2title = "3b: Floating Point"
11A mold for the floating-point arithmetic using the base of two, the formula is
14Produces either a float (`$f`), an infinity of other sign (`$i`),
15or not-a-number (`$n`). `s` refers to sign, the `flag`s `&` or `|`; `e` to
16exponent, a signed decimal; and `a` to the significand, an unsigned integer.
34 > (add:fl [%f & --33 2] [%f | --23 2])
35 [%f s=%.y e=-79 a=10.374.452.512.267.829.421.849.019.032.797.184]
37 > (add:fl [%i &] [%i &])
40 > (add:fl [%n ~] [%i |])
43 > (sun:fl 961.193.554.848.514.048.973.893.027.381.506.219.443)
44 [%f s=%.y e=--17 a=7.333.324.850.834.000.007.430.214.137.126.970]
53A mold for the floating-point arithmetic using the base of 10; the formula is
56Valid values are a float (`$d`), an infinity (`$i`), or a not-a-number (`$n`).
57`s` refers to sign, the `flag`s `&` or `|`; `e` to exponent, a signed decimal;
58and `a` to the significand, an unsigned integer.
71 > `dn`[%d & --0 17.163.091.968]
72 [%d s=%.y e=--0 a=17.163.091.968]
86A mold for the floating-point arithmetic using the base of 10; the formula is
89Produces either a parsed float (`$d`), infinity of either sign (`$i`), or
95 $% [$d a=? b=[c=@ [d=@ e=@] f=? i=@]]
104 > `rn`[%d | [2 [3 4] | 17.163]]
105 [%d a=%.n b=[c=2 [d=3 e=4] f=%.n i=17.163]]
111Arbitrary-precision floating-point
113Container arm for floating-point arithmetic functions.
115- Precision (`p`): number of bits in the significand; must be at least 2.
117- Minimum exponent (`v`): minimum value of e. Default is -16.494.
118- Width (`w`): Max value of `e` minus min value of `e`. 0 is for fixed-point.
120- Rounding mode (`r`): Possible modes are nearest (`$n`), up (`$u`),
121 down (`$d`), to zero (`$z`), and away from zero (`$a`). Default value is `%n`.
122- Behavior (`d`): return denormals (`$d`), flush denormals to zero (`$z`),
123 infinite exponent range (`$i`). Default value is `%d`.
129 =+ ^- [[p=@u v=@s w=@u] r=$?($n $u $d $z $a) d=$?($d $f $i)]
130 [[113 -16.494 32.765] %n %d]
141Rounds `a` to a the nearest float that can be represented with a 113-bit
142significand. There is no term to sign the significand, meaning that a positive
143sign will always be produced.
147`a` is a cell of a signed integer and an unsigned integer.
157 |= [a=[e=@s a=@u]] ^- fn (rau a &)
163 > =a 10.161.487.211.429.486.882.397.572.894.294.017.777
165 [%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]
167 > (^rou:fl [--12 (add a 1)])
168 [%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]
170 > (^rou:fl [--12 (add a 300)])
171 [%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]
173 > (^rou:fl [--12 (add a 1.000)])
174 [%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.003]
182Rounds `a` based on what the state of of `r` in the core contained in `fl`. `t`
183is a sticky bit that represents a value less than ULP(a) = 2^(e.a) when passed
188`a` is a cell of a signed integer and an unsigned integer.
200 |= [a=[e=@s a=@u] t=?] ^- fn
202 $z (lug %fl a t) $d (lug %fl a t)
203 $a (lug %ce a t) $u (lug %ce a t)
211 > (rau:fl [-18 342.602.577] &)
212 [%f s=%.y e=-102 a=6.626.897.619.228.945.634.459.505.846.648.832]
218See `lug:fl` for possible rounding operations.
225Produces the sum of `a` and `b`. `e` is used to choose between an exact result
226(any-sized significand) or a rounded result (113-bit significand).
228There is no term to sign the significands, so a positive sign will always be
247 |= [a=[e=@s a=@u] b=[e=@s a=@u] e=?] ^- fn
248 =+ q=(dif:si e.a e.b)
249 |- ?. (syn:si q) $(b a, a b, q +(q)) :: a has larger exp
251 [%f & e.b (^add (lsh 0 (abs:si q) a.a) a.b)]
252 =+ [ma=(met 0 a.a) mb=(met 0 a.b)]
253 =+ ^= w %+ dif:si e.a %- sun:si :: expanded exp of a
254 ?: (gth prc ma) (^sub prc ma) 0
255 =+ ^= x %+ sum:si e.b (sun:si mb) :: highest exp for b
256 ?: =((cmp:si w x) --1) :: don't need to add
258 $z (lug %fl a &) $d (lug %fl a &)
259 $a (lug %lg a &) $u (lug %lg a &)
262 (rou [e.b (^add (lsh 0 (abs:si q) a.a) a.b)])
268 > (^add:fl [--33 2.718] [--23 11] %.y)
269 [%f s=%.y e=--23 a=2.783.243]
271 > (^add:fl [--33 2.718] [--23 11] %.n)
272 [%f s=%.y e=-68 a=6.890.975.897.521.519.304.902.126.405.156.864]
280Produces the difference of `a` minus `b.` `e` is used to choose between an exact
281result (any-sized significand) or a rounded result (113-bit significand).
285`a` is a cell of a signed integer and an unsigned integer.
287`b` is a cell of a signed integer and an unsigned integer.
299 |= [a=[e=@s a=@u] b=[e=@s a=@u] e=?] ^- fn
300 =+ q=(dif:si e.a e.b)
302 (fli $(b a, a b, q +(q), r swr))
303 =+ [ma=(met 0 a.a) mb=(met 0 a.b)]
304 =+ ^= w %+ dif:si e.a %- sun:si
305 ?: (gth prc ma) (^sub prc ma) 0
306 =+ ^= x %+ sum:si e.b (sun:si +(mb))
307 ?: &(!e =((cmp:si w x) --1))
309 $z (lug %sm a &) $d (lug %sm a &)
310 $a (lug %ce a &) $u (lug %ce a &)
313 =+ j=(lsh 0 (abs:si q) a.a)
315 (fli $(a.b j, j a.b, r swr))
318 ?: e [%f & e.b i] (rou [e.b i])
324 > (^sub:fl [--33 2.718] [--23 11] %.y)
325 [%f s=%.y e=--23 a=2.783.221]
327 > (^sub:fl [--33 2.718] [--63 11] %.y)
328 [%f s=%.n e=--33 a=11.811.157.346]
336Produces the product of `a` multiplied by `b`. There is no term to sign the
337significands, so a positive sign will always be produced.
341`a` is a cell of a signed integer and an unsigned integer.
343`b` is a cell of a signed integer and an unsigned integer.
353 |= [a=[e=@s a=@u] b=[e=@s a=@u]] ^- fn
354 (rou (sum:si e.a e.b) (^mul a.a a.b))
360 > (^mul:fl [--3 2.718] [--23 11])
361 [%f s=%.y e=-72 a=9.475.054.411.405.900.661.487.108.108.582.912]
369Produces the quotient of `a` divided by `b`. There is no term to sign the
370significands, so a positive sign will always be produced.
374`a` is a cell of a signed integer and an unsigned integer.
376`b` is a cell of a signed integer and an unsigned integer.
386 |= [a=[e=@s a=@u] b=[e=@s a=@u]] ^- fn
387 =+ [ma=(met 0 a.a) mb=(met 0 a.b)]
388 =+ v=(dif:si (sun:si ma) (sun:si +((^add mb prc))))
390 a(e (sum:si v e.a), a (lsh 0 (abs:si v) a.a))
391 =+ [j=(dif:si e.a e.b) q=(dvr a.a a.b)]
392 (rau [j p.q] =(q.q 0))
398 > (^div:fl [--13 2.718] [--23 11])
399 [%f s=%.y e=-115 a=10.023.198.055.040.952.765.870.659.817.343.907]
407Produces the square root of `a`.
411`a` is a cell of a signed integer and an unsigned integer.
421 |= [a=[e=@s a=@u]] ^- fn
423 =+ [w=(met 0 a.a) x=(^mul +(prc) 2)]
424 =+ ?:((^lth w x) (^sub x w) 0)
425 =+ ?: =((dis - 1) (dis (abs:si e.a) 1)) -
427 a(e (dif:si e.a (sun:si -)), a (lsh 0 - a.a))
428 =+ [y=(^sqt a.a) z=(fra:si e.a --2)]
429 (rau [z p.y] =(q.y 0))
435 > (^sqt:fl [-18 342.602.577])
436 [%f s=%.y e=-107 a=5.865.903.143.604.945.574.132.671.852.050.553]
444Tests if `a` is less than `b`.
448`a` is a cell of a signed integer and an unsigned integer.
450`b` is a cell of a signed integer and an unsigned integer.
460 |= [a=[e=@s a=@u] b=[e=@s a=@u]] ^- ?
461 ?: =(e.a e.b) (^lth a.a a.b)
462 =+ c=(cmp:si (ibl a) (ibl b))
463 ?: =(c -1) & ?: =(c --1) |
464 ?: =((cmp:si e.a e.b) -1)
465 (^lth (rsh 0 (abs:si (dif:si e.a e.b)) a.a) a.b)
466 (^lth (lsh 0 (abs:si (dif:si e.a e.b)) a.a) a.b)
472 > (^lth:fl [-4 684] [--0 35])
475 > (^lth:fl [-4 684] [--0 90])
484Tests if `a` is equal to `b`.
488`a` is a cell of a signed integer and an unsigned integer.
490`b` is a cell of a signed integer and an unsigned integer.
500 |= [a=[e=@s a=@u] b=[e=@s a=@u]] ^- ?
501 ?. =((ibl a) (ibl b)) |
502 ?: =((cmp:si e.a e.b) -1)
503 =((lsh 0 (abs:si (dif:si e.a e.b)) a.b) a.a)
504 =((lsh 0 (abs:si (dif:si e.a e.b)) a.a) a.b)
510 > (^equ:fl [-4 480] [-0 50])
513 > (^equ:fl [-4 480] [-0 30])
520Integer binary logarithm
522Produces the lowest power to which the number 2 must be raised to obtain `a` or
527`a` is a cell of a signed integer and an unsigned integer.
537 |= [a=[e=@s a=@u]] ^- @s
538 (sum:si (sun:si (dec (met 0 a.a))) e.a)
544 > (ibl:fl [-18 342.602.577])
551Change representation to odd
553Produces another representation of the floating point `a` where the significand
554is odd. Every floating-point number has a unique representation of this kind. If
555the significand of `a` is already odd, nothing changes.
559`a` is a cell of a signed integer and an unsigned integer.
563A cell of a signed integer and an unsigned integer.
570 |- ?: =((end 0 1 a.a) 1) a
571 $(a.a (rsh 0 1 a.a), e.a (sum:si e.a --1))
577 > (uni:fl [-8 342.602.578])
580 > (uni:fl [-8 342.602.577])
589Produces the fully precise form of `a`, or the denormalized form of `a`.
593`a` is a cell of a signed integer and an unsigned integer.
597A cell of a signed integer and an unsigned integer.
606 =+ ?: =(den %i) (^sub prc ma)
608 =+ w=(dif:si e.a emn)
609 ?: (syn:si w) (abs:si w) 0
610 (min q (^sub prc ma))
611 a(e (dif:si e.a (sun:si -)), a (lsh 0 - a.a))
618 [e=-106 a=7.788.445.287.802.241.442.795.744.493.830.144]
620 > (xpd:fl [-8 342.602.577])
621 [e=-92 a=6.626.897.619.228.945.634.459.505.846.648.832]
623 > (xpd:fl [-92 6.626.897.619.228.945.634.459.505.846.648.832])
624 [e=-92 a=6.626.897.619.228.945.634.459.505.846.648.832]
630Central rounding mechanism
632Performs various rounding operations on `a`. An operation is chosen based on the
633value selected for `t`. `s` is a sticky bit that represents a value less than
636Possible rounding operations:
641- Nearest (`$ne`) -- Rounds ties away from 0 if the number is even, rounds
642 toward 0 if the number is odd.
646`t` is one of the following: `$fl`, `$ce`, `$sm`, `$lg`, `$ne`, `$na`, or `$nt`.
648`a` is a cell of a signed integer and an unsigned integer.
660 :: central rounding mechanism
661 :: can perform: floor, ceiling, smaller, larger,
662 :: nearest (round ties to: even, away from 0, toward 0)
663 :: s is sticky bit: represents a value less than ulp(a) = 2^(e.a)
667 |= [t=$?($fl $ce $sm $lg $ne $na $nt) a=[e=@s a=@u] s=?] ^- fn
670 ?. =(den %f) - :: flush denormals
672 ?: =((met 0 ->+>) prc) - [%f & zer]
675 ?> |(s (gth m prc)) :: require precision
677 ?: (gth m prc) (^sub m prc) 0 :: reduce precision
678 %- abs:si ?: =(den %i) --0 :: enforce min. exp
679 ?: =((cmp:si e.a emn) -1) (dif:si emn e.a) --0
680 =^ b a :- (end 0 q a.a)
681 a(e (sum:si e.a (sun:si q)), a (rsh 0 q a.a))
690 $ne ?: s [%f & ?:((lte b (bex (dec q))) zer spd)]
691 [%f & ?:((^lth b (bex (dec q))) zer spd)]
692 $nt ?: s [%f & ?:((lte b (bex (dec q))) zer spd)]
693 [%f & ?:((^lth b (bex (dec q))) zer spd)]
694 $na [%f & ?:((^lth b (bex (dec q))) zer spd)]
704 ?: &(=(e.a emn) !=(den %i)) a(a (dec a.a))
705 =+ y=(dec (^mul a.a 2))
706 ?. (lte (met 0 y) prc) a(a (dec a.a))
708 $ce ?: &(=(b 0) s) a a(a +(a.a))
711 ?: &(=(b y) s) :: round halfs to even
712 ?~ (dis a.a 1) a a(a +(a.a))
713 ?: (^lth b y) a a(a +(a.a))
716 ?: (^lth b y) a a(a +(a.a))
719 ?: =(b y) ?: s a a(a +(a.a))
720 ?: (^lth b y) a a(a +(a.a))
723 =. a ?. =((met 0 a.a) +(prc)) a
724 a(a (rsh 0 1 a.a), e (sum:si e.a --1))
727 ?: =(den %i) [%f & a]
728 ?: =((cmp:si emx e.a) -1) [%i &] [%f & a] :: enforce max. exp
737Produces the decimal form of `a` using the Dragon4 algorithm. Guarantees
738accurate results for rounded floats.
742`a` is a cell of a signed integer and an unsigned integer.
746A cell of a signed integer and an unsigned integer.
753 |= [a=[e=@s a=@u]] ^- [@s @u]
756 =+ r=(lsh 0 ?:((syn:si e.a) (abs:si e.a) 0) a.a)
757 =+ s=(lsh 0 ?.((syn:si e.a) (abs:si e.a) 0) 1)
758 =+ mn=(lsh 0 ?:((syn:si e.a) (abs:si e.a) 0) 1)
761 ?& =(a.a (bex (dec prc))) :: if next smallest
762 |(!=(e.a emn) =(den %i)) :: float is half ULP,
763 == :: tighten lower bound
770 =+ [k=--0 q=(^div (^add s 9) 10)]
778 |- ?: (gte (^add (^mul r 2) mp) (^mul s 2))
779 $(s (^mul s 10), k (sum:si k --1))
781 |- :: r/s+o = a*10^-k
782 =+ v=(dvr (^mul r 10) s)
790 =+ l=(^lth (^mul r 2) mn) :: in lower bound
791 =+ ^= h :: in upper bound
792 ?| (^lth (^mul s 2) mp)
793 (gth (^mul r 2) (^sub (^mul s 2) mp))
796 $(o (^add (^mul o 10) u))
797 =+ q=&(h |(!l (gth (^mul r 2) s)))
798 =. o (^add (^mul o 10) ?:(q +(u) u))
806 [%f s=%.y e=-95 a=8.640.464.947.480.640.129.276.716.135.743.488]
807 > (^drg:fl [e=-95 a=8.640.464.947.480.640.129.276.716.135.743.488])
810 > (sun:fl 102.057.673.128.349)
811 [%f s=%.y e=-66 a=7.530.527.107.827.833.883.675.587.233.447.936]
812 > (^drg:fl [e=-66 a=7.530.527.107.827.833.883.675.587.233.447.936])
813 [--0 102.057.673.128.349]
821Rounds float `a` to the nearest decimal float with an exponent of 0.
825`a` is a cell of a signed integer and an unsigned integer.
835 |= [a=[e=@s a=@u]] ^- fn
836 ?. =((cmp:si e.a --0) -1) [%f & a]
839 ?: |(=(r %d) =(r %z)) [%f & --0 y]
841 ?: |(=(r %u) =(r %a)) [%f & --0 ?~(z y +(y))]
843 ?: &(=(z i) =((dis y 1) 0)) [%f & --0 y]
844 ?: (^lth z i) [%f & --0 y] [%f & --0 +(y)]
850 > (^toj:fl [-11 7.530.107.827.833.587])
851 [%f s=%.y e=--0 a=3.676.810.462.809]
853 > (^toj:fl [-11 7.530.107.827.833.589])
854 [%f s=%.y e=--0 a=3.676.810.462.809]
862Produces `a` if `a` is a is of floating-point representation. If `a` is another
863case of `fn`, such as infinity or not-a-number, a crash is produced.
871A cell of a signed integer and an unsigned integer.
877 |= [a=fn] ^- [$f s=? e=@s a=@u]
887 > (ned:fl [%f s=%.y e=-11 a=7.530.107.827.833.587])
888 [%f s=%.y e=-11 a=7.530.107.827.833.587]
904Multiplies `a` by 2 to the `b` power without rounding. This results in shifting
905the exponent term by `b`.
911`b` is a signed integer.
922 ?: |(?=([$n *] a) ?=([$i *] a)) a
929 > (shf:fl [[%f & -2 7] --2])
932 > (shf:fl [[%f & -2 7] -2])
935 > (shf:fl [%f & -11 7.530.107.827.833.587] --5)
936 [%f s=%.y e=-6 a=7.530.107.827.833.587]
944Produces `a` with its signed changed from positive to negative, or vice versa.
959 ?-(-.a $f a(s !s.a), $i a(s !s.a), $n a)
965 > (fli:fl [%f %.y -2 7])
968 > (fli:fl [%f %.n --2 30.617])
969 [%f s=%.y e=--2 a=30.617]
971 > (fli:fl [%f | --2 30.617])
972 [%f s=%.y e=--2 a=30.617]
980Switches the rounding mode of `r:fl`.
985 ++ swr ?+(r r $d %u, $u %d)
996 > =new-fl fl :: new fl core with changed state
997 > =new-fl new-fl(r %u)
1005Force precision of 2 or greater
1007Produces `p`, the core's precision, if `p` is greater than or equal to 2.
1008Otherwise, a crash is produced.
1013 ++ prc ?>((gth p 1) p)
1023 > =new-fl new-fl(p 1)
1027 > =new-fl new-fl(p 2)
1037Produces `d:fl`. Denormalizes if `d:fl` is `%d`. Flushes denormals to zero if
1040- Denormalizes if `d:fl` is `%d`.
1041- Flushes denormals to zero if `d:fl` is `%f`.
1042- Infinite exponent range if `$d` is `$i`.
1044The default value of `d` is `%d`.
1059 > =new-fl new-fl(d %f)
1069Produces `v:fl`, the minimum exponent. The default minimum exponent is -16.494.
1089Returns the maximum exponent of `fl`. The default maximum exponent is --16.271.
1094 ++ emx (sum:si emn (sun:si w))
1112Produces the smallest possible denormalized float.
1132Produces the smallest representable normal float.
1137 ++ spn [e=emn a=(bex (dec prc))]
1144 [e=-16.494 a=5.192.296.858.534.827.628.530.496.329.220.096]
1152Produces the largest representable normal float.
1157 ++ lfn [e=emx a=(fil 0 prc 1)]
1164 [e=--16.271 a=10.384.593.717.069.655.257.060.992.658.440.191]
1172Produces the sum of `emx:fl` plus `prc:fl`.
1177 ++ lfe (sum:si emx (sun:si prc)) :: 2^lfe is > than all
1192Produces zero represented as a float.
1214Rounds `a`. The way in which `a` is rounded depends on the value of `r:fl`.
1232 =.(r swr (fli (^rou +>.a)))
1238 > =a 10.161.487.211.429.486.882.397.572.894.294.017.777
1239 > (rou:fl [%f & --12 a])
1240 [%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]
1242 > (rou:fl [%f & --12 (add a 1)])
1243 [%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]
1245 > (rou:fl [%f & --12 (add a 300)])
1246 [%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.002]
1248 > (rou:fl [%f & --12 (add a 1.000)])
1249 [%f s=%.y e=--22 a=9.923.327.354.911.608.283.591.379.779.584.003]
1257Produces the sign of `a`.
1272 ?-(-.a $f s.a, $i s.a, $n &)
1278 > (syn:fl (sun:fl 106))
1281 > (syn:fl [%f | --0 106])
1290Produces the absolute value of `a`.
1305 ?: ?=([$f *] a) [%f & e.a a.a]
1306 ?: ?=([$i *] a) [%i &] [%n ~]
1312 > (abs:fl [%f | --0 106])
1313 [%f s=%.y e=--0 a=106]
1315 > (abs:fl [%f & --0 106])
1316 [%f s=%.y e=--0 a=106]
1324Produces the sum of `a` plus `b`.
1340 |= [a=fn b=fn] ^- fn
1341 ?: |(?=([$n *] a) ?=([$n *] b)) [%n ~]
1342 ?: |(?=([$i *] a) ?=([$i *] b))
1343 ?: &(?=([$i *] a) ?=([$i *] b))
1346 ?: |(=(a.a 0) =(a.b 0))
1347 ?. &(=(a.a 0) =(a.b 0)) %- rou ?~(a.a b a)
1348 [%f ?:(=(r %d) &(s.a s.b) |(s.a s.b)) zer]
1354 ?: s.a (^add +>.a +>.b |)
1355 =.(r swr (fli (^add +>.a +>.b |)))
1356 ?: s.a (^sub +>.a +>.b |)
1363 > (add:fl [%f & --0 106] [%f | --3 55])
1364 [%f s=%.n e=-104 a=6.774.324.807.619.657.921.598.381.929.529.344]
1372Produces the exact sum of `a` plus `b`.
1388 |= [a=fn b=fn] ^- fn
1389 ?: |(?=([$n *] a) ?=([$n *] b)) [%n ~]
1390 ?: |(?=([$i *] a) ?=([$i *] b))
1391 ?: &(?=([$i *] a) ?=([$i *] b))
1394 ?: |(=(a.a 0) =(a.b 0))
1395 ?. &(=(a.a 0) =(a.b 0)) ?~(a.a b a)
1396 [%f ?:(=(r %d) &(s.a s.b) |(s.a s.b)) zer]
1402 ?: s.a (^add +>.a +>.b &)
1403 (fli (^add +>.a +>.b &))
1404 ?: s.a (^sub +>.a +>.b &)
1412 > (ead:fl [%f & --0 106] [%f | --3 55])
1413 [%f s=%.n e=--0 a=334]
1421Produces the difference of `a` minus `b`.
1437 |= [a=fn b=fn] ^- fn (add a (fli b))
1443 > (sub:fl [%f & --13 2.718] [%f & --23 11])
1444 [%f s=%.n e=-86 a=5.416.671.014.775.224.232.595.412.796.571.648]
1452Produces the product of `a` multiplied by `b`.
1468 |= [a=fn b=fn] ^- fn
1469 ?: |(?=([$n *] a) ?=([$n *] b)) [%n ~]
1473 ?: =(a.b 0) [%n ~] [%i =(s.a s.b)]
1475 ?: =(a.a 0) [%n ~] [%i =(s.a s.b)]
1476 ?: |(=(a.a 0) =(a.b 0)) [%f =(s.a s.b) zer]
1477 ?: =(s.a s.b) (^mul +>.a +>.b)
1478 =.(r swr (fli (^mul +>.a +>.b)))
1486Produces the exact product of `a` multiplied by `b`.
1502 |= [a=fn b=fn] ^- fn
1503 ?: |(?=([$n *] a) ?=([$n *] b)) [%n ~]
1507 ?: =(a.b 0) [%n ~] [%i =(s.a s.b)]
1509 ?: =(a.a 0) [%n ~] [%i =(s.a s.b)]
1510 ?: |(=(a.a 0) =(a.b 0)) [%f =(s.a s.b) zer]
1511 [%f =(s.a s.b) (sum:si e.a e.b) (^^mul a.a a.b)]
1519Produces the quotient of `a` divided by `b`.
1535 |= [a=fn b=fn] ^- fn
1536 ?: |(?=([$n *] a) ?=([$n *] b)) [%n ~]
1538 ?: ?=([$i *] b) [%n ~] [%i =(s.a s.b)]
1539 ?: ?=([$i *] b) [%f =(s.a s.b) zer]
1540 ?: =(a.a 0) ?: =(a.b 0) [%n ~] [%f =(s.a s.b) zer]
1541 ?: =(a.b 0) [%i =(s.a s.b)]
1542 ?: =(s.a s.b) (^div +>.a +>.b)
1543 =.(r swr (fli (^div +>.a +>.b)))
1551Produces the sum of `c` plus the product of `a` multiplied by `b`;
1570 |= [a=fn b=fn c=fn] ^- fn
1577 > (fma:fl [%f & --13 2.718] [%f & --23 11] [%f & --13 2.718])
1578 [%f s=%.y e=-62 a=9.475.054.514.089.037.465.004.673.635.188.736]=
1586Produces the square root of `a`.
1601 ?: ?=([$n *] a) [%n ~]
1602 ?: ?=([$i *] a) ?:(s.a a [%n ~])
1604 ?: s.a (^sqt +>.a) [%n ~]
1610 > (sqt:fl [%f s=%.y e=-18 a=342.602.577])
1611 [%f s=%.y e=-107 a=5.865.903.143.604.945.574.132.671.852.050.553]
1619Produces the inverse of `a` by dividing `1` by `a`.
1634 (div [%f & --0 1] a)
1640 > (inv:fl [%f s=%.y e=--0 a=10])
1641 [%f s=%.y e=-116 a=8.307.674.973.655.724.205.648.794.126.752.154]
1642 > (drg:fl [%f s=%.y e=-116 a=8.307.674.973.655.724.205.648.794.126.752.154])
1645 > (inv:fl [%f s=%.y e=--1 a=10])
1646 [%f s=%.y e=-117 a=8.307.674.973.655.724.205.648.794.126.752.154]
1647 > (drg:fl [%f s=%.y e=-117 a=8.307.674.973.655.724.205.648.794.126.752.154])
1650 > (inv:fl [%f s=%.y e=--2 a=10])
1651 [%f s=%.y e=-118 a=8.307.674.973.655.724.205.648.794.126.752.154]
1652 > (drg:fl [%f s=%.y e=-118 a=8.307.674.973.655.724.205.648.794.126.752.154])
1653 [%d s=%.y e=-3 a=25]
1659Signed integer to float
1661Produces `a` in floating-point representation.
1665`a` is an unsigned integer.
1683 [%f s=%.y e=--0 a=0]
1685 > (sun:fl 5.048.729)
1686 [%f s=%.y e=-90 a=6.250.023.776.601.238.669.911.180.544.311.296]
1695Signed integer to float
1697Produces the floating-point representation of `a`, an unsigned integer.
1701`a` is an unsigned integer.
1713 (rou [%f -.b --0 +.b])
1720 [%f s=%.y e=-106 a=8.112.963.841.460.668.169.578.900.514.406.400]
1723 [%f s=%.n e=-106 a=8.112.963.841.460.668.169.578.900.514.406.400]
1731Tests if `a` is less than `b`. Returns `~` in the event of `a` or `b` being a
1748 |= [a=fn b=fn] ^- (unit ?)
1749 ?: |(?=([$n *] a) ?=([$n *] b)) ~ :- ~
1751 ?: ?=([$i *] a) !s.a ?: ?=([$i *] b) s.b
1752 ?: |(=(a.a 0) =(a.b 0))
1753 ?: &(=(a.a 0) =(a.b 0)) |
1754 ?: =(a.a 0) s.b !s.a
1756 ?: s.a (^lth +>.a +>.b) (^lth +>.b +>.a)
1762 > (lth:fl (sun:fl 116) (sun:fl 4.820))
1765 > (lth:fl (sun:fl 218.116) (sun:fl 4.820))
1768 > (lth:fl (sun:fl 218.116) [%n ~])
1777Tests whether `a` is less than or equal to `b`. Returns `~` in the event of `a`
1778or `b` being a NaN (`[%n ~]`).
1794 |= [a=fn b=fn] ^- (unit ?)
1795 %+ bind (lth b a) |= a=? !a
1801 > (lte:fl (sun:fl 102) [%f %.y -5 973.655.724])
1804 > (lte:fl (sun:fl 102) [%f %.y -24 973.655.724])
1807 > (lte:fl [%f %.y --2 25] (sun:fl 100))
1810 > (lte:fl [%f %.y --2 25] [%f %.y --3 2])
1813 > (lte:fl [%f %.y --2 25] [%n ~])
1822Tests if `a` is equal to `b`. Returns `~` in the event of `a` or `b` being a NaN
1839 |= [a=fn b=fn] ^- (unit ?)
1840 ?: |(?=([$n *] a) ?=([$n *] b)) ~ :- ~
1842 ?: |(?=([$i *] a) ?=([$i *] b)) |
1843 ?: |(=(a.a 0) =(a.b 0))
1844 ?: &(=(a.a 0) =(a.b 0)) & |
1845 ?: |(=(e.a e.b) !=(s.a s.b)) |
1852 > (equ:fl [%f %.y --2 25] (sun:fl 100))
1855 > (equ:fl [%f %.y --2 25] (sun:fl 101))
1862Greater or equal than
1864Tests whether `a` is greater than or equal to `b`. Returns `~` in the event of
1865`a` or `b` being a NaN (`[%n ~]`).
1881 |= [a=fn b=fn] ^- (unit ?) (lte b a)
1887 > (gte:fl [%f %.y --2 25] (sun:fl 100))
1890 > (gth:fl [%f %.y --6 73.989] [%f %.y --5 919.599])
1893 > (gth:fl [%f %.y --6 73.989] [%n ~])
1902Tests whether `a` is greater than `b`. Returns `~` in the event of `a` or `b`
1903being a NaN (`[%n ~]`).
1919 |= [a=fn b=fn] ^- (unit ?) (lth b a)
1925 > (gth:fl [%f %.y --2 25] (sun:fl 100))
1928 > (gth:fl [%f %.y --6 73.989] [%f %.y --5 119.599])
1937Produces the decimal form of `a` using the Dragon4 algorithm. Guarantees
1938accurate results for rounded floats.
1953 ?: ?=([$n *] a) [%n ~]
1954 ?: ?=([$i *] a) [%i s.a]
1955 ?~ a.a [%d s.a --0 0]
1956 [%d s.a (^drg +>.a)]
1962 > (drg:fl [%f | --6 73.989])
1963 [%d s=%.n e=--0 a=4.735.296]
1971Converts decimal `a` to `fn`.
1986 ?: ?=([$n *] a) [%n ~]
1987 ?: ?=([$i *] a) [%i s.a]
1991 (mul [%f s.a --0 a.a] [%f & e.a (pow 5 q)])
1992 (div [%f s.a --0 a.a] [%f & (sun:si q) (pow 5 q)])
1998 > (grd:fl [%d s=%.n e=--0 a=73.989])
1999 [%f s=%.n e=-96 a=5.862.012.516.267.904.074.208.723.341.410.304]
2001 > (grd:fl [%d s=%.n e=--0 a=100])
2002 [%f s=%.n e=-106 a=8.112.963.841.460.668.169.578.900.514.406.400]
2008Round to signed integer
2010Rounds `a` to the nearest signed integer.
2024 |= [a=fn] ^- (unit @s)
2026 ?. ?=([$f *] b) ~ :- ~
2027 =+ c=(^^mul (bex (abs:si e.b)) a.b)
2034 > (toi:fl [%f s=%.y e=-78 a=8.112.963.841.460.668.169.578.900.514.406.400])
2035 [~ u=--26.843.545.600]
2043Rounds `a` to the nearest decimal integer.
2061 =.(r swr (fli (^toj +>.a)))
2068 > (toj:fl [%f s=%.y e=-78 a=8.112.963.841.460.668.169.578.900.514.406.400])
2069 [%f s=%.y e=--0 a=26.843.545.600]
2071 > (toj:fl [%f s=%.y e=-78 a=8.112.963.841.460.668.169.578.900.514])
2072 [%f s=%.y e=--0 a=26.844]
2074 > (toj:fl [%f s=%.y e=-78 a=8.112.963.841.460])
2075 [%f s=%.y e=--0 a=0]
2077 > (toj:fl [%f s=%.y e=-9 a=9.002])
2078 [%f s=%.y e=--0 a=16]
2086Container core for IEEE-754 formatting operations.
2088- `w` is width: The number of bits in the exponent field.
2089- `p` is precision: The number of bits in the significand field.
2090- `w` is bias: Added to exponent when storing.
2091- `r` is rounding mode: Possible modes are nearest (`$n`), up (`$u`),
2092 down (`$d`), to zero (`$z`), and away from zero (`$a`). Default value is `%a`.
2098 |_ [[w=@u p=@u b=@s] r=$?($n $u $d $z $a)]
2104 > =ffcore ~(. ff [8 8 0] %n)
2106 <24.ltg {{{@ud @ud @ud} r/$n} <54.tyv 119.wim 31.ohr 1.jmk $143>}>
2113`++ff` has no use outside of the functionality provided to other cores: `++rd`,
2114`++rs`, `++rq`, and `++rh`. It's not intended to be used directly; it's just
2115meant to power those cores.
2122Produces the sign bit of `++ff`.
2127 ++ sb (bex (^add w p))
2142Produces the minimum possible exponent of `ff`.
2147 ++ me (dif:si (dif:si --1 b) (sun:si p))
2162Instantiates the core `fl`, giving values to its samples based on the
2163configuration of the `ff` core.
2169 %*(. fl p +(p), v me, w (^sub (bex w) 3), d %d, r r)
2175 > ~(pa ff [11 52 --1.023] %n)
2178 { {{p/@ v/@s w/@} r/?($n $u $a $d $z) d/$d}
2179 <54.tyv 119.wim 31.ohr 1.jmk $143>
2186`++pa` exists exclusively for internal use of `++ff`, and `++ff` exists for
2187internal use in other cores.
2194Converts `a` from `@r` to `fn`.
2198`a` is a `@r`, an IEEE float.
2209 =+ [f=(cut 0 [0 p] a) e=(cut 0 [p w] a)]
2212 ?: =(f 0) [%f s --0 0] [%f s me f]
2214 ?: =(f 0) [%i s] [%n ~]
2215 =+ q=:(sum:si (sun:si e) me -1)
2216 =+ r=(^add f (bex p))
2224 [%f s=%.y e=--0 a=0]
2230`fn` to `@r`, rounding
2232Converts `a` from `fn` to `@r` and applies rounding.
2245 ++ bit |= [a=fn] (bif (rou:pa a))
2251 > (bit:ma:rd [%f | -6 202])
2258`fn` to `@r`, no rounding
2260Converts `a` from `fn` to `@r`. No rounding is applied.
2264`a` is a `@r`, an IEEE float.
2276 =+ q=(lsh 0 p (fil 0 w 1))
2277 ?: s.a q (^add q sb)
2278 ?: ?=([$n *] a) (lsh 0 (dec p) (fil 0 +(w) 1))
2279 ?~ a.a ?: s.a `@r`0 sb
2284 ?: s.a `@r`a.a (^add a.a sb)
2285 =+ q=(sum:si (dif:si e.a me) --1)
2286 =+ r=(^add (lsh 0 p (abs:si q)) (end 0 p a.a))
2287 ?: s.a r (^add r sb)
2302Produces the sign of `a`.
2306`a` is a `@r`, an IEEE float.
2317 =(0 (cut 0 [(^add p w) 1] a))
2332Produces the exponent of `a`.
2336`a` is a `@r`, an IEEE float.
2347 (dif:si (sun:si (cut 0 [p w] a)) b)
2362Produces the sum of `a` plus `b`.
2366`a` is a `@r`, an IEEE float.
2368`b` is a `@r`, an IEEE float.
2372A `@r`, an IEEE float.
2379 (bif (add:pa (sea a) (sea b)))
2385 > (add:ma:rd `@r`5 `@r`11)
2394Produces the sum of `a` plus `b`.
2398`a` is a `@r`, an IEEE float.
2400`b` is a `@r`, an IEEE float.
2404A `@r`, an IEEE float.
2411 (bif (sub:pa (sea a) (sea b)))
2417 > (sub:ma:rd `@r`5 `@r`11)
2426Produces the product of `a` multiplied by `b`.
2430`a` is a `@r`, an IEEE float.
2432`b` is a `@r`, an IEEE float.
2436A `@r`, an IEEE float.
2443 (bif (mul:pa (sea a) (sea b)))
2449 > (mul:ma:rd `@r`11 `@r`2)
2458Produces the quotient of `a` divided by `b`.
2462`a` is a `@r`, an IEEE float.
2464`b` is a `@r`, an IEEE float.
2468A `@r`, an IEEE float.
2475 (bif (div:pa (sea a) (sea b)))
2481 > (div:ma:rd `@r`175 `@r`26)
2490Produces the sum of `c` plus the product of `a` multiplied by `b`;
2495`a` is a `@r`, an IEEE float.
2497`b` is a `@r`, an IEEE float.
2499`c` is a `@r`, an IEEE float.
2503A `@r`, an IEEE float.
2510 (bif (fma:pa (sea a) (sea b) (sea c)))
2516 > (fma:ma:rd `@r`175 `@r`26 `@r`100)
2525Produces the square root of `a`.
2529`a` is a `@r`, an IEEE float.
2533A `@r`, an IEEE float.
2540 (bif (sqt:pa (sea a)))
2546 > (sqt:ma:rd `@r`175)
2555Tests whether `a` is less than `b`.
2559`a` is a `@r`, an IEEE float.
2561`b` is a `@r`, an IEEE float.
2571 |= [a=@r b=@r] (fall (lth:pa (sea a) (sea b)) |)
2577 > (lth:ma:rd `@rd`1 `@rd`2)
2580 > (lth:ma:rd `@rd`10 `@rd`2)
2587Less than or equal to
2589Tests whether `a` is less than or equal to `b`.
2593`a` is a `@r`, an IEEE float.
2595`b` is a `@r`, an IEEE float.
2605 |= [a=@r b=@r] (fall (lte:pa (sea a) (sea b)) |)
2611 > (lte:ma:rd `@rd`10 `@rd`2)
2614 > (lte:ma:rd `@rd`10 `@rd`10)
2623Tests whether `a` is equal to `b`.
2627`a` is a `@r`, an IEEE float.
2629`b` is a `@r`, an IEEE float.
2639 |= [a=@r b=@r] (fall (equ:pa (sea a) (sea b)) |)
2645 > (equ:ma:rd `@rd`10 `@rd`2)
2648 > (equ:ma:rd `@rd`10 `@rd`10)
2655Greater or equal than
2657Tests whether `a` is greater than or equal to `b`.
2661`a` is a `@r`, an IEEE float.
2663`b` is a `@r`, an IEEE float.
2673 |= [a=@r b=@r] (fall (gte:pa (sea a) (sea b)) |)
2679 > (gte:ma:rd `@rd`10 `@rd`10)
2682 > (gte:ma:rd `@rd`10 `@rd`11)
2691Tests whether `a` is greater than or equal to `b`.
2695`a` is a `@r`, an IEEE float.
2697`b` is a `@r`, an IEEE float.
2707 |= [a=@r b=@r] (fall (gth:pa (sea a) (sea b)) |)
2713 > (gth:ma:rd `@rd`10 `@rd`10)
2716 > (gth:ma:rd `@rd`10 `@rd`9)
2723Unsigned integer to `@r`
2725Converts `a` from an unsigned integer (`@u`) to `@r`.
2729`a` is `@u`, unsigned integer.
2733A `@r`, an IEEE float.
2739 |= [a=@u] (bit [%f & --0 a])
2745 > (sun:ma:rd 658.149.282)
2752Signed integer to `@r`
2754Converts `a` from a signed integer to `@r`.
2758`a` is `@s`, an unsigned integer
2762A `@r`, an IEEE float.
2768 |= [a=@s] (bit [%f (syn:si a) --0 (abs:si a)])
2783Rounds `a` to the nearest signed integer.
2787`a` is a `@r`, an IEEE float.
2797 |= [a=@r] (toi:pa (sea a))
2803 > (toi:ma:rd `@r`0x4af)
2810`@r` to decimal float
2812Converts `a` from `@r` to `dn` using the Dragon4 algorithm.
2816`a` is a `@r`, an IEEE float.
2826 |= [a=@r] (drg:pa (sea a))
2832 > (drg:ma:rd `@r`0x41c0)
2833 [%d s=%.y e=-323 a=8.316]
2835 > (drg:ma:rd (sun:ma:rd 658.149.282))
2836 [%d s=%.y e=--0 a=658.149.282]
2842Decimal float to `@r`
2844Converts `a` from `dn` to `@r`.
2852A `@r`, an IEEE float.
2858 |= [a=dn] (bif (grd:pa a))
2865 > (grd:ma:rd [%d s=%.y e=--0 a=658.149.282])
2874Converts `a` from a double-precision binary float to decimal64.
2878`a` is a `@rd`, a double-precision float.
2887 ++ rlyd |= a=@rd ^- dn (drg:rd a)
2893 > (rlyd .~2.4703e-320)
2894 [%d s=%.y e=-324 a=24.703]
2902Converts `a` from a single-precision binary float to decimal32.
2906`a` is a `@rs`, a single-precision float.
2915 ++ rlys |= a=@rs ^- dn (drg:rs a)
2921 > (rlys .1.681557e-39)
2922 [%d s=%.y e=-45 a=1.681.557]
2930Converts `a` from a half-precision binary float to decimal16.
2934`a` is a `@rh`, a half-precision float.
2943 ++ rlyh |= a=@rh ^- dn (drg:rh a)
2950 [%d s=%.y e=--1 a=3]
2958Converts `a` from a quad-precision binary float to decimal128.
2962`a` is a `@rq`, a quad-precision float.
2971 ++ rlyq |= a=@rq ^- dn (drg:rq a)
2977 > (rlyq .~~~2.2628017865927756787440310766086816e-4343)
2978 [%d s=%.y e=-4.377 a=22.628.017.865.927.756.787.440.310.766.086.816]
2986Converts `a` from a decimal float to a double-precision binary float.
2994A a `@rd`, a double-precision float.
2999 ++ ryld |= a=dn ^- @rd (grd:rd a)
3005 > (ryld [%d s=%.y e=-324 a=24.703])
3014Converts `a` from a decimal float to a single-precision binary float.
3022A a `@rs`, a single-precision float.
3027 ++ ryls |= a=dn ^- @rs (grd:rs a)
3033 > (ryls [%d s=%.y e=-324 a=24.703])
3036 > (ryls [%d s=%.y e=-32 a=24.703])
3045Converts `a` from a decimal float to a half-precision binary float.
3053A a `@rh`, a half-precision float.
3058 ++ rylh |= a=dn ^- @rh (grd:rh a)
3064 > (rylh [%d s=%.y e=--1 a=703])
3067 > (rylh [%d s=%.y e=--3 a=56])
3070 > (rylh [%d s=%.y e=--4 a=56])
3079Converts `a` from a decimal float to a quad-precision binary float.
3087A a `@rq`, a quad-precision float.
3092 ++ rylq |= a=dn ^- @rq (grd:rq a)
3098 > (rylq [%d s=%.y e=-324 a=24.703])
3107A container core for operations related to double-precision binary floats.
3109`++rd` has four rounding modes: round to nearest (`$n`), round up (`$u`), round
3110down (`$d`), and round to zero (`$z`).
3118 |_ r=$?($n $u $d $z)
3126Instantiates the core `ff`, giving values to its samples based on the
3127configuration of the `rd` core.
3133 %*(. ff w 11, p 52, b --1.023, r r)
3141 {{{w/@ud p/@ud b/@sd} r/?($n $u $d $z)} <54.tyv 119.wim 31.ohr 1.jmk $143>}
3150Converts `a` from a double-precision binary float to `fn`.
3154 |= [a=@rd] (sea:ma a)
3160 > (sea:rd .~4.94066e-319)
3161 [%f s=%.y e=-1.074 a=100.000]
3169Converts `a` from `fn` to a double-precision binary float.
3177A `@rd`, a double-precision float.
3183 |= [a=fn] ^- @rd (bit:ma a)
3189 > (bit:rd [%f s=%.y e=-1.074 a=100.000])
3198Produces the sum of `a` plus `b`.
3202`a` is a `@rd`, a double-precision float.
3204`b` is a `@rd`, a double-precision float.
3214 |= [a=@rd b=@rd] ^- @rd
3222 > (add:rd .~3.94066e12 .~9.2846e11)
3231Produces the difference of `a` minus `b`.
3235`a` is a `@rd`, a double-precision float.
3237`b` is a `@rd`, a double-precision float.
3247 |= [a=@rd b=@rd] ^- @rd
3255 > (sub:rd .~7.94069e2 .~1.2846e3)
3256 .~-4.9053099999999995e2
3264Produces the product of `a` times `b`.
3268`a` is a `@rd`, a double-precision float.
3270`b` is a `@rd`, a double-precision float.
3280 |= [a=@rd b=@rd] ^- @rd
3288 > (mul:rd .~7.94069e2 .~1.2246e3)
3289 .~9.724168973999998e5
3297Produces the quotient of `a` divided by `b`.
3301`a` is a `@rd`, a double-precision float.
3313 |= [a=@rd b=@rd] ^- @rd
3321 > (div:rd .~7.94099e2 .~1.2246e3)
3322 .~6.484558223093255e-1
3330Produces the sum of `c` plus the product of `a` multiplied by `b`;
3335`a` is a `@rd`, an IEEE float.
3352 |= [a=@rd b=@rd c=@rd] ^- @rd
3360 > (fma:rd .~7.94099e2 .~1.2246e3 .~3.94066e3)
3369Produces the square root of `a`.
3373`a` is a `@rd`, a double-precision float.
3383 |= [a=@rd] ^- @rd ~_ leaf+"rd-fail"
3390 > (sqt:rd .~3.94066e3)
3391 .~6.2774676422901614e1
3399Test whether `a` is less than `b`.
3403`a` is a `@rd`, a double-precision float.
3405`b` is a `@rd`, a double-precision float.
3423 > (lth:rd .~7.94099e2 .~1.2246e3)
3426 > (lth:rd .~7.94099e2 .~1.2246e2)
3429 > (lth:rd .~1.2246e2 .~1.2246e2)
3438Test whether `a` is less than or equal to `b`.
3442`a` is a `@rd`, a double-precision float.
3444`b` is a `@rd`, a double-precision float.
3453 ++ lte ~/ %lte :: less-equals
3462 > (lte:rd .~7.94099e2 .~1.2246e3)
3465 > (lte:rd .~7.94099e2 .~1.2246e2)
3468 > (lte:rd .~1.2246e2 .~1.2246e2)
3477Test whether `a` is equal to `b`.
3481`a` is a `@rd`, a double-precision float.
3483`b` is a `@rd`, a double-precision float.
3501 > (equ:rd .~7.94099e2 .~1.2246e3)
3504 > (equ:rd .~7.94099e2 .~1.2246e2)
3507 > (equ:rd .~1.2246e2 .~1.2246e2)
3514Greater than or equal
3516Test whether `a` is greater than or equal to `b`.
3520`a` is a `@rd`, a double-precision float.
3522`b` is a `@rd`, a double-precision float.
3540 > (gte:rd .~7.94099e2 .~1.2246e3)
3543 > (gte:rd .~7.94099e2 .~1.2246e2)
3546 > (gte:rd .~1.2246e2 .~1.2246e2)
3555Test whether `a` is greater `b`.
3559`a` is a `@rd`, a double-precision float.
3561`b` is a `@rd`, a double-precision float.
3579 > (gth:rd .~7.94099e2 .~1.2246e3)
3582 > (gth:rd .~7.94099e2 .~1.2246e2)
3585 > (gth:rd .~1.2246e2 .~1.2246e2)
3592Unsigned integer to `@rd`
3594Converts an unsigned integer `a` to `@rd`.
3598`a` is a `@u`, an unsigned integer.
3607 ++ sun |= [a=@u] ^- @rd (sun:ma a)
3620Unsigned integer to `@rd`
3622Converts a signed integer `a` to `@rd`.
3626`a` is a `@s`, a signed integer.
3635 ++ san |= [a=@s] ^- @rd (san:ma a)
3650Produces the sign of `a`.
3663 ++ sig |= [a=@rd] ^- ? (sig:ma a)
3669 > (sig:rd .~1.2246e3)
3678Produces the exponent of `a`.
3691 ++ exp |= [a=@rd] ^- @s (exp:ma a) :: get exponent
3697 > (exp:rd .~1.2246e3)
3706Rounds `a` to the nearest integer.
3719 ++ toi |= [a=@rd] ^- (unit @s) (toi:ma a)
3725 > (toi:rd .~1.2246e3)
3732`@rd` to decimal float
3734Produces the decimal form of `a` using the Dragon4 algorithm. Guarantees
3735accurate results for rounded floats.
3748 ++ drg |= [a=@rd] ^- dn (drg:ma a)
3754 > (drg:rd .~1.2246e3)
3755 [%d s=%.y e=-1 a=12.246]
3761Decimal float to `@rd`
3763Converts `a` from decimal float to `@rd`.
3776 ++ grd |= [a=dn] ^- @rd (grd:ma a) :: decimal float to @rd
3783 > (grd:rd [%d s=%.y e=-1 a=12.246])
3792A container core for operations related to single-precision binary floats.
3794`++rs` has four rounding modes: round to nearest (`$n`), round up (`$u`), round
3795down (`$d`), and round to zero (`$z`).
3803 |_ r=$?($n $u $d $z)
3811Instantiates the core `ff`, giving values to its samples based on the
3812configuration of the `rs` core.
3818 %*(. ff w 8, p 23, b --127, r r)
3826 {{{w/@ud p/@ud b/@sd} r/?($n $u $d $z)} <54.tyv 119.wim 31.ohr 1.jmk $143>}
3835Converts `a` from `@rs` to `fn`.
3839`a` is a `@rs`, an single-precision float.
3849 |= [a=@rs] (sea:ma a)
3856 [%f s=%.y e=-149 a=100]
3864Converts `a` from `fn` to `@rs`.
3872A `@rs`, a single-precision float.
3878 |= [a=fn] ^- @rs (bit:ma a)
3884 > (bit:rs [%f & -2 1.000])
3893Produces the sum of `a` plus `b`.
3897`a` is a `@rs`, a single-precision float.
3909 |= [a=@rs b=@rs] ^- @rs
3917 > (add:rs .2.5e1 .2.5e2)
3926Subtracts `a` from `b`.
3932 |= [a=@rs b=@rs] ^- @rs
3940 > (sub:rs .2.5e1 .2.5e2)
3949Produces the product of `a` multiplied by `b`.
3953`a` is a `@rs`, a single-precision float.
3965 |= [a=@rs b=@rs] ^- @rs
3973 > (mul:rs .2.5e1 .2.5e2)
3982Produces the quotient of `a` divided by `b`.
3986`a` is a `@rs`, a single-precision float.
3998 |= [a=@rs b=@rs] ^- @rs
4006 > (div:rs .4.5e1 .2.2e2)
4015Produces the sum of `c` plus the product of `a` multiplied by `b`;
4020`a` is a `@rs`, a single-precision float.
4034 |= [a=@rs b=@rs c=@rs] ^- @rs
4042 > (fma:rs .2.5e1 .2.5e2 .8.2e1)
4051Produces the square root of `a`.
4055`a` is a `@rs`, a single-precision float.
4082Test whether `a` is less than `b`.
4086`a` is a `@rs`, a single-precision float.
4106 > (lth:rs .9.9e1 .1.1e2)
4109 > (lth:rs .9.9e1 .9.9e1)
4118Test whether `a` is less than or equal to `b`.
4122`a` is a `@rs`, a single-precision float.
4142 > (lte:rs .9.9e1 .1.1e2)
4145 > (lte:rs .9.9e1 .9.9e1)
4154Test whether `a` is equal to `b`.
4158`a` is a `@rs`, a single-precision float.
4178 > (equ:rs .9.9e1 .1.1e2)
4181 > (equ:rs .9.9e1 .9.9e1)
4188Greater than or equal
4190Test whether `a` is greater than or equal to `b`.
4194`a` is a `@rs`, a single-precision float.
4214 > (gte:rs .9.9e1 .9.9e1)
4217 > (gte:rs .9.9e1 .9.2e2)
4226Test whether `a` is greater than `b`.
4230`a` is a `@rs`, a single-precision float.
4250 > (gth:rs .9.9e1 .9.2e2)
4253 > (gth:rs .9.9e1 .9.9e1)
4256 > (gth:rs .9.9e1 .1.9e1)
4263Unsigned integer to `@rs`
4265Converts `a` from an unsigned integer to `@rs`.
4269`a` is an unsigned integer.
4278 ++ sun |= [a=@u] ^- @rs (sun:ma a)
4291Signed integer to `@rs`'
4293Converts `a` from an unsigned integer to `@rs`.
4297`a` is a signed integer.
4306 ++ san |= [a=@s] ^- @rs (san:ma a)
4320Produces the sign of `a`.
4333 ++ sig |= [a=@rs] ^- ? (sig:ma a)
4351Produces the exponent of `a`.
4364 ++ exp |= [a=@rs] ^- @s (exp:ma a)
4379Rounds `a` to the nearest integer.
4392 ++ toi |= [a=@rs] ^- (unit @s) (toi:ma a)
4405`@rs` to decimal float
4407Produces the decimal form of `a` using the Dragon4 algorithm. Guarantees
4408accurate results for rounded floats.
4421 ++ drg |= [a=@rs] ^- dn (drg:ma a)
4428 [%d s=%.n e=--0 a=343]
4434Decimal float to `@rs`
4436Converts `a` from `dn` to `@rs`.
4449 ++ grd |= [a=dn] ^- @rs (grd:ma a)
4456 > (grd:rs [%d s=%.n e=--0 a=343])
4463Quadruple-precision fp
4465A container core for operations related to quadruple-precision binary floats.
4467`++rq` has four rounding modes: round to nearest (`$n`), round up (`$u`), round
4468down (`$d`), and round to zero (`$z`).
4476 |_ r=$?($n $u $d $z)
4483Instantiates the core `ff`, giving values to its samples based on the
4484configuration of the `rq` core.
4490 %*(. ff w 15, p 112, b --16.383, r r)
4498Converts `a` from `@rq` to `fn`.
4502`a` is a `@rq`, a quad-precision float.
4512 |= [a=@rq] (sea:ma a)
4518 > (sea:rq .~~~1.05102e5)
4519 [%f s=%.y e=-96 a=8.327.038.336.574.210.409.756.656.268.214.272]
4527Converts `a` from `fn` to `@rq`.
4535A `@rq`, a quad-precision float.
4541 |= [a=n] ^- @rq (bit:ma a)
4547 > (bit:rq [%f s=%.y e=-96 a=8.327.038.336.574.210.409.756.656.268.214.272])
4556Produces the sum of `a` plus `b`.
4560`a` is a `@rq`, a quad-precision float.
4572 |= [a=@rq b=@rq] ^- @rq
4580 > (add:rq .~~~-1.821e5 .~~~1.05102e5)
4589Produces the difference of `a` minus `b`.
4593`a` is a `@rq`, a quad-precision float.
4605 |= [a=@rq b=@rq] ^- @rq
4613 > (sub:rq .~~~1.821e5 .~~~1.05102e5)
4616 > (sub:rq .~~~1.821e5 .~~~-1.05102e5)
4625Produces the product of `a` times `b`.
4629`a` is a `@rq`, a quad-precision float.
4641 |= [a=@rq b=@rq] ^- @rq
4649 > (mul:rq .~~~1.821e5 .~~~-1.05102e5)
4658Produces the product of `a` divided by `b`.
4662`a` is a `@rq`, a quad-precision float.
4674 |= [a=@rq b=@rq] ^- @rq
4682 > (div:rq .~~~1.821e5 .~~~1.05102e3)
4683 .~~~1.732602614602957127361991208540275e2
4691Produces the sum of `c` plus the product of `a` multiplied by `b`;
4696`a` is a `@rq`, a quad-precision float.
4710 |= [a=@rq b=@rq c=@rq] ^- @rq
4718 > (fma:rq .~~~1.821e5 .~~~-1.05102e2 .~~~6.2044e7)
4727Produces the square root of `a`.
4731`a` is a `@rq`, a quad-precision float.
4749 > (sqt:rq .~~~6.2044e7)
4750 .~~~7.876801381271461258959876570289002e3
4758Tests whether `a` is less than `b`.
4762`a` is a `@rq`, a quad-precision float.
4783 > (lth:rq .~~~1.2044e7 (mul:rq .~~~9.02e2 .~~~7.114e3))
4786 > (lth:rq .~~~1.2044e7 (mul:rq .~~~9.02e3 .~~~7.114e3))
4795Tests whether `a` is less than or equal to `b`.
4799`a` is a `@rq`, a quad-precision float.
4819 > (lte:rq .~~~1.2044e7 (mul:rq .~~~9.02e2 .~~~7.114e3))
4822 > (lte:rq .~~~1.2044e7 (mul:rq .~~~9.02e3 .~~~7.114e3))
4825 > (lte:rq .~~~1.2044e7 .~~~1.2044e7)
4834Tests whether `a` is equal to `b`.
4838`a` is a `@rq`, a quad-precision float.
4858 > (equ:rq .~~~1.2044e7 .~~~1.2044e7)
4861 > (equ:rq .~~~2.2044e7 .~~~1.2044e7)
4868Greater than or equal
4870Tests whether `a` is greater than or equal to `b`.
4874`a` is a `@rq`, a quad-precision float.
4894 > (gte:rq .~~~1.2044e7 .~~~1.2044e7)
4897 > (gte:rq .~~~2.2044e7 .~~~1.2044e7)
4900 > (gte:rq .~~~1.2044e7 .~~~2.2044e7)
4907Tests whether `a` is greater than `b`.
4911`a` is a `@rq`, a quad-precision float.
4931 > (gth:rq .~~~1.2044e7 .~~~1.2044e7)
4934 > (gth:rq .~~~2.2044e7 .~~~1.2044e7)
4937 > (gth:rq .~~~1.2044e7 .~~~2.2044e7)
4944Unsigned integer to `@rq`
4946Converts `@` from an unsigned integer to `@rq`.
4950`a` is a `@u`, an unsigned integer.
4954A `@rq`, a quad-precision float.
4959 ++ sun |= [a=@u] ^- @rq (sun:ma a)
4972Signed integer to `rq`
4974Converts `@` from a signed integer to `@rq`.
4978`a` is a `@s`, a signed integer.
4982A `@rq`, a quad-precision float.
4987 ++ san |= [a=@s] ^- @rq (san:ma a)
5002Produces the sign of `a`.
5006`a` is a `@rq`, a quad-precision float.
5015 ++ sig |= [a=@rq] ^- ? (sig:ma a)
5021 > (sig:rq .~~~-2.05e2)
5030Gets the exponent of `a`.
5034`a` is a `@rq`, a quad-precision float.
5038A `@s`, a signed integer.
5043 ++ exp |= [a=@rq] ^- @s (exp:ma a)
5049 > (exp:rq .~~~-2.05e2)
5058Rounds `a` to the nearest integer.
5062`a` is a `@rq`, a quad-precision float.
5071 ++ toi |= [a=@rq] ^- (unit @s) (toi:ma a)
5077 > (toi:rq .~~~-2.085e2)
5080 > (toi:rq .~~~-2.08e2)
5087`@rq` to decimal float
5089Produces the decimal form of `a` using the Dragon4 algorithm. Guarantees
5090accurate results for rounded floats.
5094`a` is a `@rq`, a quad-precision float.
5103 ++ drg |= [a=@rq] ^- dn (drg:ma a) :: @rq to decimal float
5109 > (drg:rq .~~~-2.085e2)
5110 [%d s=%.n e=-1 a=2.085]
5112 > (drg:rq .~~~-2.08e2)
5113 [%d s=%.n e=--0 a=208]
5119Decimal float to `@rq`
5121Converts `a` from `dn` to `@rq`.
5131A `@rq`, a quad-precision float.
5136 ++ grd |= [a=dn] ^- @rq (grd:ma a)
5143 > (grd:rq [%d s=%.n e=--0 a=343])
5152A container core for operations related to half-precision binary floats.
5154`++rh` has four rounding modes: round to nearest (`$n`), round up (`$u`), round
5155down (`$d`), and round to zero (`$z`).
5163 |_ r=$?($n $u $d $z)
5171Instantiates the core `ff`, giving values to its samples based on the
5172configuration of the `rh` core.
5178 %*(. ff w 5, p 10, b --15, r r)
5186Converts `a` from `@rh` to `fn`.
5190`a` is a `@rh`, a half-precision float.
5200 |= [a=@rh] (sea:ma a)
5206 > (sea:rh .~~1.22e-5)
5207 [%f s=%.y e=-24 a=205]
5215Converts `a` from `fn` to `@rh`.
5223A `@rh`, a half-precision float.
5229 |= [a=fn] ^- @rh (bit:ma a)
5235 > (bit:rh [%f s=%.y e=-24 a=205])
5242Produces the sum of `a` plus `b`.
5246`a` is a `@rh`, a half-precision float.
5258 |= [a=@rh b=@rh] ^- @rh
5266 > (add:rh .~~1.82e2 .~~1.02e2)
5275Produces the difference of `a` minus `b`.
5279`a` is a `@rh`, a half-precision float.
5291 |= [a=@rh b=@rh] ^- @rh
5299 > (sub:rh .~~1.821e2 .~~1.051e2)
5302 > (sub:rh .~~1.821e2 .~~6.051e2)
5311Produces the product of `a` times `b`.
5315`a` is a `@rh`, a quad-precision float.
5327 |= [a=@rh b=@rh] ^- @rh
5335 > (mul:rh .~~1.821e1 .~~-1.05102e2)
5344Produces the product of `a` divided by `b`.
5348`a` is a `@rh`, a half-precision float.
5360 |= [a=@rh b=@rh] ^- @rh
5368 > (div:rh .~~1.821e3 .~~1.05102e2)
5377Produces the sum of `c` plus the product of `a` multiplied by `b`;
5382`a` is a `@rh`, a half-precision float.
5396 |= [a=@rh b=@rh c=@rh] ^- @rh
5404 > (fma:rh .~~1.821e4 .~~-1.05102e2 .~~6.2044e3)
5413Produces the square root of `a`.
5417`a` is a `@rh`, a half-precision float.
5435 > (sqt:rh .~~6.24e4)
5444Tests whether `a` is less than `b`.
5448`a` is a `@rh`, a half-precision float.
5468 > (lth:rh .~~1.2e5 (mul:rh .~~9.02e2 .~~7.114e2))
5471 > (lth:rh .~~1.2e3 (mul:rh .~~9.02e1 .~~7.114e2))
5480Tests whether `a` is less than or equal to `b`.
5484`a` is a `@rh`, a half-precision float.
5504 > (lte:rh .~~1.2e5 (mul:rh .~~9.02e2 .~~7.114e2))
5507 > (lte:rh .~~1.2e3 (mul:rh .~~9.02e1 .~~7.114e2))
5510 > (lte:rh .~~1.2e3 .~~1.2e3)
5519Tests whether `a` is equal to `b`.
5523`a` is a `@rh`, a half-precision float.
5543 > (equ:rh .~~1.24e4 .~~1.24e4)
5546 > (equ:rh .~~2.24e4 .~~1.24e4)
5553Greater than or equal
5555Tests whether `a` is greater than or equal to `b`.
5559`a` is a `@rh`, a half-precision float.
5579 > (gte:rh .~~1.24e4 .~~1.24e4)
5582 > (gte:rh .~~2.24e4 .~~1.24e4)
5585 > (gte:rh .~~1.24e4 .~~2.24e4)
5592Tests whether `a` is greater than `b`.
5596`a` is a `@rh`, a half-precision float.
5616 > (gth:rh .~~1.24e4 .~~1.244e4)
5619 > (gth:rh .~~2.24e4 .~~1.24e4)
5622 > (gth:rh .~~1.24e4 .~~2.24e4)
5632Converts `@` from `@rh` to `@rs`.
5636`a` is a `@rh`, a half-precision float.
5640A `@rs`, a single-precision float.
5645 ++ tos |= {a/@rh} (bit:rs (sea a))
5660Converts `@` from `@rs` to `@rh`.
5664`a` is a `@rs`, a single-precision float.
5668A `@rh`, a half-precision float.
5673 ++ fos |= {a/@rs} (bit (sea:rs a))
5686Unsigned integer to `@rh`
5688Converts `@` from an unsigned integer to `@rh`.
5692`a` is a `@u`, an unsigned integer.
5696A `@rh`, a half-precision float.
5701 ++ sun |= [a=@u] ^- @rh (sun:ma a)
5714Signed integer to `@rh`
5716Converts `@` from a signed integer to `@rh`.
5720`a` is a `@s`, a signed integer.
5724A `@rh`, a half-precision float.
5729 ++ san |= [a=@s] ^- @rh (san:ma a)
5744Produces the sign of `a`.
5748`a` is a `@rh`, a half-precision float.
5757 ++ sig |= [a=@rh] ^- ? (sig:ma a)
5763 > (sig:rh .~~-2.05e2)
5772Gets the exponent of `a`.
5776`a` is a `@rh`, a half-precision float.
5780A `@s`, a signed integer.
5785 ++ exp |= [a=@rh] ^- @s (exp:ma a)
5791 > (exp:rh .~~-2.05e2)
5800Rounds `a` to the nearest integer.
5804`a` is a `@rh`, a half-precision float.
5813 ++ toi |= [a=@rh] ^- (unit @s) (toi:ma a)
5819 > (toi:rh .~~-2.085e2)
5822 > (toi:rh .~~-2.08e2)
5829`@rh` to decimal float
5831Produces the decimal form of `a` using the Dragon4 algorithm. Guarantees
5832accurate results for rounded floats.
5836`a` is a `@rh`, a half-precision float.
5845 ++ drg |= [a=@rh] ^- dn (drg:ma a)
5851 > (drg:rh .~~-2.085e2)
5852 [%d s=%.n e=-1 a=2.085]
5854 > (drg:rh .~~-2.08e2)
5855 [%d s=%.n e=--0 a=208]
5861Decimal float to `@rh`
5863Converts `a` from `dn` to `@rh`.
5873A `@rh`, a a half-precision float.
5878 ++ grd |= [a=dn] ^- @rh (grd:ma a)
5885 > (grd:rh [%d s=%.n e=--0 a=343])
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