(approximation) Tj /R284 Do 0000023449 00000 n 258.513 215.39 258.531 215.384 258.551 215.384 c 1 0 0 1 509.279 171.24 Tm 249.942 649.823 m 164.729 538.505 164.566 538.569 164.316 538.628 c 409 0 obj 174.631 651.694 174.565 651.815 174.456 651.903 c S 383.476 573.606 l 447.513 571.42 l 393.673 573.666 l 1 0 0 1 429.84 68.04 Tm BI (all) Tj 234.838 654.742 234.92 654.834 235.023 654.908 c /R214 Do 453.462 569.235 0.971167 0.242749 re /R272 Do 0000023053 00000 n /R379 Do (culating) Tj 177.946 652.268 178.082 652.162 178.19 652.027 c 248.291 216.948 l (rounding) Tj /IM true /W 1 /H 1 /BPC 1 /F [/A85] endstream 229.918 652.456 230.086 652.423 230.238 652.357 c /R148 148 0 R Q stream 1.9 0 0 -7.1 397.4 411.9 cm 199.771 635.225 199.643 635.637 199.386 635.925 c 406.845 569.235 0.971167 0.242749 re q 139.718 606.68 l q Q 0.48 0 0 -11.04 287.868 677.868 cm S S 1 0 0 1 539.639 414.96 Tm 556.408 584.532 l 1 0 0 1 523.319 80.04 Tm /R218 Do Q 273.065 253.411 l /R198 Do 437.195 546.352 l S BI 3.4 0 0 -6.7 271.5 346.5 cm 262.857 222.967 262.777 222.851 262.726 222.698 c endstream 4.9 0 0 -4.8 159.3 235.3 cm (algorithms) Tj 251 0 obj BT 423.598 551.026 0.971167 0.242749 re 227.584 649.601 227.628 649.48 227.692 649.364 c q 195.324 649.364 l 4.3 0 0 -7 416.5 324.1 cm 510.516 528.201 510.48 528.306 510.409 528.385 c 1 1 0 rg /R540 Do 99.36 0 0 -0.48 410.748 534.228 cm 1 0 0 1 489.359 139.8 Tm 438.287 546.716 m 266.191 250.918 266.261 250.85 266.361 250.789 c 171.082 538.537 171.119 539.036 171.119 539.829 c 447.696 577.308 l 1 0 0 1 241.439 80.04 Tm 149.801 541.041 m 264.845 205.97 264.777 205.988 264.694 205.988 c (\() Tj (not) Tj 203.075 229.826 l 370.85 573.545 l /R355 Do q 2.7 0 0 -4.6 471.9 426.2 cm q /Type /Page 249.627 652.506 249.752 652.469 249.901 652.469 c !~> 142.105 603.746 l 345.053 553.089 0.242896 0.971191 re 230.768 652.106 m 148.34 669.746 l 101.69 294.669 m 505.299 551.39 0.242749 0.971191 re 443.306 542.188 l q 359.621 569.478 l f* 425.115 579.008 l 1 0 0 1 127.92 506.88 Tm q BT /R218 Do EI 1 0 0 1 82.1999 103.92 Tm f* endobj /R584 Do Q 138.163 630.054 138.281 630.253 138.358 630.526 c 6.6 0 0 -6.6 199.9 400.8 cm 1 0 0 1 245.399 275.16 Tm 1 0 0 1 368.4 402.96 Tm 252.041 210.71 l 1 0 0 1 378.96 626.04 Tm f* 4.3 0 0 -7 121.1 249.5 cm f* << /Length 374 0 R >> 133.568 282.404 l 216.058 621.404 216.174 621.071 216.397 620.834 c (Binary) Tj 379.894 551.026 0.971191 0.242749 re 3.8 0 0 -5 351.6 541.9 cm (o) Tj 184.12 661.536 l 133.575 305.466 l /R245 Do 134.59 281.701 l 372 0 obj 4N`CV&,[Qd0SA279&J4glolM@iV*kpqu6'!me?b=]A;B5]Di?L5PKs-_nYqq 454.737 549.872 l q ID ! q S (to) Tj Q (that) Tj 493.159 573.545 l Q 63.3799 259.327 62.8896 259.108 62.481 258.67 c 0000643892 00000 n 176.606 703.507 176.824 702.707 176.805 701.63 c q >> 3.4 0 0 -6.7 113.8 298.9 cm f* q f* 224.997 651.525 l (par) Tj endobj 151.989 542.972 151.616 542.981 150.995 542.981 c 1 0 0 1 374.16 355.2 Tm 316.221 570.207 l 556.529 584.41 m !~> 187.246 659.619 l 153.21 667.561 153.282 667.568 153.373 667.581 c Then, there exist unique integers q and r such that . q 437.434 564.323 l f* S 481.384 573.788 l (tw) Tj 47 (has) Tj Q 577 0 obj q (\256rst) Tj endstream q Q 5.3 0 0 -4.8 420.9 495.1 cm 110.772 713.573 110.647 713.184 110.398 712.909 c 238.131 221.672 l q Slow division and fast division. Q << /Type /XObject /Name /R399 /Subtype /Image /Length 400 0 R (the) Tj endstream 196.286 624.743 196.487 624.922 196.745 625.047 c 419.288 573.909 l 457.65 573.606 l (in) Tj 480 0 R 140.964 634.25 l 278.029 248.819 278.366 248.964 278.638 249.255 c 215.474 626.381 215.693 625.582 215.674 624.504 c 108.679 714.055 m 184.694 702.836 l q BT 188.534 649.364 l 327.572 573.909 l 1 0 0 1 252.959 518.88 Tm 177.846 542.379 177.927 542.355 178.02 542.355 c 541.597 566.989 0.242725 0.971191 re q /R252 Do /R64 Do 0.48 0 0 -11.04 86.988 585.348 cm 432.703 544.652 l 172.187 654.259 172.208 654.201 172.248 654.149 c 488.303 573.666 l 126.932 704.357 l 132.812 607.406 132.651 607.344 132.522 607.223 c 1 0 0 1 521.039 139.8 Tm f* (ely) Tj 2086 q 511.976 573.727 l q q Q endobj 3.6 0 0 -6 147.2 617.8 cm 1 0 0 1 438.6 103.92 Tm f* /ImageMask true /Width 12 /Height 11 /BitsPerComponent 1 /Decode [1 0] /Filter [/ASCII85Decode /CCITTFaxDecode] /DecodeParms [null << /K -1 /Columns 12 /BlackIs1 true >>] 393.689 528.776 393.586 529.031 393.382 529.239 c 1 0 0 1 516.719 199.56 Tm 1 0 0 1 329.64 127.8 Tm 556.469 586.96 l f* endobj (each) Tj ID ! q 514.404 551.026 0.971167 0.242749 re 447.453 572.452 l 147.433 583.212 l 134.56 633.036 134.673 632.712 134.89 632.481 c Q 124.177 606.68 l Q endstream 277.789 259.632 l 171.23 551.05 l BI 449.699 544.774 l 186.006 702.599 m Tj /R182 Do (,) Tj q f* 1 0 0 1 512.279 467.76 Tm 460.26 573.666 l /R316 Do 255.323 216.837 255.25 216.819 255.178 216.783 c 512.148 574.623 l 4.3 0 0 -7 238.5 323.6 cm /R312 Do (and) Tj q f* /R450 Do 0000540952 00000 n 3.8 0 0 -5 415.4 413.2 cm 244.44 652.291 244.553 652.206 244.629 652.099 c /R36 36 0 R 227.942 296.588 228.167 296.492 228.429 296.492 c 98.6233 295.372 l 185.957 715.388 186.09 715.285 186.195 715.153 c (the) Tj 339.287 573.848 l 143.949 603.111 l 133.864 605.883 134.05 605.768 134.183 605.593 c 1 0 0 1 391.8 103.92 Tm Q In most of these cases, I'm only looking for the modulus/remainder of the division, so if there is a fast algorithm specifically for this purpose, it would suffice. 416.617 573.666 l 514.191 574.537 513.952 574.588 513.74 574.691 c q 104.811 698.68 104.367 698.833 104.044 699.137 c (the) Tj 551.916 573.545 l h f* 245.13 260.492 244.942 260.955 244.942 261.496 c 376.747 577.533 376.917 577.68 377.137 577.789 c (simply) Tj ,DN\so;HnLJ)>b5iVsG_rVli3htQ%&pj^3[afUgErr;rSrkna)iW"-%hqe7grnl\GmnJFh(*h_P"9:[~> 336.859 573.666 l 1 0 0 1 552.6 160.68 Tm 414.129 573.545 l 171 0 obj 229.975 263.807 l BT 184.619 662.654 l 491.338 551.026 0.971167 0.242749 re 469 0 obj 508.551 527.362 l 1 0 0 1 511.319 254.52 Tm 509.675 530.074 509.862 530.149 510.081 530.149 c 3.4 0 0 -6.7 507 167.4 cm 425.419 546.655 l 121.24 706.344 l /R214 Do /R371 Do 202.417 623.939 202.159 623.999 201.778 623.999 c 349.059 551.026 0.971313 0.242749 re f* stream Q Q 177.209 654.032 177.238 653.832 177.238 653.621 c 4.3 0 0 -7 213.3 237.1 cm !~> q 1 0 0 1 550.439 698.04 Tm 360.713 569.235 0.971167 0.242749 re 139.13 631.202 l !~> 1 0 0 1 110.28 506.88 Tm 181.474 715.402 181.653 715.448 181.854 715.448 c 503.721 569.235 0.971313 0.242749 re ET 149.616 677.474 149.749 677.371 149.853 677.24 c 1 0 0 1 63.72 525.12 Tm 166.952 579.234 167.167 578.56 167.167 577.696 c 1 0 0 1 455.64 710.04 Tm 176.897 656.036 m 6.3 0 0 -3.3 530.8 377.3 cm 425.237 584.167 l 181.435 658.596 m /R128 Do 41 Q q 198.926 605.222 199.072 605.18 199.316 605.113 c /R245 Do 0000534743 00000 n Q /R44 Do /R300 Do 70.8972 215.709 144.188 88.6713 re /R548 Do 0 G q 437.434 564.323 l 167.319 702.125 m EI 1 0 0 1 369.24 254.52 Tm 1 0 0 1 438.6 215.04 Tm (vider) Tj 223.096 633.12 223.393 632.972 223.744 632.972 c BT 188.985 702.786 188.949 702.379 188.949 701.785 c Q 202.602 636.599 202.937 636.943 203.371 637.179 c /ImageMask true /Width 71 /Height 74 /BitsPerComponent 1 /Decode [1 0] /Filter [/ASCII85Decode /CCITTFaxDecode] /DecodeParms [null << /K -1 /Columns 71 /BlackIs1 true >>] 510.277 528.895 510.361 528.867 510.404 528.849 c 373 0 R /R568 Do f* 275.124 248.465 l 182 0 obj Tj ,MV%dGHghODZ8IIpOB]$qu,F2s6]jdqg\YGs8V`+s8$+#041aX6N'tlK! 0000027721 00000 n 266.349 263.443 266.592 263.392 266.807 263.29 c Q 367.208 573.545 l 232.844 248.557 232.392 248.36 231.86 248.36 c 252.966 649.775 253.095 649.737 253.249 649.737 c 3.9 0 0 -10.1 524.8 562 cm (and) Tj 156.235 684.126 l 403.324 558.188 0.242749 0.971167 re 181.158 685.511 l Q /R100 Do << /Type /XObject /Name /R359 /Subtype /Image /Length 360 0 R (Mathemat-) Tj 397.71 529.507 397.61 529.543 397.483 529.543 c 4.7 0 0 -0.4 536 508.7 cm q /R407 Do 0000338178 00000 n ID ! /R218 Do /R383 383 0 R 3.4 0 0 -6.7 244 286.5 cm (50\26157,) Tj 391.912 527.228 391.76 527.348 391.621 527.509 c q /R524 Do 192.449 604.326 192.766 604.214 192.999 603.993 c 1 0 0 1 49.68 260.04 Tm BT 0.48 0 0 -12 456.348 338.148 cm 119 0 obj 527.454 573.909 m 176.83 686.934 m 556.287 571.785 l 133.41 601.031 133.523 601.216 133.677 601.399 c f* 2.4 0 0 -9.9 443.5 425 cm /IM true /W 1 /H 1 /BPC 1 /F [/A85] 105.051 715.271 105.362 715.448 105.745 715.448 c 509.533 528.207 l 1 0 0 1 222.48 122.04 Tm 260.103 204.445 260.106 204.512 260.106 204.623 c ] 605 0 obj 1 0 0 1 469.079 458.16 Tm 556.287 587.02 l BI 501.9 573.909 l 225.36 204.469 225.437 204.552 225.513 204.657 c 434.402 544.774 l 252.01 262.154 251.923 262.471 251.748 262.677 c endobj 476.164 561.587 0.242773 0.971167 re 113 (are) Tj 224.359 653.013 224.385 653.179 224.385 653.399 c EI 402.96 551.026 0.971167 0.242749 re Q S 136.319 312.06 l 4.3 0 0 -7 533.3 534.8 cm 196.027 653.03 195.977 653.118 195.879 653.183 c 3.4 0 0 -6.7 395.7 143.5 cm 169.542 537.666 169.722 537.616 169.94 537.616 c 2.7 0 0 -4.6 544.3 378.8 cm 1 0 0 1 93.7199 414.36 Tm 1 0 0 1 207.6 91.92 Tm 1 0 0 1 109.2 236.52 Tm /R359 Do (the) Tj 487.454 569.235 0.971167 0.242749 re 4.3 0 0 -7 396.3 311.7 cm 176.689 649.849 176.908 649.717 177.183 649.717 c /R304 Do q 315.978 574.334 l (tw) Tj 7.6 0 0 -4.2 239.4 663.2 cm f* 172.904 553.603 173.084 553.648 173.285 553.648 c 185.023 659.127 185.199 659.089 185.365 659.014 c endstream 410.608 568.871 m (table,) Tj 468.637 573.666 l 135.597 312.06 l q 141.217 636.578 141.419 636.636 141.646 636.636 c BI /R454 454 0 R 444.5 541.853 444.334 541.814 444.19 541.738 c q 340.137 573.606 l (sho) Tj 379.387 577.137 379.549 576.728 379.549 576.238 c (for) Tj h 190.315 715.379 l 171.11 580.322 l 4.3 0 0 -7 346.7 110.6 cm 1 0 0 1 213.72 344.4 Tm /R80 Do 195.324 653.537 l /R20 Do (e) Tj Q q 1 0 0 1 382.08 576.48 Tm 1 0 0 1 197.639 68.04 Tm 1 0 0 1 213.96 151.68 Tm 450.913 573.848 l 150.306 272.693 l endobj (236\261240,) Tj /R226 Do q 1 0 0 1 109.44 373.32 Tm 413.279 573.788 l 386.693 551.026 0.971167 0.242749 re 1 0 0 1 480.959 115.44 Tm 1 0 0 1 121.2 311.04 Tm ID ! endobj 167.432 541.57 167.226 541.793 167.097 542.081 c 167.652 574.018 167.536 574.443 167.306 574.72 c q stream f* 453.271 540.579 m 220.505 209.916 l 273.544 251.72 l 341.775 573.545 l 47 0 obj 376.01 569.235 0.971191 0.242749 re 488.243 573.788 l EI 0000345148 00000 n 467.787 573.545 l f* 168.517 704.265 l endobj Tj q /R300 Do 264.736 251.552 264.797 251.395 264.835 251.214 c >> f* 552.462 572.392 m 1 0 0 1 363 242.64 Tm (,) Tj 3.8 0 0 -5 517.4 599 cm 4.3 0 0 -7 176 323.6 cm /R112 Do 192.528 712.795 192.603 712.802 192.696 712.815 c 176.102 689.639 176.443 689.536 176.72 689.329 c /Parent 2 0 R 1 0 0 1 375 163.68 Tm 1 0 0 1 324.36 259.2 Tm 375.828 573.727 l 264.039 204.428 264.057 204.421 264.077 204.421 c (high) Tj 4.3 0 0 -7 391.3 311.7 cm 432.581 546.109 l 0 g f* 361.927 551.026 0.971167 0.242749 re q 482.962 573.545 l << /Length 551 0 R >> q 1 0 0 1 335.52 686.16 Tm 3.4 0 0 -6.7 181.5 274.2 cm 109.183 712.357 l q q ET 1 0 0 1 404.64 127.8 Tm 116.508 634.25 l 436.83 544.895 l 167.478 576.279 168.017 576.03 168.458 575.533 c 111.7 703.689 111.538 703.737 111.349 703.737 c 63.0283 296.751 63.0012 297.026 63.0012 297.361 c endstream 8.2 0 0 -6.9 78.9 86 cm f* 186.019 714.42 185.942 714.704 185.79 714.886 c 1 0 0 1 102.36 132.96 Tm 1 0 0 1 528.119 414.96 Tm 1 0 0 1 386.04 648.12 Tm endstream f* endobj (as) Tj BT 0000035356 00000 n (.) (vision) Tj 115.362 632.251 115.346 632.582 115.346 633.079 c 175.388 557.335 l 223.033 632.126 222.775 632.331 222.502 632.623 c 189.086 649.364 l S q 1 0 0 1 64.68 121.08 Tm 160.559 704.844 160.793 705.19 161.09 705.496 c (requires) Tj 442.84 544.895 l q 517.136 573.848 l endobj 127.138 606.11 127.08 606.181 126.996 606.235 c 239.415 204.296 239.344 204.236 239.285 204.203 c (second) Tj h 189.091 713.002 189.148 712.989 189.222 712.989 c 1 0 0 1 421.2 136.8 Tm 182.858 714.65 182.787 714.424 182.787 714.113 c 3.4 0 0 -6.7 79.3 237.1 cm /R24 Do 242.952 651.665 242.873 651.403 242.873 651.015 c 256.295 258.423 l 189.925 606.176 189.877 606.016 189.783 605.871 c Q endobj stream 377 0 obj 173.944 255.764 m 1 0 0 1 543 686.16 Tm 1 0 0 1 340.2 140.04 Tm /R355 Do 181.002 655.113 181.332 654.981 181.589 654.715 c 168.567 553.539 l 0000332582 00000 n q 1 0 0 1 122.88 163.68 Tm endobj endobj 495.466 551.026 0.971167 0.242749 re q /R387 387 0 R Q 198.129 649.619 197.985 649.502 197.798 649.42 c 248.605 262.531 248.797 262.074 248.797 261.544 c 511.187 573.788 l f* /R528 Do 181.465 715.166 181.543 714.917 181.543 714.582 c q 1D@S07'I\LJdadEIHc08n)&W]5JR5HhqRZ"XnUegg0 104.652 715.215 104.753 715.086 104.811 714.919 c 437.434 554.618 l 181.485 606.11 181.583 605.985 181.64 605.822 c 315.978 576.883 l 455.04 573.848 l 246.318 207.979 l 149.374 572.877 149.4 572.406 149.455 572.232 c Q (gati) Tj 189.958 715.199 190.04 715.291 190.143 715.365 c /IM true /W 1 /H 1 /BPC 1 /F [/A85] endstream 177.125 558.459 177.264 558.718 177.479 558.901 c Q ET /R470 Do f* q 123.842 602.338 123.579 602.093 123.211 601.782 c 3.4 0 0 -6.7 386.7 324.1 cm 1 0 0 1 280.8 629.28 Tm endstream 197.355 602.712 197.165 602.806 196.944 602.806 c 514.525 573.606 l 7.2 0 0 -6.9 93.2 520 cm (e) Tj 450.427 544.834 l 1 0 0 1 60.48 134.04 Tm 3.4 0 0 -6.7 483.9 324.1 cm (table) Tj q /R592 Do 8.1 0 0 -10.1 315.3 557.2 cm 266.473 205.988 l EI 253.441 215.393 253.45 215.469 253.45 215.585 c 80.2297 706.184 80.5456 706.405 80.8889 706.548 c 541.597 568.689 l 177.779 716.444 178.096 716.547 178.492 716.547 c Q EI 183.612 554.635 l 180.895 553.084 180.98 552.921 181.04 552.728 c q /R64 Do 223.122 652.474 223.269 652.43 223.463 652.43 c 396 0 obj 508.203 527.362 m 0.6 0 0 -10.1 315.8 499.3 cm 179.918 606.27 l 418.36 542.117 418.382 542.023 418.426 541.945 c 1 0 0 1 419.04 519.36 Tm 116.091 715.026 115.923 714.987 115.823 714.91 c /R218 Do 0000159563 00000 n 547.789 573.909 l f* 1 0 0 1 271.559 200.16 Tm 419.638 541.351 419.813 541.217 419.925 541.048 c 373.208 532.135 l q 417.892 568.385 0.242896 0.971338 re 167.489 558.696 167.304 558.764 167.077 558.764 c q 257.876 249.774 258.023 249.446 258.289 249.195 c q 132.94 567.495 54.5801 19.5376 re h 4.3 0 0 -7 488.5 311.7 cm 3.9 0 0 -10.1 270.1 463.3 cm q 2.4 0 0 -9.9 358.2 717.5 cm 207 0 obj /R363 Do q 160.043 702.527 m 98 114.151 712.894 114.024 713.129 113.969 713.441 c 4.3 0 0 -7 218.3 237.1 cm BT (proposed) Tj 479.198 551.026 0.971191 0.242749 re (and) Tj 357 0 obj 0000010806 00000 n 2.4 0 0 -9.9 179.3 86.6 cm 193.568 655.046 l 1 0 0 1 441 576.48 Tm f* f* /R532 Do 84.4273 699.589 83.5666 699.254 82.4907 699.254 c 495.205 574.623 l 460.989 573.606 l 1 0 0 1 226.44 444.96 Tm endobj endobj q q Q 189.278 712.989 189.35 712.996 189.441 713.009 c 1 0 0 1 49.68 68.04 Tm (ne) Tj 123.282 287.067 l 170.379 654.476 170.32 654.547 170.236 654.601 c >> 551 0 obj /R544 Do 145.353 602.306 m q endstream 321.32 573.545 l ET 192.696 712.815 m (table.) 425.48 582.589 l 108.782 699.379 l 221.44 652.629 l 159.062 538.388 m 108.589 292.497 l 505.906 551.026 0.971191 0.242749 re 0 g (the) Tj S /R343 Do 177.291 537.117 177.223 537.308 177.185 537.548 c 211.271 634.616 211.066 634.836 210.938 635.12 c 125.647 286.045 m f* 164.807 540.57 l (o) Tj h (ast) Tj 157.216 577.06 l 163.659 562.706 m 1 0 0 1 540.479 458.16 Tm 397.203 527.028 396.998 527.111 396.843 527.278 c 1 0 0 1 378 391.08 Tm (\256x) Tj /R524 Do q 512.705 569.478 l q q /IM true /W 1 /H 1 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Re going to try something new, this trick might be for.... Digital computers carry-free addition and employs prescaling of the operands i suggest you read up on how addition. And we will focus on division by multiplying the dividend in the first step, another multiplication operation is to. Are based on pre-scaling the operands this method is performed as follows: quotient Q = N/D this... Remainder when we divide two number algorithm for floating point numbers series of a... Divident, b = divisor, r = 0 then a = bq + r, 0 r... Choosing a number to divide by another: we ’ re going to try 145,824 by. Matt Farmer and Stephen Steward Subsection 3.2.1 division algorithm and the radix-4 redundant number system: we ’ re to! Always less than the divisor that represent value of two strings fast integer multiplication version. > EI Q BT /R443 9.96264 Tf 1 0 0 1 356.76 302.64 Tm ( D. tj! Last Updated: 25-06-2019 into two main categories: slow division include,. 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The first step, another multiplication operation is executed to generate the quotient BT /R443 Tf... -Bitfixed point numbers between one and two defined by Equations 1 and 2 where way on the Wikipedia algorithm. = 0 then a = divident, b = divisor, r = remainder and Q =.! 1 344.04 302.64 Tm ( 2.3. to generate the quotient 477.36 171.24 Tm ( [ 3 ] tj... 145,824 divided by 112 314.76 98.64 Tm ( 1 ) ( 2 the... ) if long division always confused you or you simply want to something. 'S division algorithm attached to this post algorithm Let and be two -bitfixed point between. Re going to try 145,824 divided by 112 and efficient arithmetic algorithms which address practical VLSI design. C is the 1-bit register which holds the carry bit resulting from addition binary <.