1 /* 2 * Copyright 2012 Google Inc. 3 * 4 * Use of this source code is governed by a BSD-style license that can be 5 * found in the LICENSE file. 6 */ 7 #include <vector> 8 9 /* Given: 10 * Resultant[a*t^3 + b*t^2 + c*t + d - x, e*t^3 + f*t^2 + g*t + h - y, t] 11 */ 12 13 const char result1[] = 14 "-d^3 e^3 + c d^2 e^2 f - b d^2 e f^2 + a d^2 f^3 - c^2 d e^2 g + " 15 " 2 b d^2 e^2 g + b c d e f g - 3 a d^2 e f g - a c d f^2 g - " 16 " b^2 d e g^2 + 2 a c d e g^2 + a b d f g^2 - a^2 d g^3 + c^3 e^2 h - " 17 " 3 b c d e^2 h + 3 a d^2 e^2 h - b c^2 e f h + 2 b^2 d e f h + " 18 " a c d e f h + a c^2 f^2 h - 2 a b d f^2 h + b^2 c e g h - " 19 " 2 a c^2 e g h - a b d e g h - a b c f g h + 3 a^2 d f g h + " 20 " a^2 c g^2 h - b^3 e h^2 + 3 a b c e h^2 - 3 a^2 d e h^2 + " 21 " a b^2 f h^2 - 2 a^2 c f h^2 - a^2 b g h^2 + a^3 h^3 + 3 d^2 e^3 x - " 22 " 2 c d e^2 f x + 2 b d e f^2 x - 2 a d f^3 x + c^2 e^2 g x - " 23 " 4 b d e^2 g x - b c e f g x + 6 a d e f g x + a c f^2 g x + " 24 " b^2 e g^2 x - 2 a c e g^2 x - a b f g^2 x + a^2 g^3 x + " 25 " 3 b c e^2 h x - 6 a d e^2 h x - 2 b^2 e f h x - a c e f h x + " 26 " 2 a b f^2 h x + a b e g h x - 3 a^2 f g h x + 3 a^2 e h^2 x - " 27 " 3 d e^3 x^2 + c e^2 f x^2 - b e f^2 x^2 + a f^3 x^2 + " 28 " 2 b e^2 g x^2 - 3 a e f g x^2 + 3 a e^2 h x^2 + e^3 x^3 - " 29 " c^3 e^2 y + 3 b c d e^2 y - 3 a d^2 e^2 y + b c^2 e f y - " 30 " 2 b^2 d e f y - a c d e f y - a c^2 f^2 y + 2 a b d f^2 y - " 31 " b^2 c e g y + 2 a c^2 e g y + a b d e g y + a b c f g y - " 32 " 3 a^2 d f g y - a^2 c g^2 y + 2 b^3 e h y - 6 a b c e h y + " 33 " 6 a^2 d e h y - 2 a b^2 f h y + 4 a^2 c f h y + 2 a^2 b g h y - " 34 " 3 a^3 h^2 y - 3 b c e^2 x y + 6 a d e^2 x y + 2 b^2 e f x y + " 35 " a c e f x y - 2 a b f^2 x y - a b e g x y + 3 a^2 f g x y - " 36 " 6 a^2 e h x y - 3 a e^2 x^2 y - b^3 e y^2 + 3 a b c e y^2 - " 37 " 3 a^2 d e y^2 + a b^2 f y^2 - 2 a^2 c f y^2 - a^2 b g y^2 + " 38 " 3 a^3 h y^2 + 3 a^2 e x y^2 - a^3 y^3"; 39 40 const size_t len1 = sizeof(result1) - 1; 41 42 /* Given: 43 * Expand[ 44 * Det[{{a, b, c, (d - x), 0, 0}, 45 * {0, a, b, c, (d - x), 0}, 46 * {0, 0, a, b, c, (d - x)}, 47 * {e, f, g, (h - y), 0, 0}, 48 * {0, e, f, g, (h - y), 0}, 49 * {0, 0, e, f, g, (h - y)}}]] 50 */ 51 // result1 and result2 are the same. 102 factors: 52 const char result2[] = 53 "-d^3 e^3 + c d^2 e^2 f - b d^2 e f^2 + a d^2 f^3 - c^2 d e^2 g + " 54 " 2 b d^2 e^2 g + b c d e f g - 3 a d^2 e f g - a c d f^2 g - " 55 " b^2 d e g^2 + 2 a c d e g^2 + a b d f g^2 - a^2 d g^3 + c^3 e^2 h - " 56 " 3 b c d e^2 h + 3 a d^2 e^2 h - b c^2 e f h + 2 b^2 d e f h + " 57 " a c d e f h + a c^2 f^2 h - 2 a b d f^2 h + b^2 c e g h - " 58 " 2 a c^2 e g h - a b d e g h - a b c f g h + 3 a^2 d f g h + " 59 " a^2 c g^2 h - b^3 e h^2 + 3 a b c e h^2 - 3 a^2 d e h^2 + " 60 " a b^2 f h^2 - 2 a^2 c f h^2 - a^2 b g h^2 + a^3 h^3 + 3 d^2 e^3 x - " 61 " 2 c d e^2 f x + 2 b d e f^2 x - 2 a d f^3 x + c^2 e^2 g x - " 62 " 4 b d e^2 g x - b c e f g x + 6 a d e f g x + a c f^2 g x + " 63 " b^2 e g^2 x - 2 a c e g^2 x - a b f g^2 x + a^2 g^3 x + " 64 " 3 b c e^2 h x - 6 a d e^2 h x - 2 b^2 e f h x - a c e f h x + " 65 " 2 a b f^2 h x + a b e g h x - 3 a^2 f g h x + 3 a^2 e h^2 x - " 66 " 3 d e^3 x^2 + c e^2 f x^2 - b e f^2 x^2 + a f^3 x^2 + " 67 " 2 b e^2 g x^2 - 3 a e f g x^2 + 3 a e^2 h x^2 + e^3 x^3 - " 68 " c^3 e^2 y + 3 b c d e^2 y - 3 a d^2 e^2 y + b c^2 e f y - " 69 " 2 b^2 d e f y - a c d e f y - a c^2 f^2 y + 2 a b d f^2 y - " 70 " b^2 c e g y + 2 a c^2 e g y + a b d e g y + a b c f g y - " 71 " 3 a^2 d f g y - a^2 c g^2 y + 2 b^3 e h y - 6 a b c e h y + " 72 " 6 a^2 d e h y - 2 a b^2 f h y + 4 a^2 c f h y + 2 a^2 b g h y - " 73 " 3 a^3 h^2 y - 3 b c e^2 x y + 6 a d e^2 x y + 2 b^2 e f x y + " 74 " a c e f x y - 2 a b f^2 x y - a b e g x y + 3 a^2 f g x y - " 75 " 6 a^2 e h x y - 3 a e^2 x^2 y - b^3 e y^2 + 3 a b c e y^2 - " 76 " 3 a^2 d e y^2 + a b^2 f y^2 - 2 a^2 c f y^2 - a^2 b g y^2 + " 77 " 3 a^3 h y^2 + 3 a^2 e x y^2 - a^3 y^3"; 78 79 const size_t len2 = sizeof(result2) - 1; 80 81 /* Given: r1 = Resultant[ 82 * a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - x, 83 * e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, t] 84 * Collect[r1, {x, y}, Simplify] 85 * CForm[%] 86 * then use regex to replace Power\(([a-h]),3\) with \1*\1*\1 87 * and Power\(([a-h]),2\) with \1*\1 88 * yields: 89 90 d*d*d*e*e*e - 3*d*d*(3*c*e*e*f + 3*b*e*(-3*f*f + 2*e*g) + a*(9*f*f*f - 9*e*f*g + e*e*h)) - 91 h*(27*c*c*c*e*e - 27*c*c*(3*b*e*f - 3*a*f*f + 2*a*e*g) + 92 h*(-27*b*b*b*e + 27*a*b*b*f - 9*a*a*b*g + a*a*a*h) + 93 9*c*(9*b*b*e*g + a*b*(-9*f*g + 3*e*h) + a*a*(3*g*g - 2*f*h))) + 94 3*d*(9*c*c*e*e*g + 9*b*b*e*(3*g*g - 2*f*h) + 3*a*b*(-9*f*g*g + 6*f*f*h + e*g*h) + 95 a*a*(9*g*g*g - 9*f*g*h + e*h*h) + 3*c*(3*b*e*(-3*f*g + e*h) + a*(9*f*f*g - 6*e*g*g - e*f*h))) 96 97 - Power(e - 3*f + 3*g - h,3)*Power(x,3) 98 99 + 3*(6*b*d*d*e*e - d*d*d*e*e + 18*b*b*d*e*f - 18*b*d*d*e*f - 100 9*b*d*d*f*f - 54*b*b*d*e*g + 12*b*d*d*e*g - 27*b*b*d*g*g - 18*b*b*b*e*h + 18*b*b*d*e*h + 101 18*b*b*d*f*h + a*a*a*h*h - 9*b*b*b*h*h + 9*c*c*c*e*(e + 2*h) + 102 a*a*(-3*b*h*(2*g + h) + d*(-27*g*g + 9*g*h - h*(2*e + h) + 9*f*(g + h))) + 103 a*(9*b*b*h*(2*f + h) - 3*b*d*(6*f*f - 6*f*(3*g - 2*h) + g*(-9*g + h) + e*(g + h)) + 104 d*d*(e*e + 9*f*(3*f - g) + e*(-9*f - 9*g + 2*h))) - 105 9*c*c*(d*e*(e + 2*g) + 3*b*(f*h + e*(f + h)) + a*(-3*f*f - 6*f*h + 2*(g*h + e*(g + h)))) + 106 3*c*(d*d*e*(e + 2*f) + a*a*(3*g*g + 6*g*h - 2*h*(2*f + h)) + 9*b*b*(g*h + e*(g + h)) + 107 a*d*(-9*f*f - 18*f*g + 6*g*g + f*h + e*(f + 12*g + h)) + 108 b*(d*(-3*e*e + 9*f*g + e*(9*f + 9*g - 6*h)) + 3*a*(h*(2*e - 3*g + h) - 3*f*(g + h)))))*y 109 110 - 3*(18*c*c*c*e - 18*c*c*d*e + 6*c*d*d*e - d*d*d*e + 3*c*d*d*f - 9*c*c*d*g + a*a*a*h + 9*c*c*c*h - 111 9*b*b*b*(e + 2*h) - a*a*(d*(e - 9*f + 18*g - 7*h) + 3*c*(2*f - 6*g + h)) + 112 a*(-9*c*c*(2*e - 6*f + 2*g - h) + d*d*(-7*e + 18*f - 9*g + h) + 3*c*d*(7*e - 17*f + 3*g + h)) + 113 9*b*b*(3*c*(e + g + h) + a*(f + 2*h) - d*(e - 2*(f - 3*g + h))) - 114 3*b*(-(d*d*(e - 6*f + 2*g)) - 3*c*d*(e + 3*f + 3*g - h) + 9*c*c*(e + f + h) + a*a*(g + 2*h) + 115 a*(c*(-3*e + 9*f + 9*g + 3*h) + d*(e + 3*f - 17*g + 7*h))))*Power(y,2) 116 117 + Power(a - 3*b + 3*c - d,3)*Power(y,3) 118 119 + Power(x,2)*(-3*(-9*b*e*f*f + 9*a*f*f*f + 6*b*e*e*g - 9*a*e*f*g + 27*b*e*f*g - 27*a*f*f*g + 18*a*e*g*g - 54*b*e*g*g + 120 27*a*f*g*g + 27*b*f*g*g - 18*a*g*g*g + a*e*e*h - 9*b*e*e*h + 3*a*e*f*h + 9*b*e*f*h + 9*a*f*f*h - 121 18*b*f*f*h - 21*a*e*g*h + 51*b*e*g*h - 9*a*f*g*h - 27*b*f*g*h + 18*a*g*g*h + 7*a*e*h*h - 18*b*e*h*h - 3*a*f*h*h + 122 18*b*f*h*h - 6*a*g*h*h - 3*b*g*h*h + a*h*h*h + 123 3*c*(-9*f*f*(g - 2*h) + 3*g*g*h - f*h*(9*g + 2*h) + e*e*(f - 6*g + 6*h) + 124 e*(9*f*g + 6*g*g - 17*f*h - 3*g*h + 3*h*h)) - 125 d*(e*e*e + e*e*(-6*f - 3*g + 7*h) - 9*(2*f - g)*(f*f + g*g - f*(g + h)) + 126 e*(18*f*f + 9*g*g + 3*g*h + h*h - 3*f*(3*g + 7*h)))) ) 127 128 + Power(x,2)*(3*(a - 3*b + 3*c - d)*Power(e - 3*f + 3*g - h,2)*y) 129 130 + x*(-3*(27*b*b*e*g*g - 27*a*b*f*g*g + 9*a*a*g*g*g - 18*b*b*e*f*h + 18*a*b*f*f*h + 3*a*b*e*g*h - 131 27*b*b*e*g*h - 9*a*a*f*g*h + 27*a*b*f*g*h - 9*a*a*g*g*h + a*a*e*h*h - 9*a*b*e*h*h + 132 27*b*b*e*h*h + 6*a*a*f*h*h - 18*a*b*f*h*h - 9*b*b*f*h*h + 3*a*a*g*h*h + 133 6*a*b*g*h*h - a*a*h*h*h + 9*c*c*(e*e*(g - 3*h) - 3*f*f*h + e*(3*f + 2*g)*h) + 134 d*d*(e*e*e - 9*f*f*f + 9*e*f*(f + g) - e*e*(3*f + 6*g + h)) + 135 d*(-3*c*(-9*f*f*g + e*e*(2*f - 6*g - 3*h) + e*(9*f*g + 6*g*g + f*h)) + 136 a*(-18*f*f*f - 18*e*g*g + 18*g*g*g - 2*e*e*h + 3*e*g*h + 2*e*h*h + 9*f*f*(3*g + 2*h) + 137 3*f*(6*e*g - 9*g*g - e*h - 6*g*h)) - 3*b*(9*f*g*g + e*e*(4*g - 3*h) - 6*f*f*h - 138 e*(6*f*f + g*(18*g + h) - 3*f*(3*g + 4*h)))) + 139 3*c*(3*b*(e*e*h + 3*f*g*h - e*(3*f*g - 6*f*h + 6*g*h + h*h)) + 140 a*(9*f*f*(g - 2*h) + f*h*(-e + 9*g + 4*h) - 3*(2*g*g*h + e*(2*g*g - 4*g*h + h*h))))) ) 141 142 + x*3*(-2*a*d*e*e - 7*d*d*e*e + 15*a*d*e*f + 21*d*d*e*f - 9*a*d*f*f - 18*d*d*f*f - 15*a*d*e*g - 143 3*d*d*e*g - 9*a*a*f*g + 9*d*d*f*g + 18*a*a*g*g + 9*a*d*g*g + 2*a*a*e*h - 2*d*d*e*h + 144 3*a*a*f*h + 15*a*d*f*h - 21*a*a*g*h - 15*a*d*g*h + 7*a*a*h*h + 2*a*d*h*h - 145 9*c*c*(2*e*e + 3*f*f + 3*f*h - 2*g*h + e*(-3*f - 4*g + h)) + 146 9*b*b*(3*g*g - 3*g*h + 2*h*(-2*f + h) + e*(-2*f + 3*g + h)) + 147 3*b*(3*c*(e*e + 3*e*(f - 3*g) + (9*f - 3*g - h)*h) + a*(6*f*f + e*g - 9*f*g - 9*g*g - 5*e*h + 9*f*h + 14*g*h - 7*h*h) + 148 d*(-e*e + 12*f*f - 27*f*g + e*(-9*f + 20*g - 5*h) + g*(9*g + h))) + 149 3*c*(a*(-(e*f) - 9*f*f + 27*f*g - 12*g*g + 5*e*h - 20*f*h + 9*g*h + h*h) + 150 d*(7*e*e + 9*f*f + 9*f*g - 6*g*g - f*h + e*(-14*f - 9*g + 5*h))))*y 151 152 - x*3*Power(a - 3*b + 3*c - d,2)*(e - 3*f + 3*g - h)*Power(y,2) 153 154 */ 155 156 const int factors = 8; 157 158 struct coeff { 159 int s; // constant and coefficient sign 160 int n[factors]; // 0 or power of a (1, 2, or 3) for a through h 161 }; 162 163 enum { 164 xxx_coeff, 165 xxy_coeff, 166 xyy_coeff, 167 yyy_coeff, 168 xx_coeff, 169 xy_coeff, 170 yy_coeff, 171 x_coeff, 172 y_coeff, 173 c_coeff, 174 coeff_count 175 }; 176 177 typedef std::vector<coeff> coeffs; 178 typedef std::vector<coeffs> n_coeffs; 179 180 static char skipSpace(const char* str, size_t& index) { 181 do { 182 ++index; 183 } while (str[index] == ' '); 184 return str[index]; 185 } 186 187 static char backSkipSpace(const char* str, size_t& end) { 188 while (str[end - 1] == ' ') { 189 --end; 190 } 191 return str[end - 1]; 192 } 193 194 static void match(const char* str, size_t len, coeffs& co, const char pattern[]) { 195 size_t patternLen = strlen(pattern); 196 size_t index = 0; 197 while (index < len) { 198 char ch = str[index]; 199 if (ch != '-' && ch != '+') { 200 printf("missing sign\n"); 201 } 202 size_t end = index + 1; 203 while (str[end] != '+' && str[end] != '-' && ++end < len) { 204 ; 205 } 206 backSkipSpace(str, end); 207 size_t idx = index; 208 index = end; 209 skipSpace(str, index); 210 if (!strncmp(&str[end - patternLen], pattern, patternLen) == 0) { 211 continue; 212 } 213 size_t endCoeff = end - patternLen; 214 char last = backSkipSpace(str, endCoeff); 215 if (last == '2' || last == '3') { 216 last = str[endCoeff - 3]; // skip ^2 217 } 218 if (last == 'x' || last == 'y') { 219 continue; 220 } 221 coeff c; 222 c.s = str[idx] == '-' ? -1 : 1; 223 bzero(c.n, sizeof(c.n)); 224 ch = skipSpace(str, idx); 225 if (ch >= '2' && ch <= '6') { 226 c.s *= ch - '0'; 227 ch = skipSpace(str, idx); 228 } 229 while (idx < endCoeff) { 230 char x = str[idx]; 231 if (x < 'a' || x > 'a' + factors) { 232 printf("expected factor\n"); 233 } 234 idx++; 235 int pow = 1; 236 if (str[idx] == '^') { 237 idx++; 238 char exp = str[idx]; 239 if (exp < '2' || exp > '3') { 240 printf("expected exponent\n"); 241 } 242 pow = exp - '0'; 243 } 244 skipSpace(str, idx); 245 c.n[x - 'a'] = pow; 246 } 247 co.push_back(c); 248 } 249 } 250 251 void cubecode_test(int test); 252 253 void cubecode_test(int test) { 254 const char* str = test ? result2 : result1; 255 size_t len = strlen(str); 256 n_coeffs c(coeff_count); 257 match(str, len, c[xxx_coeff], "x^3"); // 1 factor 258 match(str, len, c[xxy_coeff], "x^2 y"); // 1 factor 259 match(str, len, c[xyy_coeff], "x y^2"); // 1 factor 260 match(str, len, c[yyy_coeff], "y^3"); // 1 factor 261 match(str, len, c[xx_coeff], "x^2"); // 7 factors 262 match(str, len, c[xy_coeff], "x y"); // 8 factors 263 match(str, len, c[yy_coeff], "y^2"); // 7 factors 264 match(str, len, c[x_coeff], "x"); // 21 factors 265 match(str, len, c[y_coeff], "y"); // 21 factors 266 match(str, len, c[c_coeff], ""); // 34 factors : total 102 267 #define COMPUTE_MOST_FREQUENT_EXPRESSION_TRIPLETS 0 268 #define WRITE_AS_NONOPTIMIZED_C_CODE 0 269 #if COMPUTE_MOST_FREQUENT_EXPRESSION_TRIPLETS 270 int count[factors][factors][factors]; 271 bzero(count, sizeof(count)); 272 #endif 273 #if WRITE_AS_NONOPTIMIZED_C_CODE 274 printf("// start of generated code"); 275 #endif 276 for (n_coeffs::iterator it = c.begin(); it < c.end(); ++it) { 277 coeffs& co = *it; 278 #if WRITE_AS_NONOPTIMIZED_C_CODE 279 printf("\nstatic double calc_%c(double a, double b, double c, double d," 280 "\n double e, double f, double g, double h) {" 281 "\n return" 282 "\n ", 'A' + (it - c.begin())); 283 if (co[0].s > 0) { 284 printf(" "); 285 } 286 if (abs(co[0].s) == 1) { 287 printf(" "); 288 } 289 #endif 290 for (coeffs::iterator ct = co.begin(); ct < co.end(); ++ct) { 291 const coeff& cf = *ct; 292 #if WRITE_AS_NONOPTIMIZED_C_CODE 293 printf(" "); 294 bool firstFactor = false; 295 if (ct - co.begin() > 0 || cf.s < 0) { 296 printf("%c", cf.s < 0 ? '-' : '+'); 297 } 298 if (ct - co.begin() > 0) { 299 printf(" "); 300 } 301 if (abs(cf.s) > 1) { 302 printf("%d * ", abs(cf.s)); 303 } else { 304 if (ct - co.begin() > 0) { 305 printf(" "); 306 } 307 } 308 #endif 309 for (int x = 0; x < factors; ++x) { 310 if (cf.n[x] == 0) { 311 continue; 312 } 313 #if WRITE_AS_NONOPTIMIZED_C_CODE 314 for (int y = 0 ; y < cf.n[x]; ++y) { 315 if (y > 0 || firstFactor) { 316 printf(" * "); 317 } 318 printf("%c", 'a' + x); 319 } 320 firstFactor = true; 321 #endif 322 #if COMPUTE_MOST_FREQUENT_EXPRESSION_TRIPLETS 323 for (int y = x; y < factors; ++y) { 324 if (cf.n[y] == 0) { 325 continue; 326 } 327 if (x == y && cf.n[y] == 1) { 328 continue; 329 } 330 for (int z = y; z < factors; ++z) { 331 if (cf.n[z] == 0) { 332 continue; 333 } 334 if ((x == z || y == z) && cf.n[z] == 1) { 335 continue; 336 } 337 if (x == y && y == z && cf.n[z] == 2) { 338 continue; 339 } 340 count[x][y][z]++; 341 } 342 } 343 #endif 344 } 345 #if WRITE_AS_NONOPTIMIZED_C_CODE 346 if (ct + 1 < co.end()) { 347 printf("\n"); 348 } 349 #endif 350 } 351 #if WRITE_AS_NONOPTIMIZED_C_CODE 352 printf(";\n}\n"); 353 #endif 354 } 355 #if WRITE_AS_NONOPTIMIZED_C_CODE 356 printf("// end of generated code\n"); 357 #endif 358 #if COMPUTE_MOST_FREQUENT_EXPRESSION_TRIPLETS 359 const int bestCount = 20; 360 int best[bestCount][4]; 361 bzero(best, sizeof(best)); 362 for (int x = 0; x < factors; ++x) { 363 for (int y = x; y < factors; ++y) { 364 for (int z = y; z < factors; ++z) { 365 if (!count[x][y][z]) { 366 continue; 367 } 368 for (int w = 0; w < bestCount; ++w) { 369 if (best[w][0] < count[x][y][z]) { 370 best[w][0] = count[x][y][z]; 371 best[w][1] = x; 372 best[w][2] = y; 373 best[w][3] = z; 374 break; 375 } 376 } 377 } 378 } 379 } 380 for (int w = 0; w < bestCount; ++w) { 381 printf("%c%c%c=%d\n", 'a' + best[w][1], 'a' + best[w][2], 382 'a' + best[w][3], best[w][0]); 383 } 384 #endif 385 #if WRITE_AS_NONOPTIMIZED_C_CODE 386 printf("\n"); 387 #endif 388 } 389 390 /* results: variable triplets used 10 or more times: 391 aah=14 392 ade=14 393 aeh=14 394 dee=14 395 bce=13 396 beg=13 397 beh=12 398 bbe=11 399 bef=11 400 cee=11 401 cef=11 402 def=11 403 ceh=10 404 deg=10 405 */ 406
