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This program is designed to compute the sine function using a power series. A very limited floating-point format is used to demonstrate some of the problems that can occur when using floating point. The program computes each term in the power series and displays the result. It continues computing terms until the last term is so small that it doesn't contribute to the final result. For comparison purposes, the result of the library function sin is displayed as well as the computed sine. The program is invoked by: sine value where value is an angle in radians. For example, to compute sin(0) we use the command: % sine 0 x**1 0.000E+00 1! 1.000E+00 x**1/1! 0.000E+00 1 term computed sin(0.000E+00)= 0.000E+00 Actual sin(0)=0 And to compute sin( ) we use the command: % sine 3.141 x**1 3.141E+00 1! 1.000E+00 x**1/1! 3.141E+00 total 3.141E+00 x**3 3.099E+01 3! 6.000E+00 x**3/3! 5.165E+00 total -2.024E+00 x**5 3.057E+02 5! 1.200E+02 x**5/5! 2.548E+00 total 5.239E-01 x**7 3.016E+03 7! 5.040E+03 x**7/7! 5.985E-01 total -7.457E-02 x**9 2.976E+04 9! 3.629E+05 x**9/9! 8.201E-02 total 7.438E-03 x**11 2.936E+05 11! 3.992E+07 x**11/11! 7.355E-03 total 8.300E-05 x**13 2.897E+06 13! 6.227E+09 x**13/13! 4.652E-04 total 5.482E-04 x**15 2.858E+07 15! 1.308E+12 x**15/15! 2.185E-05 total 5.263E-04 x**17 2.819E+08 17! 3.557E+14 x**17/17! 7.927E-07 total 5.271E-04 x**19 2.782E+09 19! 1.217E+17 x**19/19! 2.287E-08 total 5.271E-04 x**21 2.744E+10 21! 5.109E+19 x**21/21! 5.371E-10 11 term computed sin(3.141E+00)= 5.271E-04 Actual sin(3.141)=0.000592654 Example D-1 lists the Makefile for Unix. Example D-1. sin/makefile.unx# # Makefile for many Unix compilers using the # "standard" command name CC # CC=CC CFLAGS=-g sine: sine.cpp $(CC) $(CFLAGS) -o sine sine.cpp -lm clean: rm sine Example D-2 lists the sine.cpp file. Example D-2. sin/sine.cpp/******************************************************** * sine -- compute sine using very simple floating * * arithmetic. * * * * Usage: * * sine <value> * * * * <value> is an angle in radians * * * * Format used in f.fffe+X * * * * f.fff is a 4 digit fraction * * + is a sign (+ or -) * * X is a single digit exponent * * * * sin(x) = x - x**3 + x**5 - x**7 * * ----- ---- ---- . . . . * * 3! 5! 7! * * * * Warning: This program is intended to show some of * * problems with floating point. It not intended * * to be used to produce exact values for the * * sin function. * * * * Note: Even though we specify only one-digit for the * * exponent, two are used for some calculations. * * This is due to the fact that printf has no * * format for a single digit exponent. * ********************************************************/ #include <iostream> #include <cstdlib> #include <cmath> #include <cstdio> int main(int argc, char *argv[]) { float total; // total of series so far float new_total;// newer version of total float term_top;// top part of term float term_bottom;// bottom of current term float term; // current term float exp; // exponent of current term float sign; // +1 or -1 (changes on each term) float value; // value of the argument to sin int index; // index for counting terms char *float_2_ascii(float number); // turn floating-point to ascii float fix_float(float number); // round to correct digits float factorial(float number); // compute n! if (argc != 2) { std::cerr << "Usage is:\n"; std::cerr << " sine <value>\n"; exit (8); } value = fix_float(atof(&argv[1][0])); total = 0.0; exp = 1.0; sign = 1.0; for (index = 0; /* take care of below */ ; ++index) { term_top = fix_float(pow(value, exp)); term_bottom = fix_float(factorial(exp)); term = fix_float(term_top / term_bottom); std::cout << "x**" << static_cast<int>(exp) << " " << float_2_ascii(term_top) << '\n'; std::cout << exp << "! " << float_2_ascii(term_bottom) << '\n'; std::cout << "x**" << static_cast<int>(exp) << "/" << static_cast<int>(exp) << "! " << float_2_ascii(term) << "\n"; new_total = fix_float(total + sign * term); if (new_total == total) break; total = new_total; sign = -sign; exp = exp + 2.0; std::cout <<" total " << float_2_ascii(total) << '\n'; std::cout <<'\n'; } std::cout << (index +1) << " term computed\n"; std::cout << "sin(" << float_2_ascii(value) << ")=\n"; std::cout << " " << float_2_ascii(total) << '\n'; std::cout << "Actual sin(" << atof(&argv[1][0]) << ")=" << sin(atof(&argv[1][0])) << '\n'; return (0); } /******************************************************** * float_2_ascii -- turn a floating-point string * * into ascii. * * * * Parameters * * number -- number to turn into ascii * * * * Returns * * Pointer to the string containing the number * * * * Warning: Uses static storage, so later calls * * overwrite earlier entries * ********************************************************/ char *float_2_ascii(float number) { static char result[10]; //place to put the number std::sprintf(result,"%8.3E", number); return (result); } /******************************************************** * fix_float -- turn high precision numbers into * * low precision numbers to simulate a * * very dumb floating-point structure. * * * * Parameters * * number -- number to take care of * * * * Returns * * number accurate to 5 places only * * * * Note: This works by changing a number into ascii and * * back. Very slow, but it works. * ********************************************************/ float fix_float(float number) { float result; // result of the conversion char ascii[10]; // ascii version of number std::sprintf(ascii,"%8.4e", number); std::sscanf(ascii, "%e", &result); return (result); } /******************************************************** * factorial -- compute the factorial of a number. * * * * Parameters * * number -- number to use for factorial * * * * Returns * * factorial(number) or number! * * * * Note: Even though this is a floating-point routine, * * using numbers that are not whole numbers * * does not make sense. * ********************************************************/ float factorial(float number) { if (number <= 1.0) return (number); else return (number *factorial(number - 1.0)); } |
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