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 $(if $(call _EQ,0,$(strip $(1))),$(1),$(call _reverse,$(call _trim_zeros,$(call _reverse,$(1))))))
  30
  31 # drop the last element of a list of words/digits
  32 drop_last = $(call _reverse,$(wordlist 2,$(words $(1)),$(call _reverse,$(1))))
  33
  34 ### utility function tests
  35
  36 #$(info $(filter-out 1,$(filter 1%,1 132 456)))
  37 #$(info (int_encode 13): [$(call int_encode,13)])
  38 #$(info (int_encode 156463): [$(call int_encode,156463)])
  39 #$(info (int_decode (int_encode 156463)): [$(call int_decode,$(call int_encode,156463))])
  40
  41 #$(info trim_zeros(0 0 0): [$(call trim_zeros,0 0 0)])
  42
  43
  44 ###
  45 ### comparisons
  46 ###
  47
  48 # compare two digits and return 'true' if digit 1 is less than or
  49 # equal to digit 2
  50 _lte_digit = $(strip \
  51               $(if $(call _EQ,$(1),$(2)),\
  52                 true,\
  53                 $(if $(call _EQ,0,$(1)),\
  54                   true,\
  55                   $(if $(wordlist $(1),$(2),$(LIST20_X)),\
  56                     true,\
  57                     ))))
  58
  59 # compare two lists of digits (MSB->LSB) of equal length and return
  60 # 'true' if number 1 is less than number 2
  61 _lte_digits = $(strip \
  62                 $(if $(strip $(1)),\
  63                   $(if $(call _EQ,$(word 1,$(1)),$(word 1,$(2))),\
  64                     $(call _lte_digits,$(wordlist 2,$(words $(1)),$(1)),$(wordlist 2,$(words $(2)),$(2))),\
  65                     $(if $(call _lte_digit,$(word 1,$(1)),$(word 1,$(2))),true,)),\
  66                   true))
  67
  68 ### lte/less than or equal to
  69
  70 int_lte_encoded = $(strip \
  71                     $(foreach len1,$(words $(1)),$(foreach len2,$(words $(2)),\
  72                       $(if $(call _EQ,$(len1),$(len2)),\
  73                         $(call _lte_digits,$(call _reverse,$(1)),$(call _reverse,$(2))),\
  74                         $(if $(wordlist $(len1),$(len2),$(LIST100_X)),\
  75                           true,\
  76                           )))))
  77
  78 int_lte = $(call int_lte_encoded,$(call int_encode,$(1)),$(call int_encode,$(2)))
  79
  80 ### lt/less than
  81
  82 int_lt_encoded = $(strip \
  83                    $(if $(call _EQ,$(strip $(1)),$(strip $(2))),\
  84                      ,\
  85                      $(call int_lte_encoded,$(1),$(2))))
  86
  87 int_lt = $(call int_lt_encoded,$(call int_encode,$(1)),$(call int_encode,$(2)))
  88
  89 ### gte/greater than or equal to
  90
  91 int_gte_encoded = $(strip \
  92                 $(if $(call _EQ,$(strip $(1)),$(strip $(2))),\
  93                   true,\
  94                   $(if $(call int_lte_encoded,$(1),$(2)),,true)))
  95
  96 int_gte = $(call int_gte_encoded,$(call int_encode,$(1)),$(call int_encode,$(2)))
  97
  98 ### gt/greater than
  99
 100 int_gt_encoded = $(strip \
 101                $(if $(call _EQ,$(strip $(1)),$(strip $(2))),\
 102                  ,\
 103                  $(call int_gte_encoded,$(1),$(2))))
 104
 105 int_gt = $(call int_gt_encoded,$(call int_encode,$(1)),$(call int_encode,$(2)))
 106
 107 #$(info _lte_digit,7,8: [$(call _lte_digit,7,8)])
 108 #$(info _lte_digit,8,8: [$(call _lte_digit,8,8)])
 109 #$(info _lte_digit,2,1: [$(call _lte_digit,2,1)])
 110 #$(info _lte_digit,0,0: [$(call _lte_digit,0,0)])
 111 #$(info _lte_digit,0,1: [$(call _lte_digit,0,1)])
 112 #$(info _lte_digit,1,0: [$(call _lte_digit,1,0)])
 113
 114 #$(info _lte_digits,1 2 3,1 2 4: [$(call _lte_digits,1 2 3,1 2 4)])
 115 #$(info _lte_digits,1 2 4,1 2 4: [$(call _lte_digits,1 2 4,1 2 4)])
 116 #$(info _lte_digits,1 2 5,1 2 4: [$(call _lte_digits,1 2 5,1 2 4)])
 117 #$(info _lte_digits,4 1,9 0: [$(call _lte_digits,4 1,9 0)])
 118
 119 #$(info int_lte_encoded,1,1: [$(call int_lte_encoded,1,1)])
 120 #$(info int_lte_encoded,1,2: [$(call int_lte_encoded,1,2)])
 121 #$(info int_lte_encoded,2,1: [$(call int_lte_encoded,2,1)])
 122 #$(info int_lte_encoded,0,3: [$(call int_lte_encoded,0,3)])
 123 #$(info int_lte_encoded,3,0: [$(call int_lte_encoded,3,0)])
 124 #$(info int_lte_encoded,1 4,0 9: [$(call int_lte_encoded,1 4,0 9)])
 125 #$(info int_lte_encoded,4 3 2 1,4 3 2 1: [$(call int_lte_encoded,4 3 2 1,4 3 2 1)])
 126 #$(info int_lte_encoded,5 3 2 1,4 3 2 1: [$(call int_lte_encoded,5 3 2 1,4 3 2 1)])
 127 #$(info int_lte_encoded,4 3 2 1,5 3 2 1: [$(call int_lte_encoded,4 3 2 1,5 3 2 1)])
 128
 129 #$(info int_lte,1,1: [$(call int_lte,1,1)])
 130 #$(info int_lte,1,2: [$(call int_lte,1,2)])
 131 #$(info int_lte,2,1: [$(call int_lte,2,1)])
 132 #$(info int_lte,0,3: [$(call int_lte,0,3)])
 133 #$(info int_lte,3,0: [$(call int_lte,3,0)])
 134 #$(info int_lte,1234,1234: [$(call int_lte,1234,1234)])
 135 #$(info int_lte,1235,1234: [$(call int_lte,1235,1234)])
 136 #$(info int_lte,1234,1235: [$(call int_lte,1234,1235)])
 137 #
 138 #$(info int_lt,1,1: [$(call int_lt,1,1)])
 139 #$(info int_lt,1,2: [$(call int_lt,1,2)])
 140 #$(info int_lt,2,1: [$(call int_lt,2,1)])
 141 #$(info int_lt,0,3: [$(call int_lt,0,3)])
 142 #$(info int_lt,3,0: [$(call int_lt,3,0)])
 143 #$(info int_lt,1234,1234: [$(call int_lt,1234,1234)])
 144 #$(info int_lt,1235,1234: [$(call int_lt,1235,1234)])
 145 #$(info int_lt,1234,1235: [$(call int_lt,1234,1235)])
 146 #
 147 #$(info int_gte,1,1: [$(call int_gte,1,1)])
 148 #$(info int_gte,1,2: [$(call int_gte,1,2)])
 149 #$(info int_gte,2,1: [$(call int_gte,2,1)])
 150 #$(info int_gte,0,3: [$(call int_gte,0,3)])
 151 #$(info int_gte,3,0: [$(call int_gte,3,0)])
 152 #$(info int_gte,1234,1234: [$(call int_gte,1234,1234)])
 153 #$(info int_gte,1235,1234: [$(call int_gte,1235,1234)])
 154 #$(info int_gte,1234,1235: [$(call int_gte,1234,1235)])
 155 #
 156 #$(info int_gt,1,1: [$(call int_gt,1,1)])
 157 #$(info int_gt,1,2: [$(call int_gt,1,2)])
 158 #$(info int_gt,2,1: [$(call int_gt,2,1)])
 159 #$(info int_gt,0,3: [$(call int_gt,0,3)])
 160 #$(info int_gt,3,0: [$(call int_gt,3,0)])
 161 #$(info int_gt,1234,1234: [$(call int_gt,1234,1234)])
 162 #$(info int_gt,1235,1234: [$(call int_gt,1235,1234)])
 163 #$(info int_gt,1234,1235: [$(call int_gt,1234,1235)])
 164
 165
 166 ###
 167 ### addition
 168 ###
 169
 170
 171 # add_digits_with_carry
 172 _add_digit = $(words $(if $(strip $(1)),$(wordlist 1,$(1),$(LIST20_X)),) \
 173                      $(if $(strip $(2)),$(wordlist 1,$(2),$(LIST20_X)),))
 174
 175 # add one to a single digit
 176 _inc_digit = $(words $(wordlist 1,$(if $(1),$(1),0),$(LIST20_X)) x)
 177
 178 # add two encoded numbers digit by digit without resolving carries
 179 # (each digit will be larger than 9 if there is a carry value)
 180 _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))),)
 181
 182 # take the result of _add and resolve the carry values digit by digit
 183 _resolve_carries = $(strip \
 184              $(if $(1),\
 185                $(foreach num,$(word 1,$(1)),\
 186                  $(if $(filter-out 1,$(filter 1%,$(num))),\
 187                    $(call _resolve_carries,$(call _inc_digit,$(word 2,$(1))) $(wordlist 3,$(words $(1)),$(1)),$(2) $(patsubst 1%,%,$(num))),\
 188                    $(call _resolve_carries,$(wordlist 2,$(words $(1)),$(1)),$(2) $(num)))),\
 189                $(2)))
 190
 191 # add two encoded numbers, returns encoded number
 192 int_add_encoded = $(call _resolve_carries,$(call _add,$(1),$(2)))
 193
 194 # add two unencoded numbers, returns unencoded number
 195 int_add = $(call int_decode,$(call int_add_encoded,$(call int_encode,$(1)),$(call int_encode,$(2))))
 196
 197 ### addition tests
 198
 199 #$(info _add_digit(7,6,1): [$(call _add_digit,7,6,1)])
 200 #$(info _add_digit(7,6,0): [$(call _add_digit,7,6,0)])
 201 #$(info _add_digit(7,6,0): [$(call _add_digit,7,6,0)])
 202 #$(info _carries(12 14 15): [$(call _carries,12 14 15)])
 203 #$(info _inc_digit(0): $(call _inc_digit,0))
 204 #$(info _inc_digit(1): $(call _inc_digit,1))
 205 #$(info _inc_digit(9): $(call _inc_digit,9))
 206 #$(info _inc_digit(18): $(call _inc_digit,18))
 207 #$(info int_add_encoded(0,0): [$(call int_add_encoded,0,0)])
 208
 209 #$(info int_add(1,2): [$(call int_add,1,2)])
 210 #$(info int_add(9,9): [$(call int_add,9,9)])
 211 #$(info int_add(0,9): [$(call int_add,0,9)])
 212 #$(info int_add(9,0): [$(call int_add,9,0)])
 213 #$(info int_add(0,0): [$(call int_add,0,0)])
 214 #$(info int_add(123,456): [$(call int_add,123,456)])
 215 #$(info int_add(678,789): [$(call int_add,678,789)])
 216 #$(info int_add(1,12): [$(call int_add,1,12)])
 217 #$(info int_add(123,5): [$(call int_add,123,5)])
 218 #$(info int_add(123456,9): [$(call int_add,123456,9)])
 219 #$(info int_add(999999991,9): [$(call int_add,999999991,9)])
 220
 221 ###
 222 ### subtraction
 223 ###
 224
 225 _get_zeros = $(if $(call _EQ,0,$(word 1,$(1))),$(call _get_zeros,$(wordlist 2,$(words $(1)),$(1)),$(2) 0),$(2))
 226
 227 # return a 9's complement of a single digit
 228 _complement9 = $(strip \
 229                  $(if $(call _EQ,0,$(1)),9,\
 230                  $(if $(call _EQ,1,$(1)),8,\
 231                  $(if $(call _EQ,2,$(1)),7,\
 232                  $(if $(call _EQ,3,$(1)),6,\
 233                  $(if $(call _EQ,4,$(1)),5,\
 234                  $(if $(call _EQ,5,$(1)),4,\
 235                  $(if $(call _EQ,6,$(1)),3,\
 236                  $(if $(call _EQ,7,$(1)),2,\
 237                  $(if $(call _EQ,8,$(1)),1,\
 238                  $(if $(call _EQ,9,$(1)),0)))))))))))
 239
 240 # return a 10's complement of a single digit
 241 _complement10 = $(call _inc_digit,$(call _complement9,$(1)))
 242
 243 #
 244 _complement_rest = $(if $(strip $(1)),\
 245                      $(strip \
 246                        $(call _complement10,$(word 1,$(1))) \
 247                        $(foreach digit,$(wordlist 2,$(words $(1)),$(1)),\
 248                          $(call _complement9,$(digit)))),)
 249
 250 # return the complement of a number
 251 _complement = $(strip $(call _get_zeros,$(1)) \
 252                       $(call _complement_rest,$(wordlist $(call _inc_digit,$(words $(call _get_zeros,$(1)))),$(words $(1)),$(1))))
 253
 254 # subtracted encoded number 2 from encoded number 1 and return and
 255 # encoded number result
 256 int_sub_encoded = $(strip \
 257                 $(if $(call _EQ,0,$(strip $(2))),\
 258                   $(1),\
 259                   $(call trim_zeros,\
 260                     $(call drop_last,\
 261                       $(call int_add_encoded,\
 262                         $(1),\
 263                         $(wordlist 1,$(words $(1)),$(call _complement,$(2)) $(LIST100_9)))))))
 264
 265 # subtract unencoded number 2 from unencoded number 1 and return
 266 # unencoded result
 267 int_sub = $(call int_decode,$(call int_sub_encoded,$(call int_encode,$(1)),$(call int_encode,$(2))))
 268
 269 ### subtraction tests
 270
 271 #$(info _get_zeros(5 7): [$(call _get_zeros,5 7)])
 272 #$(info _get_zeros(0 0 0 2): [$(call _get_zeros,0 0 0 2)])
 273 #$(info _get_zeros(0 0 0 2 5): [$(call _get_zeros,0 0 0 2 5)])
 274
 275 #$(info _complement(0): [$(call _complement,0)])
 276 #$(info _complement(1): [$(call _complement,1)])
 277 #$(info _complement(9): [$(call _complement,9)])
 278 #$(info _complement(5 7): [$(call _complement,5 7)])
 279 #$(info _complement(0 0 0 2): [$(call _complement,0 0 0 2)])
 280 #$(info _complement(0 0 0 5 4 3 2 1): [$(call _complement,0 0 0 5 4 3 2 1)])
 281
 282 #$(info int_sub_encoded(0 0 1, 3 1): [$(call int_sub_encoded,0 0 1,3 1)])
 283 #$(info int_sub_encoded(2, 2): [$(call int_sub_encoded,2,2)])
 284
 285 #$(info int_sub(2,1): [$(call int_sub,2,1)])
 286 #$(info int_sub(2,0): [$(call int_sub,2,0)])
 287 #$(info int_sub(2,2): [$(call int_sub,2,2)])
 288 #$(info int_sub(100,13): [$(call int_sub,100,13)])
 289 #$(info int_sub(100,99): [$(call int_sub,100,99)])
 290 #$(info int_sub(91,19): [$(call int_sub,91,19)])
 291
 292
 293 ###
 294 ### multiplication
 295 ###
 296
 297 # multiply two digits
 298 #_mult_digit = $(words $(foreach x,$(1),$(2)))
 299 _mult_digit = $(strip \
 300                 $(words $(foreach x,$(wordlist 1,$(1),$(LIST20_X)),\
 301                         $(wordlist 1,$(2),$(LIST20_X)))))
 302
 303 # multipy every digit of number 1 with number 2
 304 # params: digits, digit, indent_zeros, results
 305 _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))
 306
 307 # multiply every digit of number 2 with every digit of number 1 adding
 308 # correct zero padding to the end of each result
 309 # params: digits, digits, indent_zeros, results
 310 _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))
 311
 312 # add up a bunch of unencoded numbers. Basically reduce into the first number
 313 _add_many = $(if $(word 2,$(1)),$(call _add_many,$(call int_add,$(word 1,$(1)),$(word 2,$(1))) $(wordlist  3,$(words $(1)),$(1))),$(1))
 314
 315 # multiply two encoded numbers, returns encoded number
 316 int_mult_encoded = $(call trim_zeros,$(call int_encode,$(call _add_many,$(call _mult_each,$(1),$(2)))))
 317
 318 # multiply two unencoded numbers, returns unencoded number
 319 int_mult = $(call int_decode,$(call int_mult_encoded,$(call int_encode,$(1)),$(call int_encode,$(2))))
 320
 321 #$(info _mult_digit(8,6): [$(call _mult_digit,8,6)])
 322 #$(info _mult_digit(7,6): [$(call _mult_digit,7,6)])
 323 #$(info _mult_row(8,6): [$(call _mult_row,8,6)])
 324 #$(info _mult_row(8 7,6): [$(call _mult_row,8 7,6)])
 325 #$(info _mult_row(8 7 3,6): [$(call _mult_row,8 7 3,6)])
 326 #$(info _mult_each(8 7 6, 4 3 2): [$(call _mult_each,8 7 6,4 3 2)])
 327 #$(info _add_many(123 234 345 456): [$(call _add_many,123 234 345 456)])
 328
 329 #$(info int_mult_encoded(8 7 3,6): [$(call int_mult_encoded,8 7 3,6)])
 330 #$(info int_mult_encoded(8 7 3,0): [$(call int_mult_encoded,8 7 3,0)])
 331
 332 #$(info int_mult(378,6): [$(call int_mult,378,6)])
 333 #$(info int_mult(678,234): [$(call int_mult,678,234)])
 334 #$(info int_mult(1,23456): [$(call int_mult,1,23456)])
 335 #$(info int_mult(0,23456): [$(call int_mult,0,23456)])
 336 #$(info int_mult(0,0): [$(call int_mult,0,0)])
 337
 338 ###
 339 ### division
 340 ###
 341
 342 # return list of zeros needed to pad number 2 to the same length as number 1
 343 _zero_pad = $(strip $(wordlist 1,$(call int_sub,$(words $(1)),$(words $(2))),$(LIST100_0)))
 344
 345 # num1, num2, zero pad, result_accumulator
 346 # algorithm:
 347 # - B = pad with zeros to make same digit length as A
 348 # - loop
 349 #    - if (B <= A)
 350 #        - A = subtract B from A
 351 #        - C = C + 10^(B pad.length)
 352 #    - else
 353 #        - if B.length < origin B.length: break
 354 #        - chop least significant digit of B
 355 _div = $(strip \
 356          $(if $(call int_lte_encoded,$(3) $(2),$(1)),\
 357            $(call _div,$(call int_sub_encoded,$(1),$(3) $(2)),$(2),$(3),$(call int_add_encoded,$(4),$(3) 1)),\
 358            $(if $(3),\
 359              $(call _div,$(1),$(2),$(wordlist 2,$(words $(3)),$(3)),$(4)),\
 360              $(4))))
 361
 362 # divide two encoded numbers, returns encoded number
 363 int_div_encoded = $(strip \
 364                 $(if $(call _EQ,0,$(1)),\
 365                   0,\
 366                   $(if $(call _EQ,$(1),$(2)),\
 367                     1,\
 368                     $(if $(call int_gt_encoded,$(2),$(1)),\
 369                       0,\
 370                       $(call _div,$(1),$(2),$(call _zero_pad,$(1),$(2)),0)))))
 371
 372 # divide two unencoded numbers, returns unencoded number
 373 int_div = $(call int_decode,$(call int_div_encoded,$(call int_encode,$(1)),$(call int_encode,$(2))))
 374
 375 ### division tests
 376
 377 #$(info _zero_pad(1 2 3 4,1 3): [$(call _zero_pad,1 2 3 4,1 3)])
 378 #$(info _zero_pad(1 2,1 3): [$(call _zero_pad,1 2,1 3)])
 379 #$(info _zero_pad(2,1 3): [$(call _zero_pad,1 2,1 3)])
 380 #
 381 #$(info int_div_encoded(2,1): [$(call int_div_encoded,2,1)])
 382 #$(info int_div_encoded(3,1): [$(call int_div_encoded,3,1)])
 383 #$(info int_div_encoded(3,2): [$(call int_div_encoded,3,2)])
 384 #$(info int_div_encoded(0,7): [$(call int_div_encoded,0,7)])
 385 #$(info int_div_encoded(0 3,0 2): [$(call int_div_encoded,0 3,0 2)])
 386 #$(info int_div_encoded(0 3,5): [$(call int_div_encoded,0 3,5)])
 387 #
 388 #$(info int_div(5,1): [$(call int_div,5,1)])
 389 #$(info int_div(5,2): [$(call int_div,5,2)])
 390 #$(info int_div(123,7): [$(call int_div,123,7)])
 391 #$(info int_div(100,7): [$(call int_div,100,7)])
 392
 393
 394 ### combination tests
 395
 396 # (/ (- (+ 515 (* 222 311)) 300) 41)
 397 #$(info int_mult,222,311: [$(call int_mult,222,311)])
 398 #$(info int_add(515,69042): [$(call int_add,515,69042)])
 399 #$(info int_sub(69557,300): [$(call int_sub,69557,300)])
 400 #$(info int_div(69257,41): [$(call int_div,69257,41)])
 401
 402 ###############################################################
 403
 404 all:
 405         @true
 406
 407 endif
 408
 409 # vim: ts=2 et