =.(r swr (fli (^rou +>.a)))

Examples

> =a 10.161.487.211.429.486.882.397.572.894.294.017.777
> (rou:fl [%f & --12 a])
[%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]

> (rou:fl [%f & --12 (add a 1)])
[%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]

> (rou:fl [%f & --12 (add a 300)])
[%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]

> (rou:fl [%f & --12 (add a 1.000)])
[%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.003]

++syn:fl

Get sign

Produces the sign of a.

Accepts

a is an fn.

Produes

An fn.

Source

++  syn
  |=  [a=fn]  ^-  ?
  ?-(-.a %f s.a, %i s.a, %n &)

Examples

> (syn:fl (sun:fl 106))
%.y

> (syn:fl [%f | --0 106])
%.n

++abs:fl

Absolute value

Produces the absolute value of a.

Accepts

a is an fn.

Produes

An fn.

Sources

++  abs
  |=  [a=fn]  ^-  fn
  ?:  ?=([%f *] a)  [%f & e.a a.a]
  ?:  ?=([%i *] a)  [%i &]  [%n ~]

Examples

> (abs:fl [%f | --0 106])
[%f s=%.y e=--0 a=106]

> (abs:fl [%f & --0 106])
[%f s=%.y e=--0 a=106]

++add:fl

Add

Produces the sum of a plus b.

Accepts

a is an fn.

b is an fn.

Produces

An fn.

Source

++  add
  |=  [a=fn b=fn]  ^-  fn
  ?:  |(?=([%n *] a) ?=([%n *] b))  [%n ~]
  ?:  |(?=([%i *] a) ?=([%i *] b))
    ?:  &(?=([%i *] a) ?=([%i *] b))
      ?:  =(a b)  a  [%n ~]
    ?:  ?=([%i *] a)  a  b
  ?:  |(=(a.a 0) =(a.b 0))
    ?.  &(=(a.a 0) =(a.b 0))  %-  rou  ?~(a.a b a)
    [%f ?:(=(r %d) &(s.a s.b) |(s.a s.b)) zer]
  %-  |=  [a=fn]
      ?.  ?=([%f *] a)  a
      ?.  =(a.a 0)  a
      [%f !=(r %d) zer]
  ?:  =(s.a s.b)
    ?:  s.a  (^add +>.a +>.b |)
    =.(r swr (fli (^add +>.a +>.b |)))
  ?:  s.a  (^sub +>.a +>.b |)
  (^sub +>.b +>.a |)

Examples

> (add:fl [%f & --0 106] [%f | --3 55])
[%f s=%.n e=-104 a=6.774.324.807.619.657.921.598.381.929.529.344]

++ead:fl

Exact add

Produces the exact sum of a plus b.

Accepts

a is an fn.

b is an fn.

Produces

An fn.

Source

++  ead
  |=  [a=fn b=fn]  ^-  fn
  ?:  |(?=([%n *] a) ?=([%n *] b))  [%n ~]
  ?:  |(?=([%i *] a) ?=([%i *] b))
    ?:  &(?=([%i *] a) ?=([%i *] b))
      ?:  =(a b)  a  [%n ~]
    ?:  ?=([%i *] a)  a  b
  ?:  |(=(a.a 0) =(a.b 0))
    ?.  &(=(a.a 0) =(a.b 0))  ?~(a.a b a)
    [%f ?:(=(r %d) &(s.a s.b) |(s.a s.b)) zer]
  %-  |=  [a=fn]
      ?.  ?=([%f *] a)  a
      ?.  =(a.a 0)  a
      [%f !=(r %d) zer]
  ?:  =(s.a s.b)
    ?:  s.a  (^add +>.a +>.b &)
    (fli (^add +>.a +>.b &))
  ?:  s.a  (^sub +>.a +>.b &)
  (^sub +>.b +>.a &)

Examples

> (ead:fl [%f & --0 106] [%f | --3 55])
[%f s=%.n e=--0 a=334]

++sub:fl

Subtract

Produces the difference of a minus b.

Accepts

a is an fn.

b is an fn.

Produces

An fn.

Source

++  sub
  |=  [a=fn b=fn]  ^-  fn  (add a (fli b))

Examples

> (sub:fl [%f & --13 2.718] [%f & --23 11])
[%f s=%.n e=-86 a=5.416.671.014.775.224.232.595.412.796.571.648]

++mul:fl

Multiply

Produces the product of a multiplied by b.

Accepts

a is an fn.

b is an fn.

Produces

An fn.

Source

++  mul
  |=  [a=fn b=fn]  ^-  fn
  ?:  |(?=([%n *] a) ?=([%n *] b))  [%n ~]
  ?:  ?=([%i *] a)
    ?:  ?=([%i *] b)
      [%i =(s.a s.b)]
    ?:  =(a.b 0)  [%n ~]  [%i =(s.a s.b)]
  ?:  ?=([%i *] b)
    ?:  =(a.a 0)  [%n ~]  [%i =(s.a s.b)]
  ?:  |(=(a.a 0) =(a.b 0))  [%f =(s.a s.b) zer]
  ?:  =(s.a s.b)  (^mul +>.a +>.b)
  =.(r swr (fli (^mul +>.a +>.b)))

++emu:fl

Exact multiply

Produces the exact product of a multiplied by b.

Accepts

a is an fn.

b is an fn.

Produces

An fn.

Examples

++  emu
  |=  [a=fn b=fn]  ^-  fn
  ?:  |(?=([%n *] a) ?=([%n *] b))  [%n ~]
  ?:  ?=([%i *] a)
    ?:  ?=([%i *] b)
      [%i =(s.a s.b)]
    ?:  =(a.b 0)  [%n ~]  [%i =(s.a s.b)]
  ?:  ?=([%i *] b)
    ?:  =(a.a 0)  [%n ~]  [%i =(s.a s.b)]
  ?:  |(=(a.a 0) =(a.b 0))  [%f =(s.a s.b) zer]
  [%f =(s.a s.b) (sum:si e.a e.b) (^^mul a.a a.b)]

++div:fl

Divide

Produces the quotient of a divided by b.

Accepts

a is an fn.

b is an fn.

Produces

An fn.

Source

++  div
  |=  [a=fn b=fn]  ^-  fn
  ?:  |(?=([%n *] a) ?=([%n *] b))  [%n ~]
  ?:  ?=([%i *] a)
    ?:  ?=([%i *] b)  [%n ~]  [%i =(s.a s.b)]
  ?:  ?=([%i *] b)  [%f =(s.a s.b) zer]
  ?:  =(a.a 0)  ?:  =(a.b 0)  [%n ~]  [%f =(s.a s.b) zer]
  ?:  =(a.b 0)  [%i =(s.a s.b)]
  ?:  =(s.a s.b)  (^div +>.a +>.b)
  =.(r swr (fli (^div +>.a +>.b)))

++fma:fl

Fused multiply-add

Produces the sum of c plus the product of a multiplied by b; (a * b) + c.

Accepts

a is an fn.

b is an fn.

c is an fn.

Produces

An fn.

Source

++  fma
  |=  [a=fn b=fn c=fn]  ^-  fn
  (add (emu a b) c)

Examples

> (fma:fl [%f & --13 2.718] [%f & --23 11] [%f & --13 2.718])
[%f s=%.y e=-62 a=9.475.054.514.089.037.465.004.673.635.188.736]=

++sqt:fl

Square root

Produces the square root of a.

Accepts

a is an fn.

Produces

An fn.

Source

++  sqt
  |=  [a=fn]  ^-  fn
  ?:  ?=([%n *] a)  [%n ~]
  ?:  ?=([%i *] a)  ?:(s.a a [%n ~])
  ?~  a.a  [%f s.a zer]
  ?:  s.a  (^sqt +>.a)  [%n ~]

Examples

> (sqt:fl [%f s=%.y e=-18 a=342.602.577])
[%f s=%.y e=-107 a=5.865.903.143.604.945.574.132.671.852.050.553]

++inv:fl

Inverse

Produces the inverse of a by dividing 1 by a.

Accepts

a is an fn.

Produces

An fn.

Source

++  inv
  |=  [a=fn]  ^-  fn
  (div [%f & --0 1] a)

Examples

> (inv:fl [%f s=%.y e=--0 a=10])
[%f s=%.y e=-116 a=8.307.674.973.655.724.205.648.794.126.752.154]
> (drg:fl [%f s=%.y e=-116 a=8.307.674.973.655.724.205.648.794.126.752.154])
[%d s=%.y e=-1 a=1]

> (inv:fl [%f s=%.y e=--1 a=10])
[%f s=%.y e=-117 a=8.307.674.973.655.724.205.648.794.126.752.154]
> (drg:fl [%f s=%.y e=-117 a=8.307.674.973.655.724.205.648.794.126.752.154])
[%d s=%.y e=-2 a=5]

> (inv:fl [%f s=%.y e=--2 a=10])
[%f s=%.y e=-118 a=8.307.674.973.655.724.205.648.794.126.752.154]
> (drg:fl [%f s=%.y e=-118 a=8.307.674.973.655.724.205.648.794.126.752.154])
[%d s=%.y e=-3 a=25]

++sun:fl

Signed integer to float

Produces a in floating-point representation.

Accepts

a is an unsigned integer.

Produces

An fn.

Source

++  sun
  |=  [a=@u]  ^-  fn
  (rou [%f & --0 a])

Examples

> (sun:fl 0)
[%f s=%.y e=--0 a=0]

> (sun:fl 5.048.729)
[%f s=%.y e=-90 a=6.250.023.776.601.238.669.911.180.544.311.296]

> (sun:fl -100)
! exit

++san:fl

Signed integer to float

Produces the floating-point representation of a, an unsigned integer.

Accepts

a is an unsigned integer.

Produces

An fn.

Source

++  san
  |=  [a=@s]  ^-  fn
  =+  b=(old:si a)
  (rou [%f -.b --0 +.b])

Examples

> (san:fl --100)
[%f s=%.y e=-106 a=8.112.963.841.460.668.169.578.900.514.406.400]

> (san:fl -100)
[%f s=%.n e=-106 a=8.112.963.841.460.668.169.578.900.514.406.400]

++lth:fl

Less than

Tests if a is less than b. Returns ~ in the event of a or b being a NaN ([%n ~]).

Accepts

a is an fn.

b is an fn.

Produces

A unit of flag.

Source

++  lth
  |=  [a=fn b=fn]  ^-  (unit ?)
  ?:  |(?=([%n *] a) ?=([%n *] b))  ~  :-  ~
  ?:  =(a b)  |
  ?:  ?=([%i *] a)  !s.a  ?:  ?=([%i *] b)  s.b
  ?:  |(=(a.a 0) =(a.b 0))
    ?:  &(=(a.a 0) =(a.b 0))  |
    ?:  =(a.a 0)  s.b  !s.a
  ?:  !=(s.a s.b)  s.b
  ?:  s.a  (^lth +>.a +>.b)  (^lth +>.b +>.a)

Examples

> (lth:fl (sun:fl 116) (sun:fl 4.820))
[~ u=%.y]

> (lth:fl (sun:fl 218.116) (sun:fl 4.820))
[~ u=%.n]

> (lth:fl (sun:fl 218.116) [%n ~])
~

++lte:fl

Less than or equal

Tests whether a is less than or equal to b. Returns ~ in the event of a or b being a NaN ([%n ~]).

Accepts

a is an fn.

b is an fn.

Produces

A unit of flag.

Source

++  lte
  |=  [a=fn b=fn]  ^-  (unit ?)
  %+  bind  (lth b a)  |=  a=?  !a

Examples

> (lte:fl (sun:fl 102) [%f %.y -5 973.655.724])
[~ u=%.y]

> (lte:fl (sun:fl 102) [%f %.y -24 973.655.724])
[~ u=%.n]

> (lte:fl [%f %.y --2 25] (sun:fl 100))
[~ u=%.y]

> (lte:fl [%f %.y --2 25] [%f %.y --3 2])
[~ u=%.n]

> (lte:fl [%f %.y --2 25] [%n ~])
~

++equ:fl

Equals

Tests if a is equal to b. Returns ~ in the event of a or b being a NaN ([%n ~]).

Accepts

a is an fn.

b is an fn.

Produces

An unit of flag.

Source

++  equ
  |=  [a=fn b=fn]  ^-  (unit ?)
  ?:  |(?=([%n *] a) ?=([%n *] b))  ~  :-  ~
  ?:  =(a b)  &
  ?:  |(?=([%i *] a) ?=([%i *] b))  |
  ?:  |(=(a.a 0) =(a.b 0))
    ?:  &(=(a.a 0) =(a.b 0))  &  |
  ?:  |(=(e.a e.b) !=(s.a s.b))  |
  (^equ +>.a +>.b)

Examples

> (equ:fl [%f %.y --2 25] (sun:fl 100))
[~ u=%.y]

> (equ:fl [%f %.y --2 25] (sun:fl 101))
[~ u=%.n]

++gte:fl

Greater or equal than

Tests whether a is greater than or equal to b. Returns ~ in the event of a or b being a NaN ([%n ~]).

Accepts

a is an fn.

b is an fn.

Produces

An unit of flag.

Source

++  gte
  |=  [a=fn b=fn]  ^-  (unit ?)  (lte b a)

Examples

> (gte:fl [%f %.y --2 25] (sun:fl 100))
[~ u=%.y]

> (gth:fl [%f %.y --6 73.989] [%f %.y --5 919.599])
[~ u=%.n]

> (gth:fl [%f %.y --6 73.989] [%n ~])
~

++gth:fl

Greater than

Tests whether a is greater than b. Returns ~ in the event of a or b being a NaN ([%n ~]).

Accepts

a is an fn.

b is an fn.

Produces

An unit of flag.

Source

++  gth
  |=  [a=fn b=fn]  ^-  (unit ?)  (lth b a)

Examples

> (gth:fl [%f %.y --2 25] (sun:fl 100))
[~ u=%.n]

> (gth:fl [%f %.y --6 73.989] [%f %.y --5 119.599])
[~ u=%.y]

++drg:fl

Float to decimal

Produces the decimal form of a using the Dragon4 algorithm. Guarantees accurate results for rounded floats.

Accepts

a is an fn.

Produces

A dn.

Source

++  drg
  |=  [a=fn]  ^-  dn
  ?:  ?=([%n *] a)  [%n ~]
  ?:  ?=([%i *] a)  [%i s.a]
  ?~  a.a  [%d s.a --0 0]
  [%d s.a (^drg +>.a)]

Examples

> (drg:fl [%f | --6 73.989])
[%d s=%.n e=--0 a=4.735.296]

++grd:fl

Decimal to float

Converts decimal a to fn.

Accepts

a is a dn.

Produces

An fn.

Source

++  grd
  |=  [a=dn]  ^-  fn
  ?:  ?=([%n *] a)  [%n ~]
  ?:  ?=([%i *] a)  [%i s.a]
  =>  .(r %n)
  =+  q=(abs:si e.a)
  ?:  (syn:si e.a)
    (mul [%f s.a --0 a.a] [%f & e.a (pow 5 q)])
  (div [%f s.a --0 a.a] [%f & (sun:si q) (pow 5 q)])

Examples

> (grd:fl [%d s=%.n e=--0 a=73.989])
[%f s=%.n e=-96 a=5.862.012.516.267.904.074.208.723.341.410.304]

> (grd:fl [%d s=%.n e=--0 a=100])
[%f s=%.n e=-106 a=8.112.963.841.460.668.169.578.900.514.406.400]

++toi:fl

Round to signed integer

Rounds a to the nearest signed integer.

Accepts

a is an fn.

Produces

A unit of @s.

Source

++  toi
  |=  [a=fn]  ^-  (unit @s)
  =+  b=(toj a)
  ?.  ?=([%f *] b)  ~  :-  ~
  =+  c=(^^mul (bex (abs:si e.b)) a.b)
  (new:si s.b c)

Examples

> (toi:fl [%f s=%.y e=-78 a=8.112.963.841.460.668.169.578.900.514.406.400])
[~ u=--26.843.545.600]

++toj:fl

Round to integer fn

Rounds a to the nearest decimal integer.

Accepts

a is an fn.

Produces

A unit of @s.

Source

++  toj
  |=  [a=fn]  ^-  fn
  ?.  ?=([%f *] a)  a
  ?~  a.a  [%f s.a zer]
  ?:  s.a  (^toj +>.a)
  =.(r swr (fli (^toj +>.a)))

Examples

> (toj:fl [%f s=%.y e=-78 a=8.112.963.841.460.668.169.578.900.514.406.400])
[%f s=%.y e=--0 a=26.843.545.600]

> (toj:fl [%f s=%.y e=-78 a=8.112.963.841.460.668.169.578.900.514])
[%f s=%.y e=--0 a=26.844]

> (toj:fl [%f s=%.y e=-78 a=8.112.963.841.460])
[%f s=%.y e=--0 a=0]

> (toj:fl [%f s=%.y e=-9 a=9.002])
[%f s=%.y e=--0 a=16]

++ff

IEEE-754 Formatting

Container core for IEEE-754 formatting operations.

• w is width: The number of bits in the exponent field.
• p is precision: The number of bits in the significand field.
• w is bias: Added to exponent when storing.
• r is rounding mode: Possible modes are nearest (%n), up (%u), down (%d), to zero (%z), and away from zero (%a). Default value is %a.

Source

++  ff
  |_  [[w=@u p=@u b=@s] r=$?(%n %u %d %z %a)]

Examples

> =ffcore ~(. ff [8 8 0] %n)
> ffcore
<24.ltg {{{@ud @ud @ud} r/$n} <54.tyv 119.wim 31.ohr 1.jmk $143>}>

Discussion

++ff has no use outside of the functionality provided to other cores: ++rd, ++rs, ++rq, and ++rh. It's not intended to be used directly; it's just meant to power those cores.

++sb:ff

Sign bit

Produces the sign bit of ++ff.

Source

++  sb  (bex (^add w p))

Examples

> sb:ff
1

++me:ff

Minimum exponent

Produces the minimum possible exponent of ff.

Source

++  me  (dif:si (dif:si --1 b) (sun:si p))

Examples

> me:ff
--1

++pa:ff

Initialize fl

Instantiates the core fl, giving values to its samples based on the configuration of the ff core.

Source

++  pa
  %*(. fl p +(p), v me, w (^sub (bex w) 3), d %d, r r)

Examples

> ~(pa ff [11 52 --1.023] %n)
< 23.qzd
  28.btz
  { {{p/@ v/@s w/@} r/?($n $u $a $d $z) d/$d}
    <54.tyv 119.wim 31.ohr 1.jmk $143>
  }
>

Discussion

++pa exists exclusively for internal use of ++ff, and ++ff exists for internal use in other cores.

++sea:ff

@r to fn

Converts a from @r to fn.

Accepts

a is a @r, an IEEE float.

Produces

A unit of @s.

Source

++  sea
  |=  [a=@r]  ^-  fn
  =+  [f=(cut 0 [0 p] a) e=(cut 0 [p w] a)]
  =+  s=(sig a)
  ?:  =(e 0)
    ?:  =(f 0)  [%f s --0 0]  [%f s me f]
  ?:  =(e (fil 0 w 1))
    ?:  =(f 0)  [%i s]  [%n ~]
  =+  q=:(sum:si (sun:si e) me -1)
  =+  r=(^add f (bex p))
  [%f s q r]

Examples

> (sea:ff `@r`0x8)
[%f s=%.y e=--0 a=0]

++bit:ff

fn to @r, rounding

Converts a from fn to @r and applies rounding.

Accepts

a is an fn.

Produces

A @r.

Source

++  bit  |=  [a=fn]  (bif (rou:pa a))

Examples

> (bit:ma:rd [%f | -6 202])
0xc009400000000000

++bif:ff

fn to @r, no rounding

Converts a from fn to @r. No rounding is applied.

Accepts

a is a @r, an IEEE float.

Produces

A flag.

Source

++  bif
  |=  [a=fn]  ^-  @r
  ?:  ?=([%i *] a)
    =+  q=(lsh [0 p] (fil 0 w 1))
    ?:  s.a  q  (^add q sb)
  ?:  ?=([%n *] a)  (lsh [0 (dec p)] (fil 0 +(w) 1))
  ?~  a.a  ?:  s.a  `@r`0  sb
  =+  ma=(met 0 a.a)
  ?.  =(ma +(p))
    ?>  =(e.a me)
    ?>  (^lth ma +(p))
    ?:  s.a  `@r`a.a  (^add a.a sb)
  =+  q=(sum:si (dif:si e.a me) --1)
  =+  r=(^add (lsh [0 p] (abs:si q)) (end [0 p] a.a))
  ?:  s.a  r  (^add r sb)