以下代码对使用相同浮点数进行除法和后续乘法的行为进行实证分析。
在没有舍入的情况下查看程序的输出时，这种行为对我来说似乎有点武断。我期待的回应不那么激动。我现在的问题是：任何人都可以制定某种解释结果的规则，或者提供其他一些深刻的见解，也可以用于四舍五入的情况。
我知道this post，这与b = 1的特殊情况有关。
为简单起见，我建议限制IEEE-754 32位单精度float（以及32位unsigned int的答案，这在这里并不重要。）
我是如何看到这个主题的：假设有一个8位灰度图像（在这种情况下为unsigned char而不是unsigned int）并将其转换为32位float图像，从而除以b = 255.0f，以便将{0,1，...，255}映射到区间[0,1]。现在当通过乘以255.0f转换回{0,1，...，255}时，是否会遇到不精确的问题？根据我的实证分析，答案是否定的（参见下面没有舍入的输出，1023超过255），但是对于252.0f，答案是肯定的。当然，可以通过添加0.5f来处理这个问题，以便在转换回unsigned char时进行舍入 - 对于一般情况应该这样做，因为通常会转换回浮点图像的处理版本而不是未被触及的。不过，我开始想知道，对于未触及的情况，是否可以省略为舍入添加0.5f。
#include <limits.h>
#include <stdio.h>
unsigned int foo(float b) {
unsigned int a;
for (a = 0; a < UINT_MAX; a++) {
unsigned int a_ = (float) a / b * b; // without rounding
//unsigned int a_ = (float) a / b * b + 0.5f; // with rounding
if (a_ != a) {
break;
}
}
return a;
}
int main() {
for (unsigned int b = 1; b < 256; b++) {
printf("%u:%u, ", b, foo((float) b));
}
return 0;
}
没有舍入的输出：
1:16777217, 2:16777217, 3:12582913, 4:16777217, 5:10485763, 6:12582913, 7:31, 8:16777217, 9:9437189, 10:10485763, 11:13, 12:12582913, 13:53, 14:31, 15:63, 16:16777217, 17:8912905, 18:9437189, 19:27, 20:10485763, 21:57, 22:13, 23:7, 24:12582913, 25:53, 26:53, 27:29, 28:31, 29:15, 30:63, 31:997, 32:16777217, 33:8650769, 34:8912905, 35:103, 36:9437189, 37:3, 38:27, 39:107, 40:10485763, 41:1, 42:57, 43:15, 44:13, 45:113, 46:7, 47:1, 48:12582913, 49:27, 50:53, 51:119, 52:53, 53:15, 54:29, 55:1, 56:31, 57:63, 58:15, 59:31, 60:63, 61:1, 62:997, 63:255, 64:16777217, 65:8519713, 66:8650769, 67:55, 68:8912905, 69:15, 70:103, 71:13, 72:9437189, 73:45, 74:3, 75:49, 76:27, 77:29, 78:107, 79:3, 80:10485763, 81:11, 82:1, 83:1, 84:57, 85:217, 86:15, 87:27, 88:13, 89:27, 90:113, 91:223, 92:7, 93:49, 94:1, 95:15, 96:12582913, 97:1, 98:27, 99:7, 100:53, 101:115, 102:119, 103:29, 104:53, 105:237, 106:15, 107:1, 108:29, 109:1, 110:1, 111:15, 112:31, 113:29, 114:63, 115:1, 116:15, 117:249, 118:31, 119:251, 120:63, 121:1, 122:1, 123:1, 124:997, 125:251, 126:255, 127:2037, 128:16777217, 129:8454209, 130:8519713, 131:21, 132:8650769, 133:5, 134:55, 135:7, 136:8912905, 137:47, 138:15, 139:9, 140:103, 141:1, 142:13, 143:41, 144:9437189, 145:73, 146:45, 147:11, 148:3, 149:3, 150:49, 151:31, 152:27, 153:413, 154:29, 155:23, 156:107, 157:161, 158:3, 159:1, 160:10485763, 161:41, 162:11, 163:31, 164:1, 165:1, 166:1, 167:45, 168:57, 169:27, 170:217, 171:23, 172:15, 173:45, 174:27, 175:29, 176:13, 177:7, 178:27, 179:25, 180:113, 181:187, 182:223, 183:3, 184:7, 185:15, 186:49, 187:59, 188:1, 189:1, 190:15, 191:3, 192:12582913, 193:13, 194:1, 195:455, 196:27, 197:13, 198:7, 199:57, 200:53, 201:7, 202:115, 203:27, 204:119, 205:15, 206:29, 207:7, 208:53, 209:1, 210:237, 211:53, 212:15, 213:7, 214:1, 215:59, 216:29, 217:59, 218:1, 219:1, 220:1, 221:481, 222:15, 223:15, 224:31, 225:61, 226:29, 227:1, 228:63, 229:29, 230:1, 231:29, 232:15, 233:121, 234:249, 235:1, 236:31, 237:1, 238:251, 239:121, 240:63, 241:501, 242:1, 243:1, 244:1, 245:1, 246:1, 247:989, 248:997, 249:1, 250:251, 251:253, 252:255, 253:507, 254:2037, 255:1023
舍入输出：
1:8388609, 2:8388609, 3:8388609, 4:8388609, 5:8388609, 6:8388609, 7:7340036, 8:8388609, 9:8388609, 10:8388609, 11:5767178, 12:8388609, 13:6815751, 14:7340036, 15:7864328, 16:8388609, 17:8388609, 18:8388609, 19:4980747, 20:8388609, 21:5505025, 22:5767178, 23:6029316, 24:8388609, 25:6553602, 26:6815751, 27:7077891, 28:7340036, 29:7602181, 30:7864328, 31:8126480, 32:8388609, 33:8388609, 34:8388609, 35:4587552, 36:8388609, 37:4849688, 38:4980747, 39:5111812, 40:8388609, 41:5373953, 42:5505025, 43:5636130, 44:5767178, 45:5898246, 46:6029316, 47:6160391, 48:8388609, 49:6422530, 50:6553602, 51:6684674, 52:6815751, 53:6946819, 54:7077891, 55:7208964, 56:7340036, 57:7471109, 58:7602181, 59:7733254, 60:7864328, 61:7995403, 62:8126480, 63:8257568, 64:8388609, 65:8388609, 66:8388609, 67:4390951, 68:8388609, 69:4522008, 70:4587552, 71:4653094, 72:8388609, 73:4784130, 74:4849688, 75:4915205, 76:4980747, 77:5046294, 78:5111812, 79:5177353, 80:8388609, 81:5308417, 82:5373953, 83:5439489, 84:5505025, 85:5570561, 86:5636130, 87:5701646, 88:5767178, 89:5832710, 90:5898246, 91:5963780, 92:6029316, 93:6094852, 94:6160391, 95:6225933, 96:8388609, 97:6356994, 98:6422530, 99:6488066, 100:6553602, 101:6619138, 102:6684674, 103:6750219, 104:6815751, 105:6881283, 106:6946819, 107:7012355, 108:7077891, 109:7143427, 110:7208964, 111:7274500, 112:7340036, 113:7405572, 114:7471109, 115:7536645, 116:7602181, 117:7667718, 118:7733254, 119:7798791, 120:7864328, 121:7929865, 122:7995403, 123:8060941, 124:8126480, 125:8192021, 126:8257568, 127:8323136, 128:8388609, 129:8388609, 130:8388609, 131:4292728, 132:8388609, 133:4358217, 134:4390951, 135:4423704, 136:8388609, 137:4489235, 138:4522008, 139:4554755, 140:4587552, 141:4620296, 142:4653094, 143:4685831, 144:8388609, 145:4751362, 146:4784130, 147:4816921, 148:4849688, 149:4882455, 150:4915205, 151:4947976, 152:4980747, 153:5013530, 154:5046294, 155:5079047, 156:5111812, 157:5144580, 158:5177353, 159:5210126, 160:8388609, 161:5275649, 162:5308417, 163:5341185, 164:5373953, 165:5406721, 166:5439489, 167:5472257, 168:5505025, 169:5537793, 170:5570561, 171:5603458, 172:5636130, 173:5668884, 174:5701646, 175:5734410, 176:5767178, 177:5799944, 178:5832710, 179:5865478, 180:5898246, 181:5931019, 182:5963780, 183:5996548, 184:6029316, 185:6062084, 186:6094852, 187:6127623, 188:6160391, 189:6193162, 190:6225933, 191:6258713, 192:8388609, 193:6324226, 194:6356994, 195:6389762, 196:6422530, 197:6455298, 198:6488066, 199:6520834, 200:6553602, 201:6586370, 202:6619138, 203:6651906, 204:6684674, 205:6717495, 206:6750219, 207:6782983, 208:6815751, 209:6848515, 210:6881283, 211:6914051, 212:6946819, 213:6979587, 214:7012355, 215:7045123, 216:7077891, 217:7110659, 218:7143427, 219:7176195, 220:7208964, 221:7241732, 222:7274500, 223:7307268, 224:7340036, 225:7372804, 226:7405572, 227:7438340, 228:7471109, 229:7503877, 230:7536645, 231:7569413, 232:7602181, 233:7634950, 234:7667718, 235:7700486, 236:7733254, 237:7766023, 238:7798791, 239:7831560, 240:7864328, 241:7897097, 242:7929865, 243:7962634, 244:7995403, 245:8028172, 246:8060941, 247:8093710, 248:8126480, 249:8159250, 250:8192021, 251:8224794, 252:8257568, 253:8290347, 254:8323136, 255:8355968
答案 0 :(得分：1)
仍然，我开始怀疑是否可以为未触及的情况省略为舍入添加0.5f。
使用b==255解决OP的部分问题。 （OP的原始问题）
灰度转换使用固定/常数分频器和固定乘法器是有道理的。不要使用相同的精确值，而是使用略有不同的值。虽然这不是＆＃34;相同的浮点数＆＃34;，但我认为它符合OP的最初目标。
使用否 a == a_，代码可以获得预期的匹配+ 0.5f最多8,355,967。
float b1 = 255.0f;
float b2 = nextafterf(255.0f, 256.0f);
unsigned int a_ = (unsigned int) ((float) a / b1 * b2);
对于1 <= b <= 255，最低a在b == 129 /