Reddit reviews Computer Approximations

The best book on all things floating-point I think is Muller et al., Handbook of Floating-Point Arithmetic (2010). It's a whopping 580 pages of real talk. Everything you need to know about (short of implementing it yourself) is examined: Floating-point formats, history, correct rounding, IEEE standard compatibility (both binary and decimal floats), radix conversion, and the basic functions addition, subtraction, multiplication, division, fused-multiply-add, and square roots. A whole chapter (130 pages) discusses the implementation of each of these functions. Transcendental functions are mentioned, but better looked up elsewhere.

Some real-world implementations with source code:

• Source code from Microchip for PIC microcontrollers: Application Note AN575 - IEEE 754 Compliant Floating Point Routines, very small and well documented assembly
  
• Portable C implementation: Berkeley SoftFloat, but no comments
  
• GNU soft-float library as used by gcc: https://github.com/gcc-mirror/gcc/tree/master/libgcc/soft-fp, via macro hell. May be hard to follow, since there are no comments because Real Programmers use the code as documentation /s
  
• The GNU Multiple Precision Arithmetic Library (libgmp), contains arbitrary precision integers, rational and floating-point routines.
  
• GNU MPFR, much more complete arbitrary precision floating-point library building on libgmp. Contains also transcendental function approximations.
  
  It may be hard to get your head around the latter libraries, especially since they handle arbitrary precision
  
  
  If you want a complete libmath, you will also need to implement elementary functions: exponentials, logarithms, trigonometric functions, etc.:
  
  
• Cecil Hastings, Approximations for Digital Computers (1955), lots of good info, the meat is in the approximations of elementary functions like logarithms and trigonometric functions. Bit dated, but fun to read.
  
• Hart et al., Computer approximations (1968), great all-rounder, like Hastings + double good. Builds a foundation of intuition on how a function works, its error analysis and contains large library of ready-to-use approximations from square roots to the error function. Methodical but dry.
  
• Jean-Michel Muller, Elementary Functions, builds on the shoulders of the giants before and introduces a lot of methods to derive approximations yourself. Also a great introduction to CORDIC, a way to compute e.g. sines using only shifts and additions.

> how trig functions in a calculator actually work [without trigonometry]

This is not really sensible. What people mean by “trigonometry” is “the study of the circular (‘trigonometric’) functions”. Anything that calculators do to compute trigonometric functions is (by definition) trigonometry.

The word trigonometry is somewhat of a misnomer. There is a great deal of interesting geometry (including metrical geometry) of triangles which we do not include under “trigonometry”, and much of what we consider “trigonometry” doesn’t really have to do with triangles per se.

As for how calculators compute sines and cosines: there are several possible methods depending on what kind of hardware is available. One method that many handheld calculators use and which used to be more common on computers than now is the CORDIC algorithm

In practice most of the time trigonometric functions these days are implemented using some domain reduction followed by a polynomial approximation.

If you are curious you could try Hart’s (1978) book Computer Approximations, https://amzn.com/0882756427/