preprocessor_math.h, SQRT(): take a better initial guess.
Now results are apparently accurate across the whole uint32 range.
At least, this test passes with all numbers being exact:
#include "preprocessor_math.h"
#include <math.h>
... in main() ...
sersendf_P(PSTR("0: %lu %lu\n"), (uint32_t)SQRT(0), (uint32_t)sqrt(0));
sersendf_P(PSTR("1: %lu %lu\n"), (uint32_t)SQRT(1), (uint32_t)sqrt(1));
sersendf_P(PSTR("2: %lu %lu\n"), (uint32_t)SQRT(2), (uint32_t)sqrt(2));
sersendf_P(PSTR("3: %lu %lu\n"), (uint32_t)SQRT(3), (uint32_t)sqrt(3));
sersendf_P(PSTR("4: %lu %lu\n"), (uint32_t)SQRT(4), (uint32_t)sqrt(4));
sersendf_P(PSTR("5: %lu %lu\n"), (uint32_t)SQRT(5), (uint32_t)sqrt(5));
sersendf_P(PSTR("6: %lu %lu\n"), (uint32_t)SQRT(6), (uint32_t)sqrt(6));
sersendf_P(PSTR("7: %lu %lu\n"), (uint32_t)SQRT(7), (uint32_t)sqrt(7));
sersendf_P(PSTR("8: %lu %lu\n"), (uint32_t)SQRT(8), (uint32_t)sqrt(8));
sersendf_P(PSTR("9: %lu %lu\n"), (uint32_t)SQRT(9), (uint32_t)sqrt(9));
sersendf_P(PSTR("10: %lu %lu\n"), (uint32_t)SQRT(10), (uint32_t)sqrt(10));
sersendf_P(PSTR("20: %lu %lu\n"), (uint32_t)SQRT(20), (uint32_t)sqrt(20));
sersendf_P(PSTR("30: %lu %lu\n"), (uint32_t)SQRT(30), (uint32_t)sqrt(30));
sersendf_P(PSTR("40: %lu %lu\n"), (uint32_t)SQRT(40), (uint32_t)sqrt(40));
sersendf_P(PSTR("50: %lu %lu\n"), (uint32_t)SQRT(50), (uint32_t)sqrt(50));
sersendf_P(PSTR("60: %lu %lu\n"), (uint32_t)SQRT(60), (uint32_t)sqrt(60));
sersendf_P(PSTR("70: %lu %lu\n"), (uint32_t)SQRT(70), (uint32_t)sqrt(70));
sersendf_P(PSTR("80: %lu %lu\n"), (uint32_t)SQRT(80), (uint32_t)sqrt(80));
sersendf_P(PSTR("90: %lu %lu\n"), (uint32_t)SQRT(90), (uint32_t)sqrt(90));
sersendf_P(PSTR("100: %lu %lu\n"), (uint32_t)SQRT(100), (uint32_t)sqrt(100));
sersendf_P(PSTR("200: %lu %lu\n"), (uint32_t)SQRT(200), (uint32_t)sqrt(200));
sersendf_P(PSTR("300: %lu %lu\n"), (uint32_t)SQRT(300), (uint32_t)sqrt(300));
sersendf_P(PSTR("400: %lu %lu\n"), (uint32_t)SQRT(400), (uint32_t)sqrt(400));
sersendf_P(PSTR("500: %lu %lu\n"), (uint32_t)SQRT(500), (uint32_t)sqrt(500));
sersendf_P(PSTR("600: %lu %lu\n"), (uint32_t)SQRT(600), (uint32_t)sqrt(600));
sersendf_P(PSTR("700: %lu %lu\n"), (uint32_t)SQRT(700), (uint32_t)sqrt(700));
sersendf_P(PSTR("800: %lu %lu\n"), (uint32_t)SQRT(800), (uint32_t)sqrt(800));
sersendf_P(PSTR("900: %lu %lu\n"), (uint32_t)SQRT(900), (uint32_t)sqrt(900));
sersendf_P(PSTR("1000: %lu %lu\n"), (uint32_t)SQRT(1000), (uint32_t)sqrt(1000));
sersendf_P(PSTR("2000: %lu %lu\n"), (uint32_t)SQRT(2000), (uint32_t)sqrt(2000));
sersendf_P(PSTR("3000: %lu %lu\n"), (uint32_t)SQRT(3000), (uint32_t)sqrt(3000));
sersendf_P(PSTR("4000: %lu %lu\n"), (uint32_t)SQRT(4000), (uint32_t)sqrt(4000));
sersendf_P(PSTR("5000: %lu %lu\n"), (uint32_t)SQRT(5000), (uint32_t)sqrt(5000));
sersendf_P(PSTR("6000: %lu %lu\n"), (uint32_t)SQRT(6000), (uint32_t)sqrt(6000));
sersendf_P(PSTR("7000: %lu %lu\n"), (uint32_t)SQRT(7000), (uint32_t)sqrt(7000));
sersendf_P(PSTR("8000: %lu %lu\n"), (uint32_t)SQRT(8000), (uint32_t)sqrt(8000));
sersendf_P(PSTR("9000: %lu %lu\n"), (uint32_t)SQRT(9000), (uint32_t)sqrt(9000));
sersendf_P(PSTR("10000: %lu %lu\n"), (uint32_t)SQRT(10000), (uint32_t)sqrt(10000));
sersendf_P(PSTR("20000: %lu %lu\n"), (uint32_t)SQRT(20000), (uint32_t)sqrt(20000));
sersendf_P(PSTR("30000: %lu %lu\n"), (uint32_t)SQRT(30000), (uint32_t)sqrt(30000));
sersendf_P(PSTR("40000: %lu %lu\n"), (uint32_t)SQRT(40000), (uint32_t)sqrt(40000));
sersendf_P(PSTR("50000: %lu %lu\n"), (uint32_t)SQRT(50000), (uint32_t)sqrt(50000));
sersendf_P(PSTR("60000: %lu %lu\n"), (uint32_t)SQRT(60000), (uint32_t)sqrt(60000));
sersendf_P(PSTR("70000: %lu %lu\n"), (uint32_t)SQRT(70000), (uint32_t)sqrt(70000));
sersendf_P(PSTR("80000: %lu %lu\n"), (uint32_t)SQRT(80000), (uint32_t)sqrt(80000));
sersendf_P(PSTR("90000: %lu %lu\n"), (uint32_t)SQRT(90000), (uint32_t)sqrt(90000));
sersendf_P(PSTR("100000: %lu %lu\n"), (uint32_t)SQRT(100000), (uint32_t)sqrt(100000));
sersendf_P(PSTR("200000: %lu %lu\n"), (uint32_t)SQRT(200000), (uint32_t)sqrt(200000));
sersendf_P(PSTR("300000: %lu %lu\n"), (uint32_t)SQRT(300000), (uint32_t)sqrt(300000));
sersendf_P(PSTR("400000: %lu %lu\n"), (uint32_t)SQRT(400000), (uint32_t)sqrt(400000));
sersendf_P(PSTR("500000: %lu %lu\n"), (uint32_t)SQRT(500000), (uint32_t)sqrt(500000));
sersendf_P(PSTR("600000: %lu %lu\n"), (uint32_t)SQRT(600000), (uint32_t)sqrt(600000));
sersendf_P(PSTR("700000: %lu %lu\n"), (uint32_t)SQRT(700000), (uint32_t)sqrt(700000));
sersendf_P(PSTR("800000: %lu %lu\n"), (uint32_t)SQRT(800000), (uint32_t)sqrt(800000));
sersendf_P(PSTR("900000: %lu %lu\n"), (uint32_t)SQRT(900000), (uint32_t)sqrt(900000));
sersendf_P(PSTR("1000000: %lu %lu\n"), (uint32_t)SQRT(1000000), (uint32_t)sqrt(1000000));
sersendf_P(PSTR("2000000: %lu %lu\n"), (uint32_t)SQRT(2000000), (uint32_t)sqrt(2000000));
sersendf_P(PSTR("3000000: %lu %lu\n"), (uint32_t)SQRT(3000000), (uint32_t)sqrt(3000000));
sersendf_P(PSTR("4000000: %lu %lu\n"), (uint32_t)SQRT(4000000), (uint32_t)sqrt(4000000));
sersendf_P(PSTR("5000000: %lu %lu\n"), (uint32_t)SQRT(5000000), (uint32_t)sqrt(5000000));
sersendf_P(PSTR("6000000: %lu %lu\n"), (uint32_t)SQRT(6000000), (uint32_t)sqrt(6000000));
sersendf_P(PSTR("7000000: %lu %lu\n"), (uint32_t)SQRT(7000000), (uint32_t)sqrt(7000000));
sersendf_P(PSTR("8000000: %lu %lu\n"), (uint32_t)SQRT(8000000), (uint32_t)sqrt(8000000));
sersendf_P(PSTR("9000000: %lu %lu\n"), (uint32_t)SQRT(9000000), (uint32_t)sqrt(9000000));
sersendf_P(PSTR("10000000: %lu %lu\n"), (uint32_t)SQRT(10000000), (uint32_t)sqrt(10000000));
sersendf_P(PSTR("20000000: %lu %lu\n"), (uint32_t)SQRT(20000000), (uint32_t)sqrt(20000000));
sersendf_P(PSTR("30000000: %lu %lu\n"), (uint32_t)SQRT(30000000), (uint32_t)sqrt(30000000));
sersendf_P(PSTR("40000000: %lu %lu\n"), (uint32_t)SQRT(40000000), (uint32_t)sqrt(40000000));
sersendf_P(PSTR("50000000: %lu %lu\n"), (uint32_t)SQRT(50000000), (uint32_t)sqrt(50000000));
sersendf_P(PSTR("60000000: %lu %lu\n"), (uint32_t)SQRT(60000000), (uint32_t)sqrt(60000000));
sersendf_P(PSTR("70000000: %lu %lu\n"), (uint32_t)SQRT(70000000), (uint32_t)sqrt(70000000));
sersendf_P(PSTR("80000000: %lu %lu\n"), (uint32_t)SQRT(80000000), (uint32_t)sqrt(80000000));
sersendf_P(PSTR("90000000: %lu %lu\n"), (uint32_t)SQRT(90000000), (uint32_t)sqrt(90000000));
sersendf_P(PSTR("100000000: %lu %lu\n"), (uint32_t)SQRT(100000000), (uint32_t)sqrt(100000000));
sersendf_P(PSTR("200000000: %lu %lu\n"), (uint32_t)SQRT(200000000), (uint32_t)sqrt(200000000));
sersendf_P(PSTR("300000000: %lu %lu\n"), (uint32_t)SQRT(300000000), (uint32_t)sqrt(300000000));
sersendf_P(PSTR("400000000: %lu %lu\n"), (uint32_t)SQRT(400000000), (uint32_t)sqrt(400000000));
sersendf_P(PSTR("500000000: %lu %lu\n"), (uint32_t)SQRT(500000000), (uint32_t)sqrt(500000000));
sersendf_P(PSTR("600000000: %lu %lu\n"), (uint32_t)SQRT(600000000), (uint32_t)sqrt(600000000));
sersendf_P(PSTR("700000000: %lu %lu\n"), (uint32_t)SQRT(700000000), (uint32_t)sqrt(700000000));
sersendf_P(PSTR("800000000: %lu %lu\n"), (uint32_t)SQRT(800000000), (uint32_t)sqrt(800000000));
sersendf_P(PSTR("900000000: %lu %lu\n"), (uint32_t)SQRT(900000000), (uint32_t)sqrt(900000000));
sersendf_P(PSTR("1000000000: %lu %lu\n"), (uint32_t)SQRT(1000000000), (uint32_t)sqrt(1000000000));
sersendf_P(PSTR("2000000000: %lu %lu\n"), (uint32_t)SQRT(2000000000), (uint32_t)sqrt(2000000000));
sersendf_P(PSTR("3000000000: %lu %lu\n"), (uint32_t)SQRT(3000000000), (uint32_t)sqrt(3000000000));
sersendf_P(PSTR("4000000000: %lu %lu\n"), (uint32_t)SQRT(4000000000), (uint32_t)sqrt(4000000000));
This commit is contained in:
Markus Hitter
parent
6f83519a1d
commit
2ad7517e27
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@ -18,17 +18,16 @@
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equals
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(uint32_t)(sqrt(i) + .5)
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These two provide identical results up to 1'861'860 (tested at runtime) and
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up to 10'000'000 at compile time. At 90'000'000, deviation is about 5%,
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increasing further at even higher numbers. This "+ .5" is for rounding and
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not crucial. Casting to other sizes is also possible.
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These two provide identical results for all tested numbers across the
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uint32 range. This "+ .5" is for rounding and not crucial. Casting to
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other sizes is also possible.
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Can be used for calculations at runtime, too, where it costs 944(!) bytes
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binary size and takes roughly 10'000(!) clock cycles.
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Can principally be used for calculations at runtime, too, but its compiled
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size is prohibitively large (more than 20kB per instance).
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Initial version found on pl.comp.lang.c, posted by Jean-Louis PATANE.
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*/
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#define SQR00(x) (((double)(x)) / 2)
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#define SQR00(x) (((x) > 65535) ? (double)65535 : (double)(x) / 2)
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#define SQR01(x) ((SQR00(x) + ((x) / SQR00(x))) / 2)
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#define SQR02(x) ((SQR01(x) + ((x) / SQR01(x))) / 2)
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#define SQR03(x) ((SQR02(x) + ((x) / SQR02(x))) / 2)
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@ -41,11 +40,10 @@
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#define SQR10(x) ((SQR09(x) + ((x) / SQR09(x))) / 2)
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#define SQR11(x) ((SQR10(x) + ((x) / SQR10(x))) / 2)
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#define SQR12(x) ((SQR11(x) + ((x) / SQR11(x))) / 2)
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// You can add more of these lines here, each additional line increases
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// accurate range by about factor 2 and costs additional 40 bytes binary size
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// in the non-constant case. But beware, the length of the preprocessed term
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// explodes, leading to several seconds compile time above about SQR13.
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#define SQRT(x) ((SQR12(x) + ((x) / SQR12(x))) / 2)
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// We use 9 iterations, note how SQR10() and up get ignored. You can add more
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// iterations here, but beware, the length of the preprocessed term
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// explodes, leading to several seconds compile time above about SQR10().
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#define SQRT(x) ((SQR09(x) + ((x) / SQR09(x))) / 2)
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#endif /* _PREPROCESSOR_MATH_H */
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