make/numbers.mk

   1 #
   2 # mal (Make a Lisp) number types
   3 #
   4
   5 ifndef __mal_numbers_included
   6 __mal_numbers_included := true
   7
   8 _TOP_DIR := $(dir $(lastword $(MAKEFILE_LIST)))
   9 include $(_TOP_DIR)util.mk
  10
  11 LIST20_X := x x x x x x x x x x x x x x x x x x x x
  12 LIST100_X := $(foreach x,$(LIST20_X),X X X X X)
  13 LIST100_0 := $(foreach x,$(LIST20_X),0 0 0 0 0)
  14 LIST100_9 := $(foreach x,$(LIST20_X),9 9 9 9 9)
  15
  16 ###
  17 ### general numeric utility functions
  18 ###
  19
  20 int_encode = $(strip $(call _reverse,\
  21                $(eval __temp := $(1))\
  22                $(foreach a,- 0 1 2 3 4 5 6 7 8 9,\
  23                  $(eval __temp := $$(subst $$a,$$a$$(SPACE),$(__temp))))$(__temp)))
  24
  25 int_decode = $(strip $(call _join,$(call _reverse,$(1))))
  26
  27 # trim extaneous zero digits off the end (front of number)
  28 _trim_zeros = $(if $(call _EQ,0,$(strip $(1))),0,$(if $(call _EQ,0,$(word 1,$(1))),$(call _trim_zeros,$(wordlist 2,$(words $(1)),$(1))),$(1)))
  29 trim_zeros = $(strip \
  30                $(if $(call _EQ,0,$(strip $(filter-out -,$(1)))),\
  31                  $(filter-out -,$(1)),\
  32                  $(call _reverse,$(call _trim_zeros,$(call _reverse,$(filter-out -,$(1))))))\
  33                $(if $(filter -,$(1)), -,))
  34
  35 # drop the last element of a list of words/digits
  36 drop_last = $(call _reverse,$(wordlist 2,$(words $(1)),$(call _reverse,$(1))))
  37
  38 ### utility function tests
  39
  40 #$(info $(filter-out 1,$(filter 1%,1 132 456)))
  41 #$(info (int_encode 13): [$(call int_encode,13)])
  42 #$(info (int_encode 156463): [$(call int_encode,156463)])
  43 #$(info (int_encode -156463): [$(call int_encode,-156463)])
  44 #$(info (int_decode (int_encode 156463)): [$(call int_decode,$(call int_encode,156463))])
  45
  46 #$(info trim_zeros(0 0 0): [$(call trim_zeros,0 0 0)])
  47 #$(info trim_zeros(0 0 0 -): [$(call trim_zeros,0 0 0 -)])
  48
  49
  50 ###
  51 ### comparisons
  52 ###
  53
  54 # compare two digits and return 'true' if digit 1 is less than or
  55 # equal to digit 2
  56 _lte_digit = $(strip \
  57               $(if $(call _EQ,$(1),$(2)),\
  58                 true,\
  59                 $(if $(call _EQ,0,$(1)),\
  60                   true,\
  61                   $(if $(wordlist $(1),$(2),$(LIST20_X)),\
  62                     true,\
  63                     ))))
  64
  65 # compare two lists of digits (MSB->LSB) of equal length and return
  66 # 'true' if number 1 is less than number 2
  67 _lte_digits = $(strip \
  68                 $(if $(strip $(1)),\
  69                   $(if $(call _EQ,$(word 1,$(1)),$(word 1,$(2))),\
  70                     $(call _lte_digits,$(wordlist 2,$(words $(1)),$(1)),$(wordlist 2,$(words $(2)),$(2))),\
  71                     $(if $(call _lte_digit,$(word 1,$(1)),$(word 1,$(2))),true,)),\
  72                   true))
  73
  74 ### lte/less than or equal to
  75
  76 _int_lte_encoded = $(strip \
  77                      $(foreach len1,$(words $(1)),$(foreach len2,$(words $(2)),\
  78                      $(if $(call _EQ,$(len1),$(len2)),\
  79                        $(call _lte_digits,$(call _reverse,$(1)),$(call _reverse,$(2))),\
  80                        $(if $(wordlist $(len1),$(len2),$(LIST100_X)),\
  81                          true,\
  82                          )))))
  83
  84 int_lte_encoded = $(strip \
  85                     $(if $(filter -,$(1)),\
  86                       $(if $(filter -,$(2)),\
  87                         $(call _int_lte_encoded,$(filter-out -,$(2)),$(filter-out -,$(1))),\
  88                         true),\
  89                       $(if $(filter -,$(2)),\
  90                         ,\
  91                         $(call _int_lte_encoded,$(1),$(2)))))
  92
  93 int_lte = $(call int_lte_encoded,$(call int_encode,$(1)),$(call int_encode,$(2)))
  94
  95 ### lt/less than
  96
  97 int_lt_encoded = $(strip \
  98                    $(if $(call _EQ,$(strip $(1)),$(strip $(2))),\
  99                      ,\
 100                      $(call int_lte_encoded,$(1),$(2))))
 101
 102 int_lt = $(call int_lt_encoded,$(call int_encode,$(1)),$(call int_encode,$(2)))
 103
 104 ### gte/greater than or equal to
 105
 106 int_gte_encoded = $(strip \
 107                 $(if $(call _EQ,$(strip $(1)),$(strip $(2))),\
 108                   true,\
 109                   $(if $(call int_lte_encoded,$(1),$(2)),,true)))
 110
 111 int_gte = $(call int_gte_encoded,$(call int_encode,$(1)),$(call int_encode,$(2)))
 112
 113 ### gt/greater than
 114
 115 int_gt_encoded = $(strip \
 116                $(if $(call _EQ,$(strip $(1)),$(strip $(2))),\
 117                  ,\
 118                  $(call int_gte_encoded,$(1),$(2))))
 119
 120 int_gt = $(call int_gt_encoded,$(call int_encode,$(1)),$(call int_encode,$(2)))
 121
 122 #$(info _lte_digit,7,8: [$(call _lte_digit,7,8)])
 123 #$(info _lte_digit,8,8: [$(call _lte_digit,8,8)])
 124 #$(info _lte_digit,2,1: [$(call _lte_digit,2,1)])
 125 #$(info _lte_digit,0,0: [$(call _lte_digit,0,0)])
 126 #$(info _lte_digit,0,1: [$(call _lte_digit,0,1)])
 127 #$(info _lte_digit,1,0: [$(call _lte_digit,1,0)])
 128
 129 #$(info _lte_digits,1 2 3,1 2 4: [$(call _lte_digits,1 2 3,1 2 4)])
 130 #$(info _lte_digits,1 2 4,1 2 4: [$(call _lte_digits,1 2 4,1 2 4)])
 131 #$(info _lte_digits,1 2 5,1 2 4: [$(call _lte_digits,1 2 5,1 2 4)])
 132 #$(info _lte_digits,4 1,9 0: [$(call _lte_digits,4 1,9 0)])
 133
 134 # The main comparison operator (others are built on this)
 135 #$(info int_lte_encoded,1,1: [$(call int_lte_encoded,1,1)])
 136 #$(info int_lte_encoded,1,2: [$(call int_lte_encoded,1,2)])
 137 #$(info int_lte_encoded,2,1: [$(call int_lte_encoded,2,1)])
 138 #$(info int_lte_encoded,0,3: [$(call int_lte_encoded,0,3)])
 139 #$(info int_lte_encoded,3,0: [$(call int_lte_encoded,3,0)])
 140 #$(info int_lte_encoded,1 4,0 9: [$(call int_lte_encoded,1 4,0 9)])
 141 #$(info int_lte_encoded,4 3 2 1,4 3 2 1: [$(call int_lte_encoded,4 3 2 1,4 3 2 1)])
 142 #$(info int_lte_encoded,5 3 2 1,4 3 2 1: [$(call int_lte_encoded,5 3 2 1,4 3 2 1)])
 143 #$(info int_lte_encoded,4 3 2 1,5 3 2 1: [$(call int_lte_encoded,4 3 2 1,5 3 2 1)])
 144 # negative numbers
 145 #$(info int_lte_encoded,7 -,7: [$(call int_lte_encoded,7 -,7)])
 146 #$(info int_lte_encoded,7,7 -: [$(call int_lte_encoded,7,7 -)])
 147 #$(info int_lte_encoded,7 -,7 -: [$(call int_lte_encoded,7 -,7 -)])
 148 #$(info int_lte_encoded,1 7 -,0 7: [$(call int_lte_encoded,1 7 -,0 7)])
 149 #$(info int_lte_encoded,1 7,0 7 -: [$(call int_lte_encoded,1 7,0 7 -)])
 150 #$(info int_lte_encoded,1 7 -,0 7 -: [$(call int_lte_encoded,1 7 -,0 7 -)])
 151 #$(info int_lte_encoded,4 3 2 1 -,4 3 2 1: [$(call int_lte_encoded,4 3 2 1 -,4 3 2 1)])
 152 #$(info int_lte_encoded,4 3 2 1,4 3 2 1 -: [$(call int_lte_encoded,4 3 2 1,4 3 2 1 -)])
 153 #$(info int_lte_encoded,4 3 2 1 -,4 3 2 1 -: [$(call int_lte_encoded,4 3 2 1 -,4 3 2 1 -)])
 154
 155 #$(info int_lte,1,1: [$(call int_lte,1,1)])
 156 #$(info int_lte,1,2: [$(call int_lte,1,2)])
 157 #$(info int_lte,2,1: [$(call int_lte,2,1)])
 158 #$(info int_lte,0,3: [$(call int_lte,0,3)])
 159 #$(info int_lte,3,0: [$(call int_lte,3,0)])
 160 #$(info int_lte,1234,1234: [$(call int_lte,1234,1234)])
 161 #$(info int_lte,1235,1234: [$(call int_lte,1235,1234)])
 162 #$(info int_lte,1234,1235: [$(call int_lte,1234,1235)])
 163 #$(info int_lte,-1234,1235: [$(call int_lte,-1234,1235)])
 164 #$(info int_lte,1234,-1235: [$(call int_lte,1234,-1235)])
 165 #$(info int_lte,-1234,-1235: [$(call int_lte,-1234,-1235)])
 166
 167 #$(info int_lt,1,1: [$(call int_lt,1,1)])
 168 #$(info int_lt,1,2: [$(call int_lt,1,2)])
 169 #$(info int_lt,2,1: [$(call int_lt,2,1)])
 170 #$(info int_lt,0,3: [$(call int_lt,0,3)])
 171 #$(info int_lt,3,0: [$(call int_lt,3,0)])
 172 #$(info int_lt,1234,1234: [$(call int_lt,1234,1234)])
 173 #$(info int_lt,1235,1234: [$(call int_lt,1235,1234)])
 174 #$(info int_lt,1234,1235: [$(call int_lt,1234,1235)])
 175 #
 176 #$(info int_gte,1,1: [$(call int_gte,1,1)])
 177 #$(info int_gte,1,2: [$(call int_gte,1,2)])
 178 #$(info int_gte,2,1: [$(call int_gte,2,1)])
 179 #$(info int_gte,0,3: [$(call int_gte,0,3)])
 180 #$(info int_gte,3,0: [$(call int_gte,3,0)])
 181 #$(info int_gte,1234,1234: [$(call int_gte,1234,1234)])
 182 #$(info int_gte,1235,1234: [$(call int_gte,1235,1234)])
 183 #$(info int_gte,1234,1235: [$(call int_gte,1234,1235)])
 184 #
 185 #$(info int_gt,1,1: [$(call int_gt,1,1)])
 186 #$(info int_gt,1,2: [$(call int_gt,1,2)])
 187 #$(info int_gt,2,1: [$(call int_gt,2,1)])
 188 #$(info int_gt,0,3: [$(call int_gt,0,3)])
 189 #$(info int_gt,3,0: [$(call int_gt,3,0)])
 190 #$(info int_gt,1234,1234: [$(call int_gt,1234,1234)])
 191 #$(info int_gt,1235,1234: [$(call int_gt,1235,1234)])
 192 #$(info int_gt,1234,1235: [$(call int_gt,1234,1235)])
 193 #$(info int_gt,-1234,1235: [$(call int_gt,-1234,1235)])
 194 #$(info int_gt,-1234,-1235: [$(call int_gt,-1234,-1235)])
 195
 196
 197 ###
 198 ### addition
 199 ###
 200
 201
 202 # add_digits_with_carry
 203 _add_digit = $(words $(if $(strip $(1)),$(wordlist 1,$(1),$(LIST20_X)),) \
 204                      $(if $(strip $(2)),$(wordlist 1,$(2),$(LIST20_X)),))
 205
 206 # add one to a single digit
 207 _inc_digit = $(words $(wordlist 1,$(if $(1),$(1),0),$(LIST20_X)) x)
 208
 209 # add two encoded numbers digit by digit without resolving carries
 210 # (each digit will be larger than 9 if there is a carry value)
 211 _add = $(if $(1)$(2),$(call _add_digit,$(word 1,$(1)),$(word 1,$(2))) $(call _add,$(wordlist 2,$(words $(1)),$(1)),$(wordlist 2,$(words $(2)),$(2))),)
 212
 213 # take the result of _add and resolve the carry values digit by digit
 214 _resolve_carries = $(strip \
 215              $(if $(1),\
 216                $(foreach num,$(word 1,$(1)),\
 217                  $(if $(filter-out 1,$(filter 1%,$(num))),\
 218                    $(call _resolve_carries,$(call _inc_digit,$(word 2,$(1))) $(wordlist 3,$(words $(1)),$(1)),$(2) $(patsubst 1%,%,$(num))),\
 219                    $(call _resolve_carries,$(wordlist 2,$(words $(1)),$(1)),$(2) $(num)))),\
 220                $(2)))
 221
 222 _negate = $(strip \
 223             $(if $(call _EQ,0,$(strip $(1))),\
 224               0,\
 225               $(if $(filter -,$(1)),$(filter-out -,$(1)),$(1) -)))
 226
 227 # add two encoded numbers, returns encoded number
 228 _int_add_encoded = $(call _resolve_carries,$(call _add,$(1),$(2)))
 229
 230 int_add_encoded = $(strip \
 231                     $(if $(filter -,$(1)),\
 232                       $(if $(filter -,$(2)),\
 233                         $(call _negate,$(call _int_add_encoded,$(filter-out -,$(1)),$(filter-out -,$(2)))),\
 234                         $(call int_sub_encoded,$(2),$(filter-out -,$(1)))),\
 235                       $(if $(filter -,$(2)),\
 236                         $(call int_sub_encoded,$(1),$(filter-out -,$(2))),\
 237                         $(call _int_add_encoded,$(1),$(2)))))
 238
 239 # add two unencoded numbers, returns unencoded number
 240 int_add = $(call int_decode,$(call int_add_encoded,$(call int_encode,$(1)),$(call int_encode,$(2))))
 241
 242 ### addition tests
 243
 244 #$(info _add_digit(7,6,1): [$(call _add_digit,7,6,1)])
 245 #$(info _add_digit(7,6,0): [$(call _add_digit,7,6,0)])
 246 #$(info _add_digit(7,6,0): [$(call _add_digit,7,6,0)])
 247 #$(info _carries(12 14 15): [$(call _carries,12 14 15)])
 248 #$(info _inc_digit(0): $(call _inc_digit,0))
 249 #$(info _inc_digit(1): $(call _inc_digit,1))
 250 #$(info _inc_digit(9): $(call _inc_digit,9))
 251 #$(info _inc_digit(18): $(call _inc_digit,18))
 252 #$(info int_add_encoded(0,0): [$(call int_add_encoded,0,0)])
 253
 254 #$(info int_add(1,2): [$(call int_add,1,2)])
 255 #$(info int_add(9,9): [$(call int_add,9,9)])
 256 #$(info int_add(0,9): [$(call int_add,0,9)])
 257 #$(info int_add(9,0): [$(call int_add,9,0)])
 258 #$(info int_add(0,0): [$(call int_add,0,0)])
 259 #$(info int_add(123,456): [$(call int_add,123,456)])
 260 #$(info int_add(678,789): [$(call int_add,678,789)])
 261 #$(info int_add(1,12): [$(call int_add,1,12)])
 262 #$(info int_add(123,5): [$(call int_add,123,5)])
 263 #$(info int_add(123456,9): [$(call int_add,123456,9)])
 264 #$(info int_add(999999991,9): [$(call int_add,999999991,9)])
 265 # negative numbers
 266 #$(info int_add(-2,2): [$(call int_add,-2,2)])
 267 #$(info int_add(-1,2): [$(call int_add,-1,2)])
 268 #$(info int_add(1,-2): [$(call int_add,1,-2)])
 269 #$(info int_add(-1,-2): [$(call int_add,-1,-2)])
 270
 271 ###
 272 ### subtraction
 273 ###
 274
 275 _get_zeros = $(if $(call _EQ,0,$(word 1,$(1))),$(call _get_zeros,$(wordlist 2,$(words $(1)),$(1)),$(2) 0),$(2))
 276
 277 # return a 9's complement of a single digit
 278 _complement9 = $(strip \
 279                  $(if $(call _EQ,0,$(1)),9,\
 280                  $(if $(call _EQ,1,$(1)),8,\
 281                  $(if $(call _EQ,2,$(1)),7,\
 282                  $(if $(call _EQ,3,$(1)),6,\
 283                  $(if $(call _EQ,4,$(1)),5,\
 284                  $(if $(call _EQ,5,$(1)),4,\
 285                  $(if $(call _EQ,6,$(1)),3,\
 286                  $(if $(call _EQ,7,$(1)),2,\
 287                  $(if $(call _EQ,8,$(1)),1,\
 288                  $(if $(call _EQ,9,$(1)),0)))))))))))
 289
 290 # return a 10's complement of a single digit
 291 _complement10 = $(call _inc_digit,$(call _complement9,$(1)))
 292
 293 #
 294 _complement_rest = $(if $(strip $(1)),\
 295                      $(strip \
 296                        $(call _complement10,$(word 1,$(1))) \
 297                        $(foreach digit,$(wordlist 2,$(words $(1)),$(1)),\
 298                          $(call _complement9,$(digit)))),)
 299
 300 # return the complement of a number
 301 _complement = $(strip $(call _get_zeros,$(1)) \
 302                       $(call _complement_rest,$(wordlist $(call _inc_digit,$(words $(call _get_zeros,$(1)))),$(words $(1)),$(1))))
 303
 304 # subtracted encoded number 2 from encoded number 1 and return and
 305 # encoded number result. both numbers must be positive but may have
 306 # a negative result
 307 __int_sub_encoded = $(strip \
 308                   $(call trim_zeros,\
 309                     $(call drop_last,\
 310                       $(call int_add_encoded,\
 311                         $(1),\
 312                         $(wordlist 1,$(words $(1)),$(call _complement,$(2)) $(LIST100_9))))))
 313
 314 _int_sub_encoded = $(strip \
 315                      $(if $(call _EQ,0,$(strip $(2))),\
 316                        $(1),\
 317                        $(if $(call _int_lte_encoded,$(2),$(1)),\
 318                          $(call __int_sub_encoded,$(1),$(2)),\
 319                          $(call _negate,$(call __int_sub_encoded,$(2),$(1))))))
 320
 321 int_sub_encoded = $(strip \
 322                     $(if $(filter -,$(1)),\
 323                       $(if $(filter -,$(2)),\
 324                         $(call _int_sub_encoded,$(filter-out -,$(2)),$(filter-out -,$(1))),\
 325                         $(call _negate,$(call _int_add_encoded,$(filter-out -,$(1)),$(2)))),\
 326                       $(if $(filter -,$(2)),\
 327                         $(call _int_add_encoded,$(1),$(filter-out -,$(2))),\
 328                         $(call _int_sub_encoded,$(1),$(2)))))
 329
 330 # subtract unencoded number 2 from unencoded number 1 and return
 331 # unencoded result
 332 int_sub = $(call int_decode,$(call int_sub_encoded,$(call int_encode,$(1)),$(call int_encode,$(2))))
 333
 334 ### subtraction tests
 335
 336 #$(info _get_zeros(5 7): [$(call _get_zeros,5 7)])
 337 #$(info _get_zeros(0 0 0 2): [$(call _get_zeros,0 0 0 2)])
 338 #$(info _get_zeros(0 0 0 2 5): [$(call _get_zeros,0 0 0 2 5)])
 339
 340 #$(info _complement(0): [$(call _complement,0)])
 341 #$(info _complement(1): [$(call _complement,1)])
 342 #$(info _complement(9): [$(call _complement,9)])
 343 #$(info _complement(5 7): [$(call _complement,5 7)])
 344 #$(info _complement(0 0 0 2): [$(call _complement,0 0 0 2)])
 345 #$(info _complement(0 0 0 5 4 3 2 1): [$(call _complement,0 0 0 5 4 3 2 1)])
 346
 347 #$(info int_sub_encoded(0 0 1, 3 1): [$(call int_sub_encoded,0 0 1,3 1)])
 348 #$(info int_sub_encoded(2, 2): [$(call int_sub_encoded,2,2)])
 349
 350 #$(info int_sub(2,1): [$(call int_sub,2,1)])
 351 #$(info int_sub(2,0): [$(call int_sub,2,0)])
 352 #$(info int_sub(2,2): [$(call int_sub,2,2)])
 353 #$(info int_sub(100,13): [$(call int_sub,100,13)])
 354 #$(info int_sub(100,99): [$(call int_sub,100,99)])
 355 #$(info int_sub(91,19): [$(call int_sub,91,19)])
 356 # negative numbers
 357 #$(info int_sub(1,2): [$(call int_sub,1,2)])
 358 #$(info int_sub(-1,2): [$(call int_sub,-1,2)])
 359 #$(info int_sub(1,-2): [$(call int_sub,1,-2)])
 360 #$(info int_sub(-1,-2): [$(call int_sub,-1,-2)])
 361 #$(info int_sub(-2,-1): [$(call int_sub,-2,-1)])
 362 #$(info int_sub(19,91): [$(call int_sub,19,91)])
 363 #$(info int_sub(91,-19): [$(call int_sub,91,-19)])
 364 #$(info int_sub(-91,19): [$(call int_sub,-91,19)])
 365 #$(info int_sub(-91,-19): [$(call int_sub,-91,-19)])
 366
 367
 368 ###
 369 ### multiplication
 370 ###
 371
 372 # multiply two digits
 373 #_mult_digit = $(words $(foreach x,$(1),$(2)))
 374 _mult_digit = $(strip \
 375                 $(words $(foreach x,$(wordlist 1,$(1),$(LIST20_X)),\
 376                         $(wordlist 1,$(2),$(LIST20_X)))))
 377
 378 # multipy every digit of number 1 with number 2
 379 # params: digits, digit, indent_zeros, results
 380 _mult_row = $(if $(strip $(1)),$(call _mult_row,$(wordlist 2,$(words $(1)),$(1)),$(2),$(3)0,$(4) $(call _mult_digit,$(word 1,$(1)),$(2))$(3)),$(4))
 381
 382 # multiply every digit of number 2 with every digit of number 1 adding
 383 # correct zero padding to the end of each result
 384 # params: digits, digits, indent_zeros, results
 385 _mult_each = $(if $(strip $(2)),$(call _mult_each,$(1),$(wordlist 2,$(words $(2)),$(2)),$(3)0,$(4) $(call _mult_row,$(1),$(word 1,$(2)),$(3))),$(4))
 386
 387 # add up a bunch of unencoded numbers. Basically reduce into the first number
 388 _add_many = $(if $(word 2,$(1)),$(call _add_many,$(call int_add,$(word 1,$(1)),$(word 2,$(1))) $(wordlist  3,$(words $(1)),$(1))),$(1))
 389
 390 # multiply two encoded numbers, returns encoded number
 391 _int_mult_encoded = $(call trim_zeros,$(call int_encode,$(call _add_many,$(call _mult_each,$(1),$(2)))))
 392
 393 int_mult_encoded = $(strip \
 394              $(if $(filter -,$(1)),\
 395                $(if $(filter -,$(2)),\
 396                  $(call _int_mult_encoded,$(filter-out -,$(1)),$(filter-out -,$(2))),\
 397                  $(call _negate,$(call _int_mult_encoded,$(filter-out -,$(1)),$(2)))),\
 398                $(if $(filter -,$(2)),\
 399                  $(call _negate,$(call _int_mult_encoded,$(1),$(filter-out -,$(2)))),\
 400                  $(call _int_mult_encoded,$(1),$(2)))))
 401
 402 # multiply two unencoded numbers, returns unencoded number
 403 int_mult = $(call int_decode,$(call int_mult_encoded,$(call int_encode,$(1)),$(call int_encode,$(2))))
 404
 405 #$(info _mult_digit(8,6): [$(call _mult_digit,8,6)])
 406 #$(info _mult_digit(7,6): [$(call _mult_digit,7,6)])
 407 #$(info _mult_row(8,6): [$(call _mult_row,8,6)])
 408 #$(info _mult_row(8 7,6): [$(call _mult_row,8 7,6)])
 409 #$(info _mult_row(8 7 3,6): [$(call _mult_row,8 7 3,6)])
 410 #$(info _mult_each(8 7 6, 4 3 2): [$(call _mult_each,8 7 6,4 3 2)])
 411 #$(info _add_many(123 234 345 456): [$(call _add_many,123 234 345 456)])
 412
 413 #$(info int_mult_encoded(8 7 3,6): [$(call int_mult_encoded,8 7 3,6)])
 414 #$(info int_mult_encoded(8 7 3,0): [$(call int_mult_encoded,8 7 3,0)])
 415
 416 #$(info int_mult(378,6): [$(call int_mult,378,6)])
 417 #$(info int_mult(678,234): [$(call int_mult,678,234)])
 418 #$(info int_mult(1,23456): [$(call int_mult,1,23456)])
 419 #$(info int_mult(0,23456): [$(call int_mult,0,23456)])
 420 #$(info int_mult(0,0): [$(call int_mult,0,0)])
 421 # negative numbers
 422 #$(info int_mult(-378,6): [$(call int_mult,-378,6)])
 423 #$(info int_mult(678,-234): [$(call int_mult,678,-234)])
 424 #$(info int_mult(-1,-23456): [$(call int_mult,-1,-23456)])
 425 #$(info int_mult(0,-23456): [$(call int_mult,0,-23456)])
 426
 427 ###
 428 ### division
 429 ###
 430
 431 # return list of zeros needed to pad number 2 to the same length as number 1
 432 _zero_pad = $(strip $(wordlist 1,$(call int_sub,$(words $(1)),$(words $(2))),$(LIST100_0)))
 433
 434 # num1, num2, zero pad, result_accumulator
 435 # algorithm:
 436 # - B = pad with zeros to make same digit length as A
 437 # - loop
 438 #    - if (B <= A)
 439 #        - A = subtract B from A
 440 #        - C = C + 10^(B pad.length)
 441 #    - else
 442 #        - if B.length < origin B.length: break
 443 #        - chop least significant digit of B
 444 _div = $(strip \
 445          $(if $(call int_lte_encoded,$(3) $(2),$(1)),\
 446            $(call _div,$(call int_sub_encoded,$(1),$(3) $(2)),$(2),$(3),$(call int_add_encoded,$(4),$(3) 1)),\
 447            $(if $(3),\
 448              $(call _div,$(1),$(2),$(wordlist 2,$(words $(3)),$(3)),$(4)),\
 449              $(4))))
 450
 451 # divide two encoded numbers, returns encoded number
 452 _int_div_encoded = $(strip \
 453                 $(if $(call _EQ,0,$(1)),\
 454                   0,\
 455                   $(if $(call _EQ,$(1),$(2)),\
 456                     1,\
 457                     $(if $(call int_gt_encoded,$(2),$(1)),\
 458                       0,\
 459                       $(call _div,$(1),$(2),$(call _zero_pad,$(1),$(2)),0)))))
 460
 461 int_div_encoded = $(strip \
 462              $(if $(filter -,$(1)),\
 463                $(if $(filter -,$(2)),\
 464                  $(call _int_div_encoded,$(filter-out -,$(1)),$(filter-out -,$(2))),\
 465                  $(call _negate,$(call _int_div_encoded,$(filter-out -,$(1)),$(2)))),\
 466                $(if $(filter -,$(2)),\
 467                  $(call _negate,$(call _int_div_encoded,$(1),$(filter-out -,$(2)))),\
 468                  $(call _int_div_encoded,$(1),$(2)))))
 469
 470 # divide two unencoded numbers, returns unencoded number
 471 int_div = $(call int_decode,$(call int_div_encoded,$(call int_encode,$(1)),$(call int_encode,$(2))))
 472
 473 ### division tests
 474
 475 #$(info _zero_pad(1 2 3 4,1 3): [$(call _zero_pad,1 2 3 4,1 3)])
 476 #$(info _zero_pad(1 2,1 3): [$(call _zero_pad,1 2,1 3)])
 477 #$(info _zero_pad(2,1 3): [$(call _zero_pad,1 2,1 3)])
 478 #
 479 #$(info int_div_encoded(2,1): [$(call int_div_encoded,2,1)])
 480 #$(info int_div_encoded(3,1): [$(call int_div_encoded,3,1)])
 481 #$(info int_div_encoded(3,2): [$(call int_div_encoded,3,2)])
 482 #$(info int_div_encoded(0,7): [$(call int_div_encoded,0,7)])
 483 #$(info int_div_encoded(0 3,0 2): [$(call int_div_encoded,0 3,0 2)])
 484 #$(info int_div_encoded(0 3,5): [$(call int_div_encoded,0 3,5)])
 485 #
 486 #$(info int_div(5,1): [$(call int_div,5,1)])
 487 #$(info int_div(5,2): [$(call int_div,5,2)])
 488 #$(info int_div(123,7): [$(call int_div,123,7)])
 489 #$(info int_div(100,7): [$(call int_div,100,7)])
 490 # negative numbers
 491 #$(info int_div(-5,1): [$(call int_div,-5,1)])
 492 #$(info int_div(5,-2): [$(call int_div,5,-2)])
 493 #$(info int_div(-123,-7): [$(call int_div,-123,-7)])
 494
 495
 496 ### combination tests
 497
 498 # (/ (- (+ 515 (* 222 311)) 300) 41) = 1689
 499 #$(info int_mult,222,311: [$(call int_mult,222,311)])
 500 #$(info int_add(515,69042): [$(call int_add,515,69042)])
 501 #$(info int_sub(69557,300): [$(call int_sub,69557,300)])
 502 #$(info int_div(69257,41): [$(call int_div,69257,41)])
 503 # (/ (- (+ 515 (* -222 311)) 300) 41) = -1678
 504 #$(info int_mult,-222,311: [$(call int_mult,-222,311)])
 505 #$(info int_add(515,-69042): [$(call int_add,515,-69042)])
 506 #$(info int_sub(-68527,300): [$(call int_sub,-68527,300)])
 507 #$(info int_div(-68827,41): [$(call int_div,-68827,41)])
 508
 509 ###############################################################
 510
 511 all:
 512         @true
 513
 514 endif
 515
 516 # vim: ts=2 et