## +24 ## A New Dragon in the Den: Fast Conversion From Floating-Point Numbers ## CASSIO NERI ## 2024 September 15 - 20 WARNING: This PDF poorly reproduces the slides shown at the talk. You are strongly beast. It's not rocket science. ## A New Dragon in the Den ## Fast conversion from floating-point numbers ## Cassio Neri (Independent Researcher) CppCon 2024 - Aurora # Statistical mechanics of Decimal fixed-point representation ## 1234 ## Decimal floating-point representation ## exponent ### 10 ¹ x 2.34 mantissa ## Decimal floating-point representation exponent ### 10 ¹ x 2.34 mantissa 0

## +24 ## Cross-Platform Floating-Point Determinism Out of the Box ## SHERRY IGNATCHENKO ET AL ## Authors  ## All the 0c9de70c3d925/p2_4.jpg) Guy Davidson gd@6it.dev Advising on C++ standard with regards to floating-point, idea about intrinsics, reviews  Mykhailo Borovyk mbo@6it.dev Implementing fixed-point math with floating-point fallback  Vladyslav

assignment versus mutation 4.3 Stylistic Conventions 5 Integers and Floating-Point Numbers 5.1 Integers 5.2 Floating-Point Numbers 5.3 Arbitrary Precision Arithmetic 5.4 Numeric Literal Coefficients Precedence and Associativity 6.8 Numerical Conversions 7 Complex and Rational Numbers 7.1 Complex Numbers 7.2 Rational Numbers 8 Strings 8.1 Characters 8.2 String Basics 8.3 Unicode and UTF-8 53 467 35.24 Tweaks ..... 467 35.25 Performance Annotations ..... 468 35.26 Treat Subnormal Numbers as Zeros ..... 471 35.27 @code_warntype ..... 472 35.28 Performance of captured variable ...

both be mapped to 0 in the quantized domain. Keeping all that in mind, it is easy to see that floating-point $ x_{min} $ should map to 0, and $ x_{max} $ should map to $ 2^{b}-1 $ . How do we map the precision domain. Visually inspecting figure 2-4, can you work out the formula for mapping a given floating-point value (x) to a quantized value ( $ x_{q} $ ). Assume that you are given values of $ x_{min} [Image](/uploads/documents/5/f/8/a/5f8af6d4905642e38ad2d0bbe54cd375/p6_1.jpg) Figure 2-4: Quantizing floating-point continuous values to discrete unsigned values. The continuous values range from $ x_{min} $

Conventions 9 4 Integers and Floating-Point Numbers 11 4.1 Integers 12 Overflow behavior 14 Division errors 15 4.2 Floating-Point Numbers 15 Floating-point zero 16 Special floating-point values 17 Machine epsilon and hyperbolic functions 35 Special functions 35 6 Complex and Rational Numbers 37 6.1 Complex Numbers 37 6.2 Rational Numbers 40 7 Strings 43 7.1 Characters 44 7.2 String Basics 45 7.3 Unicode and UTF-8 Configuration 1167 80.6 References 1169 REPL 1169 TerminalMenus 1169 81 Random Numbers 1175 81.1 Random numbers module 1175 81.2 Random generation functions 1175 81.3 Subsequences, permutations and

Conventions 9 4 Integers and Floating-Point Numbers 11 4.1 Integers 12 Overflow behavior 14 Division errors 15 4.2 Floating-Point Numbers 15 Floating-point zero 16 Special floating-point values 17 Machine epsilon and hyperbolic functions 35 Special functions 35 6 Complex and Rational Numbers 37 6.1 Complex Numbers 37 6.2 Rational Numbers 40 7 Strings 43 7.1 Characters 44 7.2 String Basics 45 7.3 Unicode and UTF-8 CONTENTS Throwing Julia Exceptions . . . . . 34.22 Performance Annotations 384 34.23 Treat Subnormal Numbers as Zeros 387 34.24 @code_warntype 388 34.25 Performance of captured variable 390 34.26 Checking for

Allowed Variable Names 7 3.2 Stylistic Conventions 8 4 Integers and Floating-Point Numbers 9 4.1 Integers 10 4.2 Floating-Point Numbers 13 4.3 Arbitrary Precision Arithmetic 18 4.4 Numeric Literal Coefficients and Associativity 28 5.8 Numerical Conversions 30 6 Complex and Rational Numbers 33 6.1 Complex Numbers 33 6.2 Rational Numbers 36 7 Strings 39 7.1 Characters 40 7.2 String Basics 41 7.3 Unicode and UTF-8 deprecation warnings 378 34.22 Tweaks 378 34.23 Performance Annotations 379 34.24 Treat Subnormal Numbers as Zeros 381 34.25 @code_warntype 383 34.26 Performance of captured variable 384 CONTENTS 35 Workflow

assignment versus mutation 16 4.3 Stylistic Conventions 17 5 Integers and Floating-Point Numbers 18 5.1 Integers 19 5.2 Floating-Point Numbers 23 5.3 Arbitrary Precision Arithmetic 28 5.4 Numeric Literal Coefficients and Associativity 40 6.8 Numerical Conversions 41 7 Complex and Rational Numbers 45 7.1 Complex Numbers 45 7.2 Rational Numbers 48 8 Strings 52 8.1 Characters 53 CONTENTS 8.2 String Basics 55 8.3 Unicode 1717 90.6 Numbered prompt 1718 90.7 TerminalMenus 1719 90.8 References 1722 91 Random Numbers 1729 91.1 Random numbers module 1730 91.2 Random generation functions 1730 CONTENTS 91.3 Subsequences, permutations

td>RVPNErd,rs1,rs2Two Optional Floating-Point Instruction Extensions: RVF & RVDCategoryNameFmtRV32(F/D) body> RISC-V calling convention and five optional extensions: 8 RV32M; 11 RV32A; 34 floating-point instructions each for 32- and 64-bit data (RV32F, RV32D); and 53 RV32V. Using regex notation D. RV32 $ \{F|D\} $ adds registers f0-f31, whose width matches the widest precision, and a floating-point control and status register fcsr. RV32V adds vector registers v0-v31, vector predicate registers

versus mutation 12 4.3 Stylistic Conventions 14 ## 5 Integers and Floating-Point Numbers 15 5.1 Integers 16 5.2 Floating-Point Numbers 20 5.3 Arbitrary Precision Arithmetic 25 5.4 Numeric Literal Coefficients Associativity 37 6.8 Numerical Conversions 38 ## 7 Complex and Rational Numbers 42 7.1 Complex Numbers 42 7.2 Rational Numbers 45 ## 8 Strings 49 8.1 Characters 50 8.2 String Basics 52 ## CONTENTS deprecation warnings 466 35.24 Tweaks 466 35.25 Performance Annotations 467 35.26 Treat Subnormal Numbers as Zeros 470 35.27 @code_warntype 471 35.28 Performance of captured variable 473 35.29 Multithreading