num/ryu.fz
# This file is part of the Fuzion language implementation.
#
# The Fuzion language implementation is free software: you can redistribute it
# and/or modify it under the terms of the GNU General Public License as published
# by the Free Software Foundation, version 3 of the License.
#
# The Fuzion language implementation is distributed in the hope that it will be
# useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
# License for more details.
#
# You should have received a copy of the GNU General Public License along with The
# Fuzion language implementation. If not, see <https://www.gnu.org/licenses/>.
# -----------------------------------------------------------------------
#
# Tokiwa Software GmbH, Germany
#
# Source code of Fuzion standard library feature ryū
#
# Author: Michael Lill (michael.lill@tokiwa.software)
#
# -----------------------------------------------------------------------
#
# An implementation of Ryū, a fast float to string conversion algorithm invented by Ulf Adams.
# For more information see the original repository: https://github.com/ulfjack/ryu
# and the paper: https://dl.acm.org/citation.cfm?doid=3296979.3192369
#
# NYI lacks documentation
#
public ryū(public T type : float) is
five := T.from_u32 5
# constructor for rounding mode
# see below for predefined rounding modes
#
public rounding_mode(accept_upper_bound, accept_lower_bound bool -> bool) is
# rounding mode 'conservative', never accepts lower/upper bound
#
public rounding_conservative := (rounding_mode (is_even)->false (is_even)->false)
# rounding mode 'even', chooses the closest even number.
# examples: 0.5 => 0
# 1.5 => 2
#
public rounding_even := (rounding_mode (is_even)->is_even (is_even)->is_even)
# convert float to string using conservative rounding
#
# if the absolute value
# is smaller than 0.001
# or greater than 9_999_999
# scientific notation will be used.
#
public as_string(val T) String =>
as_string val val.use_scientific_notation rounding_conservative
# convert float to string
public as_string(val T, scientific_notation bool, rounding_mode rounding_mode) String =>
# trivial cases first
if val.is_NaN then "NaN"
else if (val = T.positive_infinity) then "Infinity"
else if (val = T.negative_infinity) then "-Infinity"
else if (val = T.zero)
if val.is_sign_bit_set then "-0.0" else "0.0"
else
as_string_nontrivial val scientific_notation rounding_mode
# convert float to string for nontrivial case
as_string_nontrivial(val T, scientific_notation bool, rounding_mode rounding_mode) String =>
(smaller_halfway_point, mv, larger_halfway_point, mm_shift, is_even) := step_2 val
(dv, dp, dm, dm_is_trailing_zeros, dv_is_trailing_zeros, e10) := step_3 (val.exponent-T.mantissa_bits-2).as_i64 smaller_halfway_point mv larger_halfway_point mm_shift is_even rounding_mode
(output, output_length, exponent_decimal) := step_4 dv dp dm dm_is_trailing_zeros dv_is_trailing_zeros e10 is_even scientific_notation rounding_mode
step_5 output output_length exponent_decimal (val < T.zero) scientific_notation
# step 2: get the interval of legal decimal representations
step_2(val T) tuple i64 i64 i64 bool bool =>
m := val.mantissa.as_i64
mv := m << 2
mm_shift := m != (i64 1 << T.mantissa_bits.as_i64) || val.exponent_biased <= 1
(mv - (if mm_shift then 2 else 1), mv, mv + 2, mm_shift, (m & 1) = 0)
# step 3: convert to decimal power base
step_3(exponent, smaller_halfway_point, mv, larger_halfway_point i64, mm_shift bool, is_even bool, rounding_mode rounding_mode) tuple i64 i64 i64 bool bool i64 =>
# calculates: ⌈log₂ 5ᵉ⌉
#
pow5bits(e i64)
pre debug: (e >= 0)
debug: (e <= 3528)
post
debug: result = ((five ** T.from_i64 e).log T.two).as_i64
=>
(e * 1217359) >> 19
# computes ⌊log₁₀ 2ᵉ⌋
#
floor_log₁₀_pow_2(e i64) i64
pre debug: (e >= 0)
debug: (e <= 1650)
post
debug: result = ((T.two ** T.from_i64 e).log T.ten).floor.as_i64
=>
e*78913 >> 18
# computes ⌊log₁₀ 5ᵉ⌋
#
floor_log₁₀_pow_5 (e i64) i64
pre debug: (e >= 0)
debug: (e <= 2620)
post
( (five ** T.from_i64 e).log T.ten = T.infinity
|| result = ((five ** T.from_i64 e).log T.ten).floor.as_i64)
=>
e*732923 >> 20
# max 3*31 = 124
pow5_bitcount => i64 121
# max 3*31 = 124
pow5_inv_bitcount => i64 122
/*********** code to generate the tables below
pow5_quarter_bitcount => i64 31
mask => ((i64 1 << pow5_quarter_bitcount) - 1).as_uint
# 5⁰, 5¹, 5², .. , 5^325
powers_of_5_cache_key(data array uint) is
powers_of_5 => (cache powers_of_5_cache_key ()->(powers_of_5_cache_key (array 326 i->(uint 5 ** uint i.as_u64)))).data
pow5_split_cache_key(data array (array i64)) is
pow5_split => (cache pow5_split_cache_key (() ->
pow5_split_cache_key (array 326 (i -> array 4 (j ->
shift := powers_of_5[i].highest_bit.as_i64 - pow5_bitcount + (i64 3 - j.as_i64) * pow5_quarter_bitcount
shifted := if shift >= 0 then powers_of_5[i] >> shift.as_uint else powers_of_5[i] << (-shift).as_uint
(shifted & mask).as_i64
))))).data
pow5_inv_split_cache_key(data array (array i64)) is
pow5_inv_split => (cache pow5_inv_split_cache_key (() ->
pow5_inv_split_cache_key (array 291 (i -> array 4 (k ->
j := powers_of_5[i].highest_bit.as_i64 - 1 + pow5_inv_bitcount
inve := (uint 1 << j.as_uint) / powers_of_5[i] + uint 1
tmp := inve >> ((3-k).as_i64 * pow5_quarter_bitcount).as_uint
(if k = 0 then tmp else tmp & mask).as_i64
))))).data
***************/
pow5_split_cache_key(val array (array i64)) is
pow5_split array (array i64) := (cache pow5_split_cache_key ()->
pow5_split_cache_key [
[134217728,0,0,0],
[167772160,0,0,0],
[209715200,0,0,0],
[262144000,0,0,0],
[163840000,0,0,0],
[204800000,0,0,0],
[256000000,0,0,0],
[160000000,0,0,0],
[200000000,0,0,0],
[250000000,0,0,0],
[156250000,0,0,0],
[195312500,0,0,0],
[244140625,0,0,0],
[152587890,1342177280,0,0],
[190734863,603979776,0,0],
[238418579,218103808,0,0],
[149011611,2015363072,0,0],
[186264514,1982332928,0,0],
[232830643,1404174336,0,0],
[145519152,609173504,0,0],
[181898940,761466880,0,0],
[227373675,951833600,0,0],
[142108547,326460544,0,0],
[177635683,2018688416,0,0],
[222044604,1986489608,0,0],
[138777878,167814181,0,0],
[173472347,1283509550,536870912,0],
[216840434,1067516025,1744830464,0],
[135525271,1204068428,285212672,0],
[169406589,968214623,356515840,0],
[211758236,1747139190,2056257536,0],
[264697796,36440340,1496580096,0],
[165436122,1096517036,2009104384,0],
[206795153,296904472,363896832,0],
[258493941,908001502,454871040,0],
[161558713,835936394,1894907136,0],
[201948391,1581791405,1294892096,0],
[252435489,1440368345,8002384,0],
[157772181,94923847,1347178770,0],
[197215226,655525721,1147102550,1073741824],
[246519032,1893148975,1970749100,268435456],
[154074395,1183218109,2037024555,1241513984],
[192592994,942151725,935667958,1015021568],
[240741243,103947832,1706455860,195035136],
[150463276,1944015587,1066534912,1195638784],
[188079096,282535836,796297728,1494548480],
[235098870,353169795,995372160,1868185600],
[146936793,1831343858,353672144,1167616000],
[183670992,678567086,1515832004,1459520000],
[229588740,848208858,821048181,1824400000],
[143492962,1603872360,1050026025,1408685456],
[179366203,931098626,1312532532,150244084],
[224207754,627002371,566923841,187805105],
[140129846,928747394,85891944,1459555470],
[175162308,87192418,1181106754,1824444338],
[218952885,108990523,402641619,1206813598],
[136845553,336554532,2130699204,485823043],
[171056941,957564078,515890357,607278804],
[213821176,1733826009,1718604770,1295969417],
[267276471,19798864,537643227,546219947],
[167047794,817680658,336027017,72952011],
[208809742,2095842646,1493775595,628060925],
[261012178,1546061484,793477670,248205245],
[163132611,1503159339,1569665367,1765741014],
[203915764,1342078262,1425210797,1670305355],
[254894705,1677597828,707771673,477268958],
[159309191,243192274,1516099119,1640470379],
[199136488,1914603079,821382075,1513717062],
[248920611,245770201,489856682,1355275415],
[155575381,2032654567,1648337706,1383918046],
[194469227,930205473,1523551221,656155734],
[243086534,625885930,293826290,1357065580],
[151929083,2001791442,720512343,1385036899],
[189911354,1965368390,1974382253,1194425212],
[237389193,1382968664,1394235992,2029902427],
[148368246,59049047,871397495,1268689017],
[185460307,1147553133,552375957,1048990359],
[231825384,897570504,1227340858,1848108861],
[144890865,560981565,767088036,1691938950],
[181113581,1238097868,1495730957,2114923688],
[226391976,2084493247,1869663697,1033041874],
[141494985,1302808279,1973846178,1987828451],
[176868732,17897613,1930436811,1411043740],
[221085915,22372017,802433278,1226933763],
[138178696,1893030702,1843698079,229962689],
[172723371,218804730,1230880774,1898066098],
[215904213,1884118649,464859144,1298840799],
[134940133,1446009611,1632714245,811775499],
[168675167,196899278,1504021894,1551590286],
[210843958,1856736834,806285544,865746033],
[263554948,1247179218,2081598754,1082182542],
[164721842,1853228835,1837870133,1213235000],
[205902303,1242794220,1760466754,2053414663],
[257377879,1016621864,53099795,1493026504],
[160861174,1440695033,33187372,664706109],
[201076468,727126967,578355127,830882636],
[251345585,908908709,186072997,501732384],
[157090990,1910245223,384731079,582018196],
[196363738,1314064704,2091526585,190651833],
[245454673,568839057,466924583,775185703],
[153409170,1697701690,1634005144,1289797432],
[191761463,1048385289,968764606,1612246790],
[239701829,773610699,1747826670,941566664],
[149813643,751942143,823956213,51608253],
[187267054,403056767,493074354,601381228],
[234083817,1577562783,79472030,1825468359],
[146302386,180670371,854976387,604046812],
[182877982,1299579788,531849572,218187604],
[228597478,550732911,664811965,272734505],
[142873423,1954820805,1220813846,438894521],
[178591779,1906655094,2062888219,1622359976],
[223239724,1846447956,1504868450,1491079058],
[139524828,80288148,2014284605,1468795323],
[174406035,100360186,370372109,225381418],
[218007543,1736062968,1536706960,818597684],
[136254714,1890345723,960441850,511623552],
[170318393,1289190330,663681400,1713271265],
[212897992,875176,1903343574,2141589081],
[266122490,1093971,231695820,1603244527],
[166326556,537554643,2023858079,2075769653],
[207908195,671943304,1992951687,2057841155],
[259885244,303058219,343705961,2035430532],
[162428277,1263153210,2093864418,466837714],
[203035346,2115812425,1543588698,1657288967],
[253794183,1571023708,318873137,997869385],
[158621364,1787196185,1273037534,1965845645],
[198276706,86511583,2128167830,1383565233],
[247845882,1181881303,2123338876,655714717],
[154903676,1275546726,2132393165,1483563522],
[193629595,1594433408,1591749633,243841667],
[242036994,1456170848,1989687041,841672995],
[151273121,1446977692,1243554400,1868222902],
[189091402,198109379,1554443001,187794979],
[236364252,1321378548,1406182839,771614636],
[147727657,1899603416,1952606098,1287565516],
[184659572,763891535,293273975,535715071],
[230824465,954864418,1977205205,132772926],
[144265290,1938967541,1772624165,351418535],
[180331613,1349967603,605167470,976144081],
[225414517,76846768,219588426,146438277],
[140884073,316464686,137242766,628394835],
[176105091,932451769,1245295281,1859235368],
[220131364,628693799,2093490014,713431474],
[137582102,1466675448,2113737626,2056507407],
[171977628,759602487,494688385,1496892435],
[214972035,949503109,81489570,260502808],
[134357522,325003987,319366437,699685167],
[167946902,1479996807,2009820782,1411477370],
[209933628,776254185,1975405066,690604889],
[262417035,970317732,858643596,1936997935],
[164010647,338013126,1610394072,136881885],
[205013308,2033129144,939250766,171102357],
[256266636,393927782,1174063457,1287619770],
[160166647,1319946688,196918748,2146939636],
[200208309,1113062448,246148436,536190897],
[250260386,1928198972,307685545,670238622],
[156412741,1741995269,1266045289,1761076419],
[195515927,566881350,2119427524,590732787],
[244394909,171730776,1575542581,738415984],
[152746818,375767191,984714113,729945446],
[190933522,1543450813,694021729,1449302720],
[238666903,855571692,1404398074,201015664],
[149166814,1340038675,1951490620,662505702],
[186458518,601306520,1902492363,828132127],
[233073147,1825374975,230631806,498294247],
[145670717,872423903,949451246,1922046640],
[182088396,1627400791,649943146,1328816476],
[227610495,2034250989,275558021,587278772],
[142256559,2076713236,440659219,635484688],
[177820699,2059020633,550824024,257484948],
[222275874,2036904879,1225400942,321856185],
[138922421,1809936461,1571181956,1811772852],
[173653027,651807841,353364710,117232417],
[217066284,277888889,978576799,1220282345],
[135666427,1247422379,1953787779,1567982833],
[169583034,1022407062,1905363812,1423107630],
[211978793,204267004,1307962941,1778884537],
[264973491,792204667,1634953677,612992936],
[165608432,226692461,753410592,651556041],
[207010540,283365576,1478634152,814445051],
[258763175,354206970,1848292690,1018056314],
[161726984,1026685724,1692053843,1173156108],
[202158730,1283357155,2115067304,929574223],
[252698413,530454620,2106963218,1161967779],
[157936508,599969594,243110187,1263100774],
[197420635,749961992,1377629558,1042005055],
[246775794,400581578,1722036948,228764495],
[154234871,787234398,1613144004,1216719633],
[192793589,447172086,942688181,1520899542],
[240991986,1095836020,104618403,290511691],
[150619991,1221768424,1139128325,2060617999],
[188274989,990339618,1423910407,965159763],
[235343736,1774795435,706146185,669578792],
[147089835,1109247147,172905909,1760664025],
[183862294,849688021,1826745123,590217295],
[229827867,2135851851,672818668,200900707],
[143642417,1066471951,152076211,1199304766],
[179553021,1869960850,1800708000,962260045],
[224441277,726838327,1177143176,1202825056],
[140275798,722709410,1541020853,751765660],
[175344747,1977128587,852534242,1476577987],
[219180934,1934539822,528796891,771980660],
[136988084,672216476,1941110793,214052456],
[171235105,840270596,278904843,804436483],
[214043881,1587209157,348631054,468674691],
[267554852,373398710,972659729,1659585188],
[167221782,1307116018,71041419,231934375],
[209027228,560153198,1162543597,1900530704],
[261284035,700191498,379437673,765050645],
[163302522,169184230,774019457,1820333933],
[204128152,1285222111,2041266146,664804680],
[255160190,1606527639,2014711770,1904747674],
[159475119,467208862,2064501224,1727338208],
[199343899,47140166,1506884707,11689112],
[249179873,1669537944,809864059,1625224126],
[155737421,238154847,506165037,747329623],
[194671776,834564471,95835384,1471032941],
[243339720,1043205588,1730406966,1838791176],
[152087325,652003493,7762530,612373573],
[190109156,1351875278,546574074,1839208790],
[237636445,1689844097,1756959417,1225269164],
[148522778,1324588017,292793267,2107970507],
[185653473,581993197,902862496,2098092222],
[232066841,1264362408,1665449033,475131630],
[145041775,2132403785,1040905645,1639134548],
[181302219,2128633819,1838002969,438305450],
[226627774,2123921362,1760632799,1084752724],
[141642359,790579939,1637266411,1483276820],
[177052949,451354012,1509712102,1317225114],
[221316186,1101063427,1887140128,572789568],
[138322616,1225035554,911027124,357993480],
[172903270,1531294443,65042081,447491850],
[216129088,840376229,1691915337,1096235725],
[135080680,525235143,1325882541,2027324608],
[168850850,656543929,1120482265,923543024],
[211063562,1894421735,1937473743,1691299692],
[263829453,1294285345,1884971267,1577253703],
[164893408,1077363797,372800674,717348108],
[206116760,1346704746,1002871754,1970426959],
[257645950,1683380933,179847869,1389291875],
[161028719,515242171,380840374,1136742878],
[201285899,107181801,2086663204,347186773],
[251607373,1744589988,997716269,433983467],
[157254608,1358804198,1697314492,539675123],
[196568260,1698505248,1047901291,674593903],
[245710325,2123131560,1309876614,306371467],
[153568953,1595392681,818672883,1802094903],
[191961192,383628115,1560212016,1715747717],
[239951490,479535144,1413394108,2144684646],
[149969681,836580377,883371318,266686080],
[187462101,1582596383,1641085059,1407099424],
[234327627,367632743,1514485412,1222003368],
[146454766,2108818656,1751859750,1837493929],
[183068458,1562281497,42341040,1223125587],
[228835573,879110047,589797212,1528906984],
[143022233,817879235,1173929625,2029308689],
[178777791,1559219956,930541120,926023125],
[223472239,1412154033,1163176400,1157528906],
[139670149,1687902638,2069162530,723455566],
[174587687,499265562,1512711338,1978061282],
[218234609,87211041,817147349,1398834779],
[136396630,1396684180,1852894373,1142707192],
[170495788,672113402,168634318,1965254903],
[213119735,840141752,1284534722,1382826804],
[266399669,513306278,1605668403,654791682],
[166499793,589251880,466671840,140809345],
[208124741,1273435762,583339800,176011681],
[260155926,2128665614,1802916574,220014602],
[162597454,793545097,589951946,1748121862],
[203246817,2065673195,1274310845,1111410503],
[254058522,971478758,1056017644,1926134041],
[158786576,1144045136,123140116,130091952],
[198483220,1430056420,153925145,162614940],
[248104025,1787570525,192406431,740139587],
[155065016,311925210,388689475,1267893610],
[193831270,389906512,1559603668,1047996100],
[242289087,1561124964,1949504585,1309995125],
[151430679,1781009471,144698542,13440585],
[189288349,1689390926,1791485913,1090542555],
[236610437,501125922,1165615567,1900049106],
[147881523,581639157,1265380641,1992837059],
[184851904,190178034,2118596714,880433588],
[231064880,237722543,1574504069,26800161],
[144415550,148576589,1789371411,285185557],
[180519437,1259462561,626101527,1967094682],
[225649296,2111199113,1319497821,1921997441],
[141030810,1319499446,19379770,1469683856],
[176288513,575632483,1097966537,763362996],
[220360641,1256411516,835587259,1491074658],
[137725400,2127434477,1595983861,663486205],
[172156751,511809449,384367090,1366228668],
[215195939,102890899,1017329775,634044011],
[134497461,1943355004,367395653,1201583875],
[168121827,818581019,459244566,2038850756],
[210152284,486355362,37184796,1474821621],
[262690355,607944202,1120222819,1843527026],
[164181472,111529670,1237010174,883768935],
[205226840,139412088,472520894,30969345],
[256533550,174265110,590651117,1112453505],
[160333468,1719528429,1979769684,963718897],
[200416836,1926889,864099369,1204648621],
[250521045,2408611,1616995123,2042681688],
[156575653,269940838,742186496,1008240599],
[195719566,874296959,2001474944,1260300749],
[244649458,19129375,1964972768,1575375936],
[152905911,548826771,2033414348,984609960],
[191132389,149162552,2004897023,1230762450],
[238915486,723324103,358637631,1001582151],
[149322178,2062690300,1029454887,1431295212],
[186652723,1504621051,1286818609,1252248103],
[233315904,1343905402,1071652349,2102181041],
[145822440,839940876,1206653630,1582298607],
[182278050,1049926095,1508317038,904131434],
[227847563,238665795,1348525386,56422469],
[142404726,2028214314,574392910,572134955],
[178005908,1461526068,1791732961,1788910518],
[222507385,1826907586,92182554,625525411],
[139067116,336510873,594485008,927824294],
[173833895,420638591,1279977172,1159780368],
[217292368,2136410975,1063100553,1449725460],
[135807730,1335256859,1469744214,100772044],
[169759663,595329250,1300309355,1199706879],
[212199579,207290651,551644870,962762687],
[265249473,1869726050,152685176,129711535],
[165780921,363272413,632299147,81069709],
[207226151,990961428,1327244845,1711949873]
]).val
pow5_inv_split_cache_key(val array (array i64)) is
pow5_inv_split array (array i64) := (cache pow5_inv_split_cache_key ()->
pow5_inv_split_cache_key [
[536870912,0,0,1],
[429496729,1288490188,1717986918,858993460],
[343597383,1460288880,1374389534,1546188227],
[274877906,2027224563,1529008357,807453852],
[439804651,237082194,1587419912,1291926163],
[351843720,1907652674,410942470,1892534390],
[281474976,1526122139,758250706,655034053],
[450359962,1582801963,1642697859,1477551214],
[360287970,407248111,1743655017,752544241],
[288230376,325798489,965427284,602035393],
[461168601,1809767771,1974180384,963256629],
[368934881,1018317487,2008841037,341108574],
[295147905,385157260,1607072829,1561377048],
[472236648,616251617,423832879,1209713087],
[377789318,1351994752,2057053222,108777011],
[302231454,1940589261,1216145848,87021609],
[483570327,1816452629,1516336627,568731303],
[386856262,594168644,1213069301,1743475232],
[309485009,1763825104,970455441,965283456],
[495176015,1533629978,693735247,255963340],
[396140812,1226903982,1413981656,1922757591],
[316912650,122529726,1990178784,1538206073],
[507060240,196047563,177808948,743142797],
[405648192,156838050,1001240617,1883004427],
[324518553,1413960629,371495764,1506403542],
[519229685,1832840277,164896493,1980748937],
[415383748,1466272221,1420407383,2014095879],
[332306998,2032011236,1136325907,322786514],
[531691198,674237600,1818121451,945955152],
[425352958,1398383539,1883993890,1615757581],
[340282366,1977700291,218704923,1722102795],
[272225893,1152663503,604460668,1377682236],
[435561429,1414764875,1396633799,915801388],
[348449143,1561308630,258313580,732641111],
[278759314,2108040363,636147593,1874603077],
[446014903,2084374392,1017836150,422384546],
[356811923,379009325,384772190,1196901096],
[285449538,1162200919,737314482,98527418],
[456719261,1430024741,750206441,1446134057],
[365375409,714523063,1029661882,2015900705],
[292300327,1001115180,823729506,753727105],
[467680523,2031281018,458973750,2064956827],
[374144419,336534625,1655669189,1222468732],
[299315535,698724430,465541892,977974986],
[478904856,1117959088,744867027,1994256706],
[383123885,464870541,166396892,1595405365],
[306499108,371896432,1851104432,1276324292],
[490398573,165537562,2102773632,2042118867],
[392318858,991423509,1252722176,1633695094],
[313855086,1652132266,1861171200,1306956075],
[502168138,1784418167,1689383732,373142802],
[401734511,139044345,922010256,298514241],
[321387608,1829222394,1596601664,238811393],
[514220174,349775453,2125065932,2100085147],
[411376139,709317092,1700052746,821074659],
[329100911,996950403,1789538926,1515853186],
[526561458,736127186,2004268823,1136874909],
[421249166,1447895208,1603415058,1768493386],
[336999333,728819437,853235317,985297980],
[269599466,1442049009,253091524,788238384],
[431359146,1448284955,834443168,1261181414],
[345087317,729131234,1526547994,149951672],
[276069853,1871795176,1221238395,549458067],
[441711766,417891904,1953981432,879132907],
[353369412,2052300442,704191686,1562299785],
[282695530,782846894,1422346808,1249839828],
[452312848,1252555031,987264704,1999743725],
[361850278,1861037484,789811763,2029291709],
[289480223,200339798,1490842870,764439908],
[463168356,2038530596,667361674,364110394],
[370534685,1201327747,963386068,2009275234],
[296427748,961062197,2059199043,2036916917],
[474284397,1108202787,288241363,252589959],
[379427518,27568770,1089586549,1490562156],
[303542014,881048475,1301165969,762952996],
[485667223,121187372,363878632,1220724793],
[388533778,955943356,2009089824,976579834],
[310827022,1623748144,1607271859,1210760597],
[497323236,880010113,424151327,648726766],
[397858589,274511360,2057307980,518981413],
[318286871,649105818,786852924,2133172049],
[509258994,179575850,399971220,1695088360],
[407407195,573157409,1608467165,926573958],
[325925756,458525927,1716270461,2029749355],
[521481209,2022131673,598549091,241121861],
[417184967,2047202068,478839272,1910884407],
[333747974,778768195,812568147,1958204256],
[533996758,2105022571,1729605766,556146431],
[427197407,395527868,1383684613,15420415],
[341757925,1604912483,1536444420,12336332],
[273406340,1283929986,2088148995,439365796],
[437450144,2054287979,334561285,273488543],
[349960115,2072927112,1985635946,1077784294],
[279968092,1658341690,729515298,3233976],
[447948948,935359786,308231017,1293664550],
[358359158,1607281288,246584814,175938181],
[286687326,2144818489,1485758040,140750545],
[458699723,425232476,659225945,1513691060],
[366959778,1199179440,527380756,1210952848],
[293567822,1818337011,851401334,1827755738],
[469708516,1191352299,1791738865,776925532],
[375766813,523585110,574397633,192043696],
[300613450,1277861547,889014836,153634957],
[480981520,2044578475,1851920467,675312661],
[384785216,1635662780,1481536373,1828740318],
[307828173,879033494,2044222558,603998795],
[492525077,976956862,693775715,1395894801],
[394020061,2070055678,1414014031,1546212571],
[315216049,1226547813,701714495,1666466786],
[504345679,673986312,1122743193,518863210],
[403476543,968685779,1327691284,415090568],
[322781234,1633942082,1921146486,1191065914],
[516449975,1325817143,1785344189,1476208732],
[413159980,1060653715,139785162,2039960445],
[330527984,848522972,111828130,772974897],
[528844775,69146566,1037918467,1666256564],
[423075820,55317253,400838044,1333005251],
[338460656,44253802,1179663894,1925397660],
[270768524,1753389960,943731115,1969814858],
[433229639,1516933747,1939466515,145226665],
[346583711,1643043727,1981069941,1404671521],
[277266969,884938252,1584855953,694240487],
[443627151,127411015,1247279336,1110784780],
[354901720,1819915730,1856816928,888627824],
[283921376,1455932584,1485453542,1569895718],
[454274202,1470498676,658738749,2082336419],
[363419362,317405481,1815481188,1665869136],
[290735489,1542414574,593391491,1762192038],
[465176783,1179373130,90432927,1531017072],
[372141426,1802491963,501843071,1654310387],
[297713141,1012496840,2119461375,1752945040],
[476341026,761001486,814157823,1516221874],
[381072821,179304459,1080822988,1212977499],
[304858256,1861430486,5664931,1399878729],
[487773210,2119295318,868057349,1810309237],
[390218568,1695436254,1553439339,159757201],
[312174855,67858814,2101744930,986799220],
[499479768,108574103,2074301699,2008375481],
[399583814,945852742,800447900,1606700385],
[319667051,1186178923,1069855050,426366849],
[511467282,1038892818,852774621,252690228],
[409173825,2119604443,1111716426,1061145642],
[327339060,1695683554,1748366600,848916514],
[523742497,565610039,1508896371,1787763151],
[418993997,1740978220,1207117097,1000713792],
[335195198,533789117,536196948,800571033],
[536312317,424565857,2146405306,421920194],
[429049853,1628142875,428634056,337536155],
[343239883,14024111,772403974,1129022383],
[274591906,870212748,617923179,1332714636],
[439347050,533346938,129683628,414356500],
[351477640,426677550,962740361,1619975389],
[281182112,341342040,770192289,866483581],
[449891379,975643994,373314203,1815870459],
[359913103,1210011924,2016638281,1023199638],
[287930482,1827002999,324820436,818559711],
[460688772,1205217880,519712698,450702077],
[368551018,105180844,2133757076,2078548580],
[294840814,943138135,418515472,1662838864],
[471745303,220530827,1099121486,83561805],
[377396242,1035418121,449800459,496346174],
[301916993,2116824685,1648330556,397076939],
[483067190,809939119,1348838701,205826373],
[386453752,647951295,1508567690,1023654558],
[309163001,1806851225,777357422,1677917105],
[494660802,2031968501,814275146,1825673909],
[395728642,766581341,1939910306,601545668],
[316582913,1901755262,692934785,1769726723],
[506532662,465828042,249702198,254582380],
[405226129,1661152622,1058755217,1492156093],
[324180903,1758418827,1276500903,1623221604],
[518689446,236489746,1183407986,1738161106],
[414951556,1907178715,1376223119,102038696],
[331961245,1096246242,1959971954,940624416],
[531137992,1753993988,1417968208,1504999066],
[424910394,544201731,1563871296,1203999253],
[339928315,864858114,2110090496,963199402],
[271942652,691886491,2117569126,1629552981],
[435108243,1536515116,1670123684,889297851],
[348086594,2088205552,1336098947,1140935011],
[278469275,2100061171,1498375887,1342244738],
[445550841,1212614226,1538407961,107933],
[356440673,540594651,1660223098,859079806],
[285152538,1291469180,1328178478,1546257304],
[456244061,1636853959,836595377,326528038],
[364995249,879986437,1957766490,1120215890],
[291996199,1133485879,1995709922,37179253],
[467193919,525087219,186658768,59486804],
[373755135,849566504,1867313932,1765576362],
[299004108,679653203,1923347875,1841957819],
[478406573,657948396,1359369682,2088139051],
[382725258,1385352176,1087495746,811517782],
[306180206,1967275200,869996597,219717496],
[489888331,141163213,962497825,1640038182],
[391910664,1830917489,340501531,23540357],
[313528531,1894230720,1990388143,448329015],
[501645651,24292046,607640651,1146823153],
[401316520,1737420555,915609250,1776451982],
[321053216,1389936444,732487400,1421161586],
[513685146,1364904851,1601476570,1414865078],
[410948117,662427151,1710677986,272898603],
[328758493,1818431910,509548929,1506809071],
[526013590,332510678,1674271746,1551901055],
[420810872,266008543,50927208,1241520844],
[336648697,1501297023,470238496,993216675],
[269318958,342044159,805687526,1653566799],
[430910333,117773925,859603313,498223231],
[344728266,953212599,1117179380,398578585],
[275782613,333073350,34750044,2036849786],
[441252181,103420630,914593531,252482550],
[353001744,1800723422,1590668284,201986040],
[282401395,1870075467,1702031356,1879575751],
[451842233,844637100,1005263252,1289334283],
[361473786,1534703139,1233707331,1460964156],
[289179029,798265782,127972406,309777866],
[462686446,2136218710,1063749309,66147855],
[370149157,1279478238,1709992906,911911743],
[296119326,164589131,1797491054,1588522854],
[473790921,1551832799,1587495498,1682643107],
[379032737,811969510,411002939,1775611215],
[303226189,1938065796,2046789270,561495513],
[485161903,1812415086,697882454,1757386280],
[388129523,161441880,558305963,1835405753],
[310503618,988146963,876141500,1468324603],
[496805789,1151538411,1831323130,1490325905],
[397444631,1350727459,176568315,1621757454],
[317955705,651085237,1429744841,867909234],
[508729128,1041736380,569604827,1818151503],
[406983302,1692382563,885180591,1884017932],
[325586642,494912591,1137641203,218724157],
[520938627,1221356876,102239006,1208952110],
[416750902,118092041,1370281394,108168229],
[333400721,1382963822,237231656,86534583],
[533441154,1353748656,379570649,1426945522],
[426752923,1512495654,1162649978,2000549877],
[341402338,2068989982,1789113442,741446442],
[273121871,366701797,1001794024,593157154],
[436994993,1875213064,1602870438,1808044905],
[349595995,211680262,2141289810,587442465],
[279676796,169344210,854038389,40457242],
[447482873,1559440925,936964692,1782718506],
[357986298,2106546199,1179068483,1855671535],
[286389039,396746770,1802248246,625543769],
[458222462,1493788292,1165610275,1430366759],
[366577970,336037174,1791481679,1573790137],
[293262376,268829739,1862682073,829535380],
[469219801,1718617772,1262304399,38766419],
[375375841,945397488,1009843519,460509865],
[300300673,326821261,378378085,1656898081],
[480481077,93417288,605404937,503553281],
[384384861,1363224019,913820679,832339354],
[307507889,661082485,2019546732,665871484],
[492012622,1916725436,1513287853,635897644],
[393610098,674386890,351636823,938214845],
[314888078,1398502971,710806188,750571876],
[503820925,1808108024,1137289901,771418272],
[403056740,1446486419,1339328650,1476128077],
[322445392,1157189135,1500959650,321909002],
[515912628,133515698,1542541981,85557674],
[412730102,965806018,375040125,1356936328],
[330184081,2061135003,729528830,226555603],
[528294531,291338898,308252668,2080475883],
[422635624,1951058036,1964589053,1234883977],
[338108499,1990343159,283181054,128913723],
[270486799,2021771256,1944531761,1391621167],
[432778879,1946343822,534270441,79110219],
[346223103,1986571787,856913082,922281634],
[276978483,300767241,256033736,737825308],
[443165573,51730856,409653978,321527033],
[354532458,900378144,327723182,1116215085],
[283625966,1579295974,1121172005,463475339],
[453801546,1667880100,75888289,2030050731],
[363041237,904807350,919704091,335550396],
[290432989,2012336069,306266543,697937046],
[464692783,1931247521,1778516658,257705815],
[371754227,256507828,1422813326,1065158111],
[297403381,1493696451,1567747390,1711119948],
[475845410,1530920863,1219905636,1019804998],
[380676328,1224736690,1834917968,815843999],
[304541062,1838782811,1897431104,652675199],
[487265700,1224065580,1317902848,1044280318],
[389812560,979252464,1054322278,1694417714],
[311850048,783401971,1272954552,1355534171],
[498960077,823946424,2036727284,450867755],
[399168061,1947647328,1629381827,790190934],
[319334449,1128621133,874008732,632152747],
[510935119,517303624,1398413971,1440941125],
[408748095,843339629,689234447,1582249629],
[326998476,674671703,980884287,1695296433],
[523197562,220481266,710421401,564990645],
[418558049,1464875201,1856827309,1740482705]
]).val
actual_shift(j i64) =>
j - 114
mul_pow5_div_pow2(m, i, j i64, table array (array i64)) =>
m_high := m >> 31
m_low := m & 0x7fffffff
bits13 := m_high * table[i.as_i32][0]
bits03 := m_low * table[i.as_i32][0]
bits12 := m_high * table[i.as_i32][1]
bits02 := m_low * table[i.as_i32][1]
bits11 := m_high * table[i.as_i32][2]
bits01 := m_low * table[i.as_i32][2]
bits10 := m_high * table[i.as_i32][3]
bits00 := m_low * table[i.as_i32][3]
((((((
((bits00 >> 31) + bits01 + bits10) >> 31)
+ bits02 + bits11) >> 31)
+ bits03 + bits12) >> 21)
+ (bits13 << 10)) >> (actual_shift j)
mul_pow5_div_pow2(m, i, j i64)
pre debug: (actual_shift j >= 0)
debug: (exponent < 0)
=>
mul_pow5_div_pow2 m i j pow5_split
mul_pow5_inv_div_pow2(m, i, j i64) i64
pre debug: (actual_shift j >= 0)
debug: (exponent >= 0)
=>
mul_pow5_div_pow2 m i j pow5_inv_split
multiple_of_power_of_5(val i64, q i64) =>
for pow5_factor := i64 0, pow5_factor+1
mod := i64 5, mod*5
while val % mod = 0 && mod*!5
else
pow5_factor >= q
if (exponent >= 0)
q := max (i64 0) (floor_log₁₀_pow_2 exponent)-1
k := pow5_inv_bitcount + pow5bits q
i := -exponent + q + k
dm := mul_pow5_inv_div_pow2 smaller_halfway_point q i
dv := mul_pow5_inv_div_pow2 mv q i
dp := mul_pow5_inv_div_pow2 larger_halfway_point q i
if q <= 21
if mv % 5 = 0
(dv, dp, dm, false, (multiple_of_power_of_5 mv q), q)
else if rounding_mode.accept_upper_bound is_even
(dv, dp, dm, (multiple_of_power_of_5 smaller_halfway_point q), false, q)
else if multiple_of_power_of_5 larger_halfway_point q
(dv, dp-1, dm, false, false, q)
else
(dv, dp, dm, false, false, q)
else
(dv, dp, dm, false, false, q)
else
q := max (i64 0) (floor_log₁₀_pow_5 -exponent)-1
i := -exponent - q
k := pow5bits i + 1 - pow5_bitcount
j := q - k
dm := mul_pow5_div_pow2 smaller_halfway_point i j
dv := mul_pow5_div_pow2 mv i j
dp := mul_pow5_div_pow2 larger_halfway_point i j
e10 := q + exponent
if q <= 1
if rounding_mode.accept_upper_bound is_even
(dv, dp, dm, mm_shift, true, e10)
else
(dv, dp - 1, dm, false, true, e10)
else if q < 63
(dv, dp, dm, false, (mv & (((i64 1) << (q - 1)) - 1)) = 0, e10)
else
(dv, dp, dm, false, false, e10)
# step 4: shortest decimal representation
step_4(dv, dp, dm i64, dm_is_trailing_zeros, dv_is_trailing_zeros bool, e10 i64, even, scientific_notation bool, rounding_mode rounding_mode) tuple i64 i64 i64 =>
# get the decimal length of this i64
decimal_length(v i64) i64 =>
if v >= 1000000000000000000 then 19
else if v >= 100000000000000000 then 18
else if v >= 10000000000000000 then 17
else if v >= 1000000000000000 then 16
else if v >= 100000000000000 then 15
else if v >= 10000000000000 then 14
else if v >= 1000000000000 then 13
else if v >= 100000000000 then 12
else if v >= 10000000000 then 11
else if v >= 1000000000 then 10
else if v >= 100000000 then 9
else if v >= 10000000 then 8
else if v >= 1000000 then 7
else if v >= 100000 then 6
else if v >= 10000 then 5
else if v >= 1000 then 4
else if v >= 100 then 3
else if v >= 10 then 2
else 1
require_more_digits(d i64) => !(d<100 && scientific_notation)
vplength := decimal_length dp
exponent := e10 + vplength - 1;
if (dm_is_trailing_zeros || dv_is_trailing_zeros)
for last_removed_digit := i64 0, dv0 % 10
dm_is_trailing_zeros0 := dm_is_trailing_zeros, dm_is_trailing_zeros0 & (dm0 % 10 = 0)
dv_is_trailing_zeros0 := dv_is_trailing_zeros, dv_is_trailing_zeros0 & (last_removed_digit = 0)
dp0 := dp, dp0 / 10
dv0 := dv, dv0 / 10
dm0 := dm, dm0 / 10
removed := i64 0, removed+1
while ((dp0 / 10 > dm0 / 10) && require_more_digits dp0)
else
t := dm_is_trailing_zeros0 && rounding_mode.accept_lower_bound even
for dv_is_trailing_zeros1 := dv_is_trailing_zeros0, dv_is_trailing_zeros1 & last_removed_digit0 = 0
last_removed_digit0 := last_removed_digit, dv1 % 10
dp1 := dp0, dp1 / 10
dv1 := dv0, dv1 / 10
dm1 := dm0, dm1 / 10
removed0 := removed, removed0+1
while t && (dm1 % 10 = 0) && require_more_digits dp1
else
outside_bounds => dv1 = dm1 && !(dm_is_trailing_zeros0 && rounding_mode.accept_lower_bound even)
is_not_point_five_exactly := !(dv_is_trailing_zeros1 && last_removed_digit0 = 5 && dv1 % 2 = 0)
must_round_up => (is_not_point_five_exactly && last_removed_digit0 >= 5)
# output
(dv1 + (if outside_bounds || must_round_up then 1 else 0),
# output length
vplength - removed0,
exponent)
else
for last_removed_digit := i64 0, dv0 % 10
dp0 := dp, dp0 / 10
dv0 := dv, dv0 / 10
dm0 := dm, dm0 / 10
removed := i64 0, removed+1
while ((dp0 / 10 > dm0 / 10) && require_more_digits dp0)
else
# output
(dv0 + (if dv0 = dm0 || (last_removed_digit >= 5) then 1 else 0),
# output length
vplength - removed,
exponent)
# step 5: construct the string
step_5(output, output_length, exponent i64, is_negative, scientific_notation bool) String
pre debug: output_length > 0
=>
# convert output to string
output_as_string(output i64, output_length i64) String =>
for o := output_as_string.this.output, o / 10
s := "{o % 10}", "{o % 10}$s"
_ in (i64 1)..(output_as_string.this.output_length-1)
do
else
s
# this float as a scientific string e.g. 3.2E2
scientific =>
for str0 := "** internal error **", str
idx in i64 1 .. output_length
output0 := output, output0 / 10
str := "{output0 % 10}", "{output0 % 10}" + (if idx = output_length then "." else "") + str
else
"{str0}E{exponent}"
# the float as a decimal string
full =>
if exponent < 0
"0.{"0" * -(exponent+1).as_i32}{output_as_string output output_length}"
else if exponent + 1 >= output_length
"{output_as_string output output_length}{"0" * (exponent+1-output_length).as_i32}.0"
else
for str0 := "** internal error **", str
idx in i64 0 .. output_length
is_dot_pos := false, output_length - idx - 1 = exponent
output0 := output, if is_dot_pos then output0 else output0 / 10
str := "{output0 % 10}", (if is_dot_pos then "." else "{output0 % 10}") + str
else
str0
if is_negative then "-" else ""
+ (if scientific_notation then scientific else full)
