## +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

and 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 ...

assignment 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 and 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 48 8.1 Characters 49 8.2 String Basics 51 8.3 Unicode and UTF-8 1495 88.6 Numbered prompt 1496 88.7 TerminalMenus 1497 88.8 References 1500 89 Random Numbers 1507 89.1 Random numbers module 1508 89.2 Random generation functions 1508 89.3 Subsequences, permutations and

## +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

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 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 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

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 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

Conventions 11 4 Integers and Floating-Point Numbers 13 4.1 Integers 14 Overflow behavior 16 Division errors 17 4.2 Floating-Point Numbers 17 Floating-point zero 18 Special floating-point values 19 Machine and hyperbolic functions 36 Special functions 36 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 operators 638 47 Numbers 639 47.1 Standard Numeric Types 639 Abstract number types 639 Concrete number types 640 47.2 Data Formats 643 47.3 General Number Functions and Constants 648 Integers 656 47.4 BigFloats

16 Integers and Floating-Point Numbers 41 16.1 Integers 42 Overflow behavior 44 Division errors 45 16.2 Floating-Point Numbers 45 Floating-point zero 46 Special floating-point values and hyperbolic functions 64 Special functions 64 18 Complex and Rational Numbers 65 18.1 Complex Numbers 65 18.2 Rational Numbers 68 19 Strings 71 19.1 Characters 72 19.2 String Basics 73 19.3 Unicode and 19Fix deprecation warnings 409 48.20Tweaks 410 48.21Performance Annotations 410 48.22Treat Subnormal Numbers as Zeros 413 48.23@code_warntype 414 48.24Performance of captured variable 415 49 Checking for equality