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1 # Q: Fixed Point Number Library
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2
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3 | Project | Q: Fixed Point Number Library |
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4 | --------- | --------------------------------- |
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5 | Author | Richard James Howe |
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6 | Copyright | 2018 Richard James Howe |
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7 | License | MIT |
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8 | Email | howe.r.j.89@gmail.com |
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9 | Website | <https://github.com/howerj/q> |
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10
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11 This is a small fixed point number library designed for embedded use. It
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12 implements all of your favorite transcendental functions, plus only the best
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13 basic operators, selected for your calculating pleasure. The format is a signed
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14 [Q16.16][], which is good enough for [Doom][] and good enough for you.
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15
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16 The default [makefile][] target builds the library and a test program, which will
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17 require a [C][] compiler (and Make). The library is small enough that is can be
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18 easily modified to suite your purpose. The dependencies are minimal; they are few
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19 string handling functions, and '[tolower][]' for numeric input. This should allow
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20 the code to ported to the platform of your choice. The 'run' make target builds
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21 the test program (called 'q') and runs it on some input. The '-h' option will
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22 spit out a more detailed help.
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23
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24 I would compile the library with the '-fwrapv' option enabled, you might
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25 be some kind of Maverick who doesn't play by no rules however.
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26
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27 The trigonometric functions, and some others, are implemented internally with
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28 [CORDIC][].
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29
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30 Of note, all operators are bounded by minimum and maximum values which are not
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31 shown in the following table and by default all arithmetic is saturating. The
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32 effective input range of a number might lower than what is possible given a
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33 mathematical functions definition - either because of the limited range of the
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34 [Q16.16][] type, or because the implementation of a function introduces too
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35 much error along some part of its' input range. Caveat Emptor (although you're
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36 not exactly paying for this library now, are you? Caveat lector perhaps).
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37
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38 ( This table needs completing, specifically the input ranges... )
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39
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40 | C Function | Operator | Input Range | Asserts | Notes |
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41 | ------------- | ----------- | ----------- | -------- | ----------------------------------------------- |
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42 | qadd(a, b) | a + b | | | Addition |
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43 | qsub(a, b) | a \- b | | | Subtraction |
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44 | qdiv(a, b) | a / b | b != 0 | Yes | Division |
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45 | qmul(a, b) | a \* b | | | Multiplication |
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46 | qrem(a, b) | a rem b | b != 0 | Yes | Remainder: remainder after division |
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47 | qmod(a, b) | a mod b | b != 0 | Yes | Modulo |
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48 | qsin(theta) | sin(theta) | | | Sine |
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49 | qcos(theta) | cos(theta) | | | Cosine |
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50 | qtan(theta) | tan(theta) | | | Tangent |
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51 | qcot(theta) | cot(theta) | | | Cotangent |
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52 | qhypot(a, b) | hypot(a, b) | | | Hypotenuse; sqrt(a\*a + b\*b) |
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53 | qasin(x) | asin(x) | abs(x) <= 1 | Yes | Arcsine |
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54 | qacos(x) | acos(x) | abs(x) <= 1 | Yes | Arccosine |
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55 | qatan(t) | atan(t) | | | Arctangent |
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56 | qsinh(a) | sinh(a) | | | Hyperbolic Sine |
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57 | qcosh(a) | cosh(a) | | | Hyperbolic Cosine |
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58 | qtanh(a) | tanh(a) | | | Hyperbolic Tangent |
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59 | qasinh(a) | asinh(a) | | | Inverse Hyperbolic Sine |
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60 | qacosh(a) | acosh(a) | | | Inverse Hyperbolic Cosine |
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61 | qatanh(a) | atanh(a) | | | Inverse Hyperbolic Tangent |
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62 | qexp(e) | exp(e) | e < ln(MAX) | No | Exponential function, High error for 'e' > ~7. |
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63 | qlog(n) | log(n) | n > 0 | Yes | Natural Logarithm |
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64 | qsqrt(x) | sqrt(x) | n >= 0 | Yes | Square Root |
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65 | qround(q) | round(q) | | | Round to nearest value |
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66 | qceil(q) | ceil(q) | | | Round up value |
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67 | qtrunc(q) | trunc(q) | | | Truncate value |
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68 | qfloor(q) | floor(q) | | | Round down value |
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69 | qnegate(a) | -a | | | Negate a number |
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70 | qabs(a) | abs(a) | | | Absolute value of a number |
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71 | qfma(a, b, c) | (a\*b)+c | | | Fused Multiply Add, uses Q32.32 internally |
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72 | qequal(a, b) | a == b | | | |
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73 | qsignum(a) | signum(a) | | | Signum function |
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74 | qsign(a) | sgn(a) | | | Sign function |
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75 | | | | | |
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76
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77 For the round/ceil/trunc/floor functions the following table from the
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78 [cplusplus.com][] helps:
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79
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80 | value | round | floor | ceil | trunc |
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81 | ----- | ----- | ----- | ---- | ----- |
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82 | 2.3 | 2.0 | 2.0 | 3.0 | 2.0 |
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83 | 3.8 | 4.0 | 3.0 | 4.0 | 3.0 |
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84 | 5.5 | 6.0 | 5.0 | 6.0 | 5.0 |
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85 | -2.3 | -2.0 | -3.0 | -2.0 | -2.0 |
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86 | -3.8 | -4.0 | -4.0 | -3.0 | -3.0 |
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87 | -5.5 | -6.0 | -6.0 | -5.0 | -5.0 |
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88
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89 Have fun with the adding, and subtracting, and stuff, I hope it goes well. It
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90 would be cool to make an [APL][] interpreter built around this library. Testing
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91 would become much easier as you could use programming language constructs to
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92 create new tests over larger ranges of numbers.
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93
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94 [APL]: https://en.wikipedia.org/wiki/APL_(programming_language)
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95 [Doom]: https://en.wikipedia.org/wiki/Doom_(1993_video_game)
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96 [tolower]: http://www.cplusplus.com/reference/cctype/tolower/
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97 [makefile]: https://en.wikipedia.org/wiki/Make_(software)
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98 [C]: https://en.wikipedia.org/wiki/C_%28programming_language%29
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99 [cplusplus.com]: http://www.cplusplus.com/reference/cmath/round/
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100 [Q16.16]: https://en.wikipedia.org/wiki/Fixed-point_arithmetic
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101 [CORDIC]: https://en.wikipedia.org/wiki/CORDIC
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102 [VHDL]: https://en.wikipedia.org/wiki/VHDL
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103 [FPGA]: https://en.wikipedia.org/wiki/Field-programmable_gate_array
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104
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105 <style type="text/css">body{margin:40px auto;max-width:850px;line-height:1.6;font-size:16px;color:#444;padding:0 10px}h1,h2,h3{line-height:1.2}</style>
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