Client Server int int ADD DIV int double end Branching 31Session? • Can we describe a valid sequence of interactions? • Or formula ?int; ?int; &(ADD: !int, DIV: !double); end 32Session? • describe a valid sequence of interactions? • Or formula ?int; ?int; &(ADD: !int, DIV: !double); end !int; !int; ⊕(ADD: ?int, DIV: ?double); end dual formula 33Session types • Formal description of “message” operation is allowed!) • Then, again receive int • ... ?int; ?int; &(ADD: !int, DIV: !double); end Client Server int int ADD DIV int double end *Name “Behavioral types” is also used. See “Behavioral

度的特点： div v = 0 • 上式对时间求导，即 d(div v)/dt = div dv/dt = 0 ；带入 dv/dt = -p 得 div grad p = 0 。 • 因此为了模拟不可压缩流我们要求保证 p 满足 div grad p = 0 ？ • 不妨假设现在 div v ≠ 0 ，然后想办法如何通过修正压强来消除他，即让 div grad p = -div v 。

bytes Mathematical operations ADD : add or subtract INC : Increment DEC : Decrement MUL : Multiply DIV : Divide Stack operations PUSH: push on stack POP : pop from stackOperators (part 2) Logical Operations reg, constantWatch out for math instructions MUL and DIV! MUL reg DIV reg MUL address DIV address MUL constant DIV constant Target of MUL and DIV is always register A! Why?Types of operands for MOV

Sub)* Add < "+" Factor Sub < "-" Factor Factor < Primary (Mul / Div)* Mul < "*" Primary Div < "/" Primary Primary < Parens / Neg / Pos / Number / Variable Parens

principles > ldea inspired by std.range.evenChunks() from the Dlanguage > Expensive computations (div mod)] are computed once per iterator > Branch-less programming for crossing border between smaller

dathskovsky@speedata.io | www.linkedin.com/in/alexdathskovsky •ADD – 1 cycle (not piped) •IMUL – 3 cycles •DIV – at least x20 time slower than imul (depending on the architecture)WHAT IS LATENCY: PIPES Alex

𝒕𝒊𝒎𝒆𝑺𝑰𝑴𝑫/𝒕𝒊𝒎𝒆𝒔𝒄𝒂𝒍𝒂𝒓 Zen II Skylake Broadwell Add 1,00 1,23 1,00 Mul 1,00 1,17 1,65 Div 1,00 1,98 4,01Performance Results: CODA test case Wallclock time speedup, lower is better, <1 is an

inline-bool-flag /fsanitize-coverage=edge /fsantize- coverage=trace-cmp /fsanitize-coverage=grace-div Link the libraries: • clang_rt.fuzzer_no_main-x86_64.lib • libsancov.libImportance of a security-focused

noexcept; template constexpr T mul_sat(T x, T y) noexcept; template constexpr T div_sat(T x, T y) noexcept; template constexpr T saturate_cast(U x) noexcept; 1

--use_fast_math 选项，那么所有对 sinf 的调用都会自动被替换成 __sinf 。 • --ftz=true 会把极小数 (denormal) 退化为 0 。 • --prec-div=false 降低除法的精度换取速度。 • --prec-sqrt=false 降低开方的精度换取速度。 • --fmad 因为非常重要，所以默认就是开启的， 会自动把 a * b + c 优化成乘加