/* Live links in the comments. What is a Single Precision number in C++? Is it a number with 8 digits? If so, does the number of digits include the sign and decimal point or without them? Is -12.345678 a single precision number? Like to see some examples of single precision numbers. Q&A: https://www.sololearn.com/Discuss/724004/?ref=app Industry standard explanation: https://stackoverflow.com/questions/30688422/is-the-most-significant-decimal-digits-precision-that-can-be-converted-to-binary */ #include #include using namespace std; int main() { cout << "A single-precision number in c++ is a float. Floats have 7.225 significant digits; In a limited way, this means that: Regardless of the decimal point, when you store a decimal number, the first 7 digits are faithful, the 8th is partially faithful and you can't count on the rest." << endl; // Show wiggle around the 8th digit float float7 = -12.345677f; // + .000001 float float8 = -12.345678f; // Q&A's example float float9 = -12.345679f; // - .000001 cout << " [>>weird area<]\n"; printf("float7: %.21f\nfloat8: %.21f\nfloat9: %.21f\n",float7,float8,float9); cout << " 8th digit ^\n\n"; cout << "You cannot reliably close the gap between two floats past the precision point.\n\n"; cout << "Gap between our three floats:\n"; printf("7 & 8: %.21f\n", float8-float7); printf("8 & 9: %.21f\n", float9-float8); cout << " ^epsilon (gap)\n\n"; cout << "Attempting to set inside the gap...\n\n"; float myFloatSplit = 0.0f; cout << "store: -12.3456778\n"; myFloatSplit =-12.3456778f; printf("result: %.21f\n", myFloatSplit); cout << "== float7 ^ Flips UP\n\n"; cout << "store: -12.3456779\n"; myFloatSplit =-12.3456779f; printf("result: %.21f\n", myFloatSplit); cout << "== float8 ^ Flips DOWN\n\n"; cout << "*************************************\n"; cout << "Lesson: 7.225 digits of precision\nmeans the 8th significant digit is\nUNRELIABLE because it can be affected\nby later digits.\n"; cout << "*************************************\n"; cout << "\n>>> Extremely technical <<<\n"; cout << "* ...by later digits means: by where you are according to epsilon. The place where it flips from float7 to float8 is 50% of epsilon, starting 'between' the digits where digit 8 of original float and digit 8 of epsilon intersect.\n\n"; cout << "0.000000953674316406250 == epsilon\n"; cout << "0.000000476837158203125 == 50% gap\n\n"; cout << "-12.345677375793457031250 | float7\n"; cout << "- 0.000000476837158203125 | + 50% gap\n"; cout << "---------^^-------------- |\n"; cout << "-12.345677852630615234375 | crossover\n\n"; cout << "===== Testing crossover =====\n"; cout << "...Setting very last digit to 4\n"; cout << "store: -12.345677852630615234374\n"; myFloatSplit = -12.345677852630615234374f; printf("result: %.21f\n", myFloatSplit); cout << "Fall BACK to float7.\n\n"; cout << "...Setting past epsilon precision\n"; cout << "store: -12.3456778526306152343749999\n"; myFloatSplit = -12.3456778526306152343749999f; printf("result: %.21f\n", myFloatSplit); cout << "Fall BACK to float7. Extra ignored.\n\n"; cout << "...Setting very last digit to 5\n"; cout << "store: -12.345677852630615234375\n"; myFloatSplit = -12.345677852630615234375f; printf("result: %.21f\n", myFloatSplit); cout << "Crossover to float8.\n\n"; cout << "At the edge of standard precision (digit 8), everything in the [weird] epsilon area falls back or forward depending on which side of 50% epsilon you land on, unless you are past the epsilon effect too.\n\n"; cout << "Note: Epsilon takes on different values by float base-2 magnitude. See my float64 internals [web] for that."; return 0; }

Single precision : digit 8 microscope