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
skipSpace(const char * str,size_t & index)180 static char skipSpace(const char* str, size_t& index) {
181 do {
182 ++index;
183 } while (str[index] == ' ');
184 return str[index];
185 }
186
backSkipSpace(const char * str,size_t & end)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
match(const char * str,size_t len,coeffs & co,const char pattern[])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
cubecode_test(int test)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