四則演算を理解する/four-arithmetic-operations

以下のイベントで発表した資料です！
https://ariaki.connpass.com/event/245907/

Transcript

1. ࢛ଇԋࢉΛཧղ͢Δ 2022/5/17 Ԭຊᕣฏ Twitter:@okaponta_
2. ࣗݾ঺հɿԬຊ ᕣฏ • େֶ͸Խֶͷݚڀ • ৽ଔ͸ۚ༥ܥͷSIer (γϯϓϨΫε) • 2021/7ʙ ϦΫϧʔτͰΤϯδχΞͯ͠·͢ • झຯɿԻήʔɺڝϓϩɺΫιΞϓϦ࡞੒ • Twitter: @okaponta_ • GitHub: @okaponta
3. ͬͦ͘͞Ͱ͕͢
4. ྼ౳ײͰམͪࠐΜͩ͜ͱ ͋Γ·ͤΜ͔ʁ
5. جૅΛษڧ͠Α͏ͱࢥ͍
6. Ҏલɺ࢛ଇԋࢉʹ͍ͭͯௐ΂·ͨ͠
7. ͷલʹɾɾɾ
8. 2ਐ๏ͱ࿦ཧԋࢉʹ͍ͭͯ(͔͚͋͠) • σδλϧίϯϐϡʔλͰ͸ిѹͷߴ௿ʹΑΓ৘ใΛදݱ͍ͯ͠Δ • ෺ཧతʹݕग़͠΍͍͔͢Β(ిѹͷࡉ͔͍ௐ੔͕೉͍͠) • ిѹͷߴ͍ঢ়ଶΛʮ1ʯɺ௿͍ঢ়ଶΛʮ0ʯͱͯ͠2ਐ๏Ͱදݱ • ࠓ͸τϥϯδελ্͕هͷεΠονϯάͷ໾ׂΛ୲͍ͬͯΔ
9. 2ਐ๏ͱ࿦ཧԋࢉʹ͍ͭͯ(͔͚͋͠) https://akizukidenshi.com/catalog/g/gI-13829/ https://sagara-works.jp/research-and-development/electronics/transistor-basic/transistor-switch/
10. 2ਐ๏ͱ࿦ཧԋࢉʹ͍ͭͯ(͔͚͋͠) • τϥϯδελΛ૊Έ߹ΘͤΔ͜ͱͰɾɾɾ • ిѹͷߴ௿Λٯʹ͢Δճ࿏Λ࡞੒Ͱ͖Δ • NOTԋࢉ • 2ͭͷిѹΛΠϯϓοτ͔ΒɺిѹΛग़ྗ͢Δճ࿏Λ࡞੒Ͱ͖Δ • NAND/OR/AND/XORԋࢉ
11. ORճ࿏ https://sagara-works.jp/research-and-development/logic
12. Ճ๏ • ͖͞΄Ͳͷ࿦ཧԋࢉͷ૊Έ߹ΘͤͰදݱͰ͖Δɻ • (1ͷҐ) A XOR B • (2ͷҐ) A AND B • ൒Ճࢉث͍ͬͯ͏Β͍͠ɻ
13. Ճ๏ • ͖͞΄Ͳͷ࿦ཧԋࢉͷ૊Έ߹ΘͤͰදݱͰ͖Δɻ • (1ͷҐ) A XOR B • (2ͷҐ) A AND B • ൒Ճࢉث͍ͬͯ͏Β͍͠ɻ • https://nandgame.com/
14. Ճ๏ • ܁Γ্͕Γʹ΋ରԠͨ͠΋ͷΛશՃࢉثͱ͍͏ • (1ͷҐ) (A XOR B) XOR C • (2ͷҐ) (A AND B) OR ((A XOR B) AND C)
15. Ճ๏ • ܁Γ্͕Γʹ΋ରԠͨ͠΋ͷΛશՃࢉثͱ͍͏ • (1ͷҐ) (A XOR B) XOR C • (2ͷҐ) (A AND B) OR ((A XOR B) AND C) • ͪͳΈʹɺNANDͷΈͰશͯͷ࿦ཧճ࿏Λද ݱͰ͖ΔΒ͍͠Ͱ͢ɻ
16. Ճ๏ • ܁Γ্͕Γʹ΋ରԠͨ͠΋ͷΛશՃࢉثͱ͍͏ • (1ͷҐ) (A XOR B) XOR C • (2ͷҐ) (A AND B) OR ((A XOR B) AND C) • ͪͳΈʹɺNANDͷΈͰશͯͷ࿦ཧճ࿏Λද ݱͰ͖ΔΒ͍͠Ͱ͢ɻ
17. ݮ๏ • ଍͠ࢉͱಉ͡Ͱ͢ʂ • ಺෦తʹෛͷ਺Λิ਺ͱ͍͏ঢ়ଶͰ؅ཧ͢Δ
18. ݮ๏ • ଍͠ࢉͱಉ͡Ͱ͢ʂ • ಺෦తʹෛͷ਺Λิ਺ͱ͍͏ঢ়ଶͰ؅ཧ͢Δ • ྫ) 5-3 • 00000101 + 11111101 • 1)00000010
19. ݮ๏ • ଍͠ࢉͱಉ͡Ͱ͢ʂ • ಺෦తʹෛͷ਺Λิ਺ͱ͍͏ঢ়ଶͰ؅ཧ͢Δ • ྫ) 5-3 • 00000101 + 11111101 • 1)00000010 ิ਺ Φʔόʔϑϩʔ͢Δ
20. ৐๏ • γϑτԋࢉͱ଍͠ࢉΛ૊Έ߹Θͤͯ଍͠ࢉΛߦ͍·͢ɻʢΒ͍͠ʣ • චࢉͱಉ͡Ͱ͢ɻ
21. ྫ) 5 × 13 0110×1101
22. ྫ) 5 × 13 0110×1101 0000110 1
23. ྫ) 5 × 13 0110×1101 0000110 1 0000000 0
24. ྫ) 5 × 13 0110×1101 0000110 1 0000000 0 0011000 1
25. ྫ) 5 × 13 0110×1101 0000110 1 0000000 0 0011000 1 0110000 1
26. ྫ) 5 × 13 0110×1101 0000110 1 0000000 0 0011000 1 0110000 1 1001110 (65)
27. γϑτԋࢉͯ͠Δ࣮ͬͯײ͍ͨ͠ package mai n func mul(x, y int) int { return x * y } func main() { mul(5, 13 ) } > go tool compile -S mul.g o 0x0000 00000 (mul.go:4) MOVD "".x(FP), R 0 0x0004 00004 (mul.go:4) MOVD "".y+8(FP), R 1 0x0008 00008 (mul.go:4) MUL R1, R0, R 0 0x000c 00012 (mul.go:4) MOVD R0, "".~r2+16(FP ) 0x0010 00016 (mul.go:4) RET (R30) ΞηϯϒϥϨϕϧͰ
    .6-໋ྩ͕͋Δ
28. γϑτԋࢉͯ͠Δ࣮ͬͯײ͍ͨ͠(ͺʔͱ2 func Benchmark_mul_2exp(b *testing.B) { var base int64 = get2exp() // 2^30 or 2^3 1 var hoge int64 = get2exp() - 1 b.ResetTimer( ) for i := 0; i < b.N; i++ { for j := 0; j < 10000; j++ { hoge = mul64(hoge, base ) } } if hoge == 1 { b.Fail( ) } } ͜͜ΛCBTFIPHFΛ͍Ζ͍Ζ ม࣮͑ͯݧͯ͠ΈΔ
29. γϑτԋࢉͯ͠Δ࣮ͬͯײ͍ͨ͠(ͺʔͱ2 > go test -bench . -benchme m goos: darwi n goarch: arm6 4 pkg: mu l Benchmark_mul_2exp-8 123480 9659 ns/op 0 B/op 0 allocs/o p Benchmark_mul_2exp_minus1-8 123620 9651 ns/op 0 B/op 0 allocs/o p Benchmark_mul_2exp_plus1-8 123568 9647 ns/op 0 B/op 0 allocs/o p Benchmark_plus_2exp-8 365474 3226 ns/op 0 B/op 0 allocs/o p PAS S ok mul 5.341s
30. ৐๏ • (ࠩΛग़ͦ͏ͱΊͬͪΌؤு͚ͬͨͲ)ͼͬ͘Γ͢Δ΄ͲมԽ͕ͳ͔ͬͨ • 2ͷྦྷ৐ϚΠφε1͕ۃ୺ʹ஗͘ͳΔͷΛظ଴͕ͨ͠ɺಉ͡ΫϩοΫ਺Ͱ ܭࢉͰ͖ΔΑ͏ʹͳͬͯΔΈ͍ͨɻ • ଍͠ࢉͷ3ഒͷ࣌ؒͰֻ͚ࢉ͸ܭࢉͰ͖Δ • ࣗ෼ͷCPU3.2GHzΈ͍ͨͳͷͰɺ1ΫϩοΫ໿0.31nsͰɺॲཧ͕ͦͷ ഒ਺ʹͳͬͯΔͷ͸໘ന͍ɻ3ΫϩοΫͰֻ͚ࢉ͕ܭࢉͰ͖ͯ·͢Ͷɻ
31. আ๏ • ஗͍ͱઐΒධ൑ • SRT๏ • ϧοΫΞοϓςʔϒϧΛ༻͍ͯෳ਺ͷܻΛ·ͱΊׂͯΓࢉ • 2ܻͣͭॲཧ͢Δͷ͕Radix-4 Divider • 4ܻͣͭॲཧ͢Δͷ͕Radix-16 Divider • ଞʹ΋χϡʔτϯ๏Λ࢖͏ํ๏΍ɺΰʔϧυγϡϛοτ๏ͳΔ΋ͷ͕͋ΔΒ ͍͠
32. আࢉ func Benchmark_div_2exp(b *testing.B) { var base int = getSmall2exp() // 2^5 or 2^ 6 var hoge int = get2exp() - 1 b.ResetTimer( ) for i := 0; i < b.N; i++ { for j := 0; j < 10000; j++ { hoge = div(hoge, base ) } } if hoge == 1 { b.Fail( ) } } ͜͜ΛมԽͤͯ͞ΈΔ
33. আࢉ > go test -bench . -benchmem goos: darwi n goarch: arm6 4 pkg: di v Benchmark_div_2exp-8 51686 22046 ns/op 0 B/op 0 allocs/o p Benchmark_div_2exp_minus1-8 53272 21935 ns/op 0 B/op 0 allocs/o p Benchmark_div_2exp_plus1-8 54561 21931 ns/op 0 B/op 0 allocs/o p Benchmark_plus_2exp-8 372552 3140 ns/op 0 B/op 0 allocs/o p PAS S ok div 5.487s
34. আࢉʢఆ਺ʣ func div2(x uint) uint { return x / 2 } func div3(x uint16) uint16 { return x / 3 } > go tool compile -S ./div-const.g o 0x0000 00000 (./div-const.go:4) MOVD "".x(FP), R 0 0x0004 00004 (./div-const.go:4) LSR $1, R0, R 0 0x0008 00008 (./div-const.go:4) MOVD R0, "".~r1+8(FP ) 0x000c 00012 (./div-const.go:4) RET (R30)
35. আࢉʢఆ਺ʣ func div2(x uint) uint { return x / 2 } func div3(x uint16) uint16 { return x / 3 } > go tool compile -S ./div-const.g o 0x0000 00000 (./div-const.go:4) MOVD "".x(FP), R 0 0x0004 00004 (./div-const.go:4) LSR $1, R0, R 0 0x0008 00008 (./div-const.go:4) MOVD R0, "".~r1+8(FP ) 0x000c 00012 (./div-const.go:4) RET (R30) ࠨγϑτԋࢉ
36. আࢉʢఆ਺ʣ func div2(x uint) uint { return x / 2 } func div3(x uint16) uint16 { return x / 3 } > go tool compile -S ./div-const.g o 0x0000 00000 (./div-const.go:8) MOVHU "".x(FP), R 0 0x0004 00004 (./div-const.go:8) MOVD $87382, R 1 0x000c 00012 (./div-const.go:8) MUL R0, R1, R 0 0x0010 00016 (./div-const.go:8) LSR $18, R0, R 0 0x0014 00020 (./div-const.go:8) MOVH R0, "".~r1+8(FP ) 0x0018 00024 (./div-const.go:8) RET (R30) ͱͷ ֻ͚ࢉʹม׵
37. আࢉʢఆ਺ʣ func div2(x uint) uint { return x / 2 } func div3(x uint16) uint16 { return x / 3 } > go tool compile -S ./div-const.g o 0x0000 00000 (./div-const.go:8) MOVHU "".x(FP), R 0 0x0004 00004 (./div-const.go:8) MOVD $87382, R 1 0x000c 00012 (./div-const.go:8) MUL R0, R1, R 0 0x0010 00016 (./div-const.go:8) LSR $18, R0, R 0 0x0014 00020 (./div-const.go:8) MOVH R0, "".~r1+8(FP ) 0x0018 00024 (./div-const.go:8) RET (R30) x 3 = x × 1 3 = x × 218 3 × 1 218 ≃ x × 218 + 2 3 × 1 218 
38. আࢉ • ༧Ίγϑτԋࢉͱೝࣝ͠ͳ͍ݶΓ͸ී௨ʹׂΓࢉ͢Δ • ࢼͨ͠ྫͩͱ͍͍ͩͨ7ΫϩοΫ͘Β͍ • தͰͲΜͳॲཧ͕ߦΘΕ͍ͯΔͷ͔͸෼͔Βͳ͔͕ͬͨɺ64bitͷ ׂΓࢉʹͯ͠΋7ΫϩοΫͰॲཧΛߦ͍͑ͯͨɻ • ఆ਺ͷ৔߹͸ίϯύΠϧ࣌ʹ͍͍ײ͡ʹม׵ͯ͘͠ΕΔ
39. ײ૝ • جૅͱࢥ࢛ͬͯଇԋࢉΛௐ΂ͯΈ͚ͨͲԞ͕ਂ͗ͨ͢ɾɾɾ • ݁ہ࢛ଇԋࢉ΋׬શʹཧղͰ͖ͳ͍ • ઌਓɺҒେ͗ͯ͢ײ৺ͪ͠Ό͏ • ͸΍ͪ͘Ͱ͝ΊΜͳ͍͞ʂʂʂ