Mulhigh generic

[albinahlback]

On Skylake with GCC 13.2.1 (cutoff-point for when to go to full multiplication is 330):

Click to expand (n from 1 to 340)

         mul_n / mulhigh || mpfr / flint
n =   1:      1.00x      ||     3.19
n =   2:      1.00x      ||     3.61
n =   3:      1.20x      ||     3.31
n =   4:      1.19x      ||     3.04
n =   5:      1.29x      ||     2.90
n =   6:      1.36x      ||     2.63
n =   7:      1.43x      ||     2.47
n =   8:      1.49x      ||     2.31
n =   9:      1.70x      ||     2.11
n =  10:      1.75x      ||     1.99
n =  11:      1.91x      ||     1.96
n =  12:      1.81x      ||     1.77
n =  13:      1.28x      ||     1.17
n =  14:      1.32x      ||     1.16
n =  15:      1.20x      ||     1.14
n =  16:      1.26x      ||     1.13
n =  17:      1.40x      ||     1.13
n =  18:      1.35x      ||     1.11
n =  19:      1.41x      ||     1.09
n =  20:      1.37x      ||     1.09
n =  21:      1.43x      ||     1.08
n =  22:      1.38x      ||     1.06
n =  23:      1.44x      ||     1.07
n =  24:      1.40x      ||     1.07
n =  25:      1.44x      ||     1.07
n =  26:      1.38x      ||     1.06
n =  27:      1.43x      ||     1.07
n =  28:      1.40x      ||     1.07
n =  29:      1.34x      ||     1.06
n =  30:      1.26x      ||     1.06
n =  31:      1.35x      ||     1.06
n =  32:      1.29x      ||     1.05
n =  33:      1.38x      ||     1.05
n =  34:      1.38x      ||     1.04
n =  35:      1.35x      ||     1.05
n =  36:      1.31x      ||     1.05
n =  37:      1.39x      ||     1.05
n =  38:      1.36x      ||     1.04
n =  39:      1.36x      ||     1.07
n =  40:      1.29x      ||     1.03
n =  41:      1.35x      ||     1.14
n =  42:      1.35x      ||     1.10
n =  43:      1.39x      ||     1.09
n =  44:      1.33x      ||     1.09
n =  45:      1.39x      ||     1.10
n =  46:      1.36x      ||     1.09
n =  47:      1.37x      ||     1.09
n =  48:      1.35x      ||     1.09
n =  49:      1.35x      ||     1.09
n =  50:      1.32x      ||     1.05
n =  51:      1.34x      ||     1.07
n =  52:      1.31x      ||     1.06
n =  53:      1.35x      ||     1.07
n =  54:      1.34x      ||     1.07
n =  55:      1.35x      ||     1.07
n =  56:      1.27x      ||     1.09
n =  57:      1.28x      ||     1.05
n =  58:      1.23x      ||     1.11
n =  59:      1.22x      ||     1.07
n =  60:      1.20x      ||     1.14
n =  61:      1.19x      ||     1.08
n =  62:      1.21x      ||     1.14
n =  63:      1.16x      ||     1.06
n =  64:      1.21x      ||     1.14
n =  65:      1.19x      ||     1.07
n =  66:      1.22x      ||     1.07
n =  67:      1.22x      ||     1.05
n =  68:      1.21x      ||     1.03
n =  69:      1.20x      ||     1.07
n =  70:      1.24x      ||     1.05
n =  71:      1.20x      ||     1.05
n =  72:      1.20x      ||     1.07
n =  73:      1.21x      ||     1.04
n =  74:      1.28x      ||     1.06
n =  75:      1.23x      ||     1.00
n =  76:      1.24x      ||     1.03
n =  77:      1.25x      ||     1.03
n =  78:      1.25x      ||     1.01
n =  79:      1.23x      ||     1.01
n =  80:      1.23x      ||     1.03
n =  81:      1.23x      ||     0.99
n =  82:      1.24x      ||     1.00
n =  83:      1.25x      ||     0.98
n =  84:      1.25x      ||     0.99
n =  85:      1.29x      ||     0.99
n =  86:      1.29x      ||     0.98
n =  87:      1.24x      ||     0.99
n =  88:      1.23x      ||     1.01
n =  89:      1.26x      ||     0.98
n =  90:      1.27x      ||     0.99
n =  91:      1.25x      ||     0.97
n =  92:      1.26x      ||     1.00
n =  93:      1.23x      ||     0.96
n =  94:      1.22x      ||     0.96
n =  95:      1.25x      ||     0.98
n =  96:      1.23x      ||     0.99
n =  97:      1.20x      ||     0.95
n =  98:      1.22x      ||     0.94
n =  99:      1.22x      ||     0.95
n = 100:      1.23x      ||     0.96
n = 101:      1.22x      ||     0.92
n = 102:      1.21x      ||     0.98
n = 103:      1.22x      ||     0.98
n = 104:      1.21x      ||     0.96
n = 105:      1.20x      ||     0.97
n = 106:      1.22x      ||     1.02
n = 107:      1.22x      ||     0.98
n = 108:      1.24x      ||     0.99
n = 109:      1.22x      ||     0.97
n = 110:      1.21x      ||     0.98
n = 111:      1.24x      ||     1.00
n = 112:      1.22x      ||     1.00
n = 113:      1.18x      ||     0.99
n = 114:      1.20x      ||     1.02
n = 115:      1.18x      ||     0.99
n = 116:      1.19x      ||     1.00
n = 117:      1.17x      ||     0.99
n = 118:      1.16x      ||     1.01
n = 119:      1.16x      ||     1.02
n = 120:      1.17x      ||     1.06
n = 121:      1.13x      ||     0.99
n = 122:      1.12x      ||     0.96
n = 123:      1.14x      ||     0.97
n = 124:      1.16x      ||     1.05
n = 125:      1.16x      ||     0.96
n = 126:      1.16x      ||     0.98
n = 127:      1.17x      ||     1.03
n = 128:      1.15x      ||     1.05
n = 129:      1.15x      ||     0.98
n = 130:      1.18x      ||     1.00
n = 131:      1.18x      ||     0.97
n = 132:      1.19x      ||     0.99
n = 133:      1.18x      ||     0.98
n = 134:      1.18x      ||     0.95
n = 135:      1.21x      ||     0.98
n = 136:      1.19x      ||     0.97
n = 137:      1.18x      ||     0.97
n = 138:      1.13x      ||     0.93
n = 139:      1.18x      ||     0.97
n = 140:      1.20x      ||     0.98
n = 141:      1.17x      ||     0.99
n = 142:      1.19x      ||     0.98
n = 143:      1.18x      ||     1.01
n = 144:      1.17x      ||     0.98
n = 145:      1.17x      ||     0.97
n = 146:      1.19x      ||     0.97
n = 147:      1.20x      ||     0.97
n = 148:      1.21x      ||     0.97
n = 149:      1.19x      ||     0.98
n = 150:      1.19x      ||     0.97
n = 151:      1.17x      ||     0.96
n = 152:      1.19x      ||     0.97
n = 153:      1.20x      ||     0.94
n = 154:      1.19x      ||     0.96
n = 155:      1.21x      ||     0.96
n = 156:      1.21x      ||     0.95
n = 157:      1.19x      ||     0.96
n = 158:      1.18x      ||     0.96
n = 159:      1.19x      ||     0.97
n = 160:      1.18x      ||     0.99
n = 161:      1.18x      ||     0.94
n = 162:      1.17x      ||     0.98
n = 163:      1.18x      ||     0.93
n = 164:      1.19x      ||     0.97
n = 165:      1.17x      ||     0.95
n = 166:      1.19x      ||     0.94
n = 167:      1.20x      ||     0.97
n = 168:      1.18x      ||     0.96
n = 169:      1.19x      ||     0.94
n = 170:      1.19x      ||     0.94
n = 171:      1.20x      ||     0.94
n = 172:      1.19x      ||     0.96
n = 173:      1.19x      ||     0.97
n = 174:      1.15x      ||     0.94
n = 175:      1.17x      ||     0.97
n = 176:      1.17x      ||     0.98
n = 177:      1.16x      ||     0.96
n = 178:      1.16x      ||     0.95
n = 179:      1.18x      ||     0.95
n = 180:      1.19x      ||     0.96
n = 181:      1.18x      ||     0.96
n = 182:      1.17x      ||     0.96
n = 183:      1.19x      ||     0.95
n = 184:      1.19x      ||     0.98
n = 185:      1.17x      ||     0.95
n = 186:      1.15x      ||     0.92
n = 187:      1.16x      ||     0.93
n = 188:      1.14x      ||     0.94
n = 189:      1.17x      ||     0.94
n = 190:      1.18x      ||     0.96
n = 191:      1.17x      ||     0.99
n = 192:      1.16x      ||     0.99
n = 193:      1.16x      ||     0.96
n = 194:      1.16x      ||     0.95
n = 195:      1.17x      ||     0.96
n = 196:      1.15x      ||     0.96
n = 197:      1.16x      ||     0.96
n = 198:      1.17x      ||     0.97
n = 199:      1.18x      ||     0.99
n = 200:      1.16x      ||     0.98
n = 201:      1.17x      ||     0.97
n = 202:      1.16x      ||     0.95
n = 203:      1.18x      ||     0.96
n = 204:      1.20x      ||     1.00
n = 205:      1.18x      ||     1.01
n = 206:      1.17x      ||     1.01
n = 207:      1.18x      ||     1.00
n = 208:      1.18x      ||     1.03
n = 209:      1.14x      ||     0.96
n = 210:      1.17x      ||     0.95
n = 211:      1.17x      ||     0.95
n = 212:      1.16x      ||     0.98
n = 213:      1.17x      ||     0.97
n = 214:      1.16x      ||     0.96
n = 215:      1.16x      ||     0.93
n = 216:      1.16x      ||     0.98
n = 217:      1.15x      ||     0.90
n = 218:      1.15x      ||     0.89
n = 219:      1.16x      ||     0.90
n = 220:      1.17x      ||     0.92
n = 221:      1.17x      ||     0.93
n = 222:      1.18x      ||     0.94
n = 223:      1.17x      ||     0.93
n = 224:      1.16x      ||     0.96
n = 225:      1.15x      ||     0.94
n = 226:      1.13x      ||     0.92
n = 227:      1.15x      ||     0.93
n = 228:      1.16x      ||     0.94
n = 229:      1.15x      ||     0.96
n = 230:      1.16x      ||     0.96
n = 231:      1.20x      ||     0.97
n = 232:      1.20x      ||     0.97
n = 233:      1.19x      ||     0.99
n = 234:      1.17x      ||     0.97
n = 235:      1.18x      ||     0.97
n = 236:      1.19x      ||     0.97
n = 237:      1.23x      ||     0.99
n = 238:      1.22x      ||     0.99
n = 239:      1.22x      ||     1.00
n = 240:      1.25x      ||     1.01
n = 241:      1.22x      ||     0.97
n = 242:      1.16x      ||     0.94
n = 243:      1.16x      ||     0.95
n = 244:      1.15x      ||     0.96
n = 245:      1.17x      ||     0.95
n = 246:      1.15x      ||     0.93
n = 247:      1.16x      ||     0.95
n = 248:      1.18x      ||     0.96
n = 249:      1.14x      ||     0.93
n = 250:      1.12x      ||     0.92
n = 251:      1.13x      ||     0.92
n = 252:      1.12x      ||     0.93
n = 253:      1.16x      ||     0.93
n = 254:      1.16x      ||     0.93
n = 255:      1.16x      ||     0.96
n = 256:      1.18x      ||     0.97
n = 257:      1.13x      ||     0.92
n = 258:      1.11x      ||     0.90
n = 259:      1.11x      ||     0.91
n = 260:      1.09x      ||     0.90
n = 261:      1.11x      ||     0.91
n = 262:      1.10x      ||     0.91
n = 263:      1.11x      ||     0.90
n = 264:      1.12x      ||     0.91
n = 265:      1.08x      ||     0.88
n = 266:      1.06x      ||     0.88
n = 267:      1.09x      ||     0.89
n = 268:      1.07x      ||     0.88
n = 269:      1.10x      ||     0.89
n = 270:      1.10x      ||     0.90
n = 271:      1.09x      ||     0.89
n = 272:      1.10x      ||     0.90
n = 273:      1.12x      ||     0.88
n = 274:      1.11x      ||     0.88
n = 275:      1.09x      ||     0.88
n = 276:      1.11x      ||     0.89
n = 277:      1.10x      ||     0.88
n = 278:      1.12x      ||     0.88
n = 279:      1.12x      ||     0.89
n = 280:      1.12x      ||     0.92
n = 281:      1.10x      ||     0.88
n = 282:      1.11x      ||     0.93
n = 283:      1.09x      ||     0.92
n = 284:      1.11x      ||     0.93
n = 285:      1.13x      ||     0.93
n = 286:      1.12x      ||     0.92
n = 287:      1.13x      ||     0.93
n = 288:      1.11x      ||     0.92
n = 289:      1.12x      ||     0.91
n = 290:      1.10x      ||     0.90
n = 291:      1.10x      ||     0.91
n = 292:      1.10x      ||     0.90
n = 293:      1.09x      ||     0.89
n = 294:      1.07x      ||     0.88
n = 295:      1.08x      ||     0.90
n = 296:      1.08x      ||     0.90
n = 297:      1.08x      ||     0.88
n = 298:      1.07x      ||     0.89
n = 299:      1.08x      ||     0.90
n = 300:      1.09x      ||     0.89
n = 301:      1.09x      ||     0.89
n = 302:      1.08x      ||     0.88
n = 303:      1.07x      ||     0.89
n = 304:      1.06x      ||     0.89
n = 305:      1.05x      ||     0.88
n = 306:      1.05x      ||     0.87
n = 307:      1.05x      ||     0.86
n = 308:      1.07x      ||     0.87
n = 309:      1.07x      ||     0.87
n = 310:      1.06x      ||     0.86
n = 311:      1.05x      ||     0.87
n = 312:      1.06x      ||     0.88
n = 313:      1.08x      ||     0.87
n = 314:      1.07x      ||     0.87
n = 315:      1.07x      ||     0.90
n = 316:      1.08x      ||     0.90
n = 317:      1.07x      ||     0.90
n = 318:      1.06x      ||     0.90
n = 319:      1.11x      ||     0.90
n = 320:      1.12x      ||     0.90
n = 321:      1.08x      ||     0.89
n = 322:      1.06x      ||     0.87
n = 323:      1.04x      ||     0.87
n = 324:      1.04x      ||     0.86
n = 325:      1.03x      ||     0.84
n = 326:      1.03x      ||     0.86
n = 327:      1.02x      ||     0.86
n = 328:      1.01x      ||     0.86
n = 329:      1.00x      ||     0.85
n = 330:      1.00x      ||     0.85
n = 331:      1.00x      ||     0.82
n = 332:      1.00x      ||     0.83
n = 333:      1.00x      ||     0.83
n = 334:      1.00x      ||     0.84
n = 335:      1.00x      ||     0.83
n = 336:      1.00x      ||     0.84
n = 337:      1.00x      ||     0.81
n = 338:      0.99x      ||     0.82
n = 339:      1.00x      ||     0.82
n = 340:      1.00x      ||     0.84

This is currently slower than MPFR above around 80 limbs as the algorithm is not quite the same. In MPFR they utilize the scrap space in rp[0...n-1] which we do not have.

Edit: I also added a function to add mpn-arrays onto another array at the same time using adcx-adox-chains. This is much faster than calling mpn_add_n twice.

[albinahlback]

So, assuming we want this in Arb, it is first above 5000 bits of precision that one need to adjust the algorithm in order to outperform MPFR.

[fredrik-johansson]

This is currently slower than MPFR above around 80 limbs as the algorithm is not quite the same. In MPFR they utilize the scrap space in rp[0...n-1] which we do not have.

Outside of the basecase range we can certainly allocate temporary scrap space too.

[fredrik-johansson]

I can try to implement the MPFR algorithm on top of your basecase to compare.

[albinahlback]

This is currently slower than MPFR above around 80 limbs as the algorithm is not quite the same. In MPFR they utilize the scrap space in rp[0...n-1] which we do not have.

Outside of the basecase range we can certainly allocate temporary scrap space too.

Indeed!

[albinahlback]

Should also set a global preprocessor constant to when full multiplication is used (so one does not need to worry about the error in this case, if that would matter).