Changeset 900 for trunk/src/trig.cpp
 Timestamp:
 Sep 4, 2011, 2:28:56 PM (8 years ago)
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 1 edited
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trunk/src/trig.cpp
r899 r900 44 44 static const double SC[] = 45 45 { 46 1.6449340668482264364724e0, // pi^2/3!47 +8.1174242528335364363700e1, // pi^4/5!48 1.9075182412208421369647e1, // pi^6/7!49 +2.6147847817654800504653e2, // pi^8/9!50 2.3460810354558236375089e3, // pi^10/11!51 +1.4842879303107100368487e4, // pi^12/13!52 6.9758736616563804745344e6, // pi^14/15!53 +2.5312174041370276513517e7, // pi^16/17!46 1.6449340668482264364724e0, // π^2/3! 47 +8.1174242528335364363700e1, // π^4/5! 48 1.9075182412208421369647e1, // π^6/7! 49 +2.6147847817654800504653e2, // π^8/9! 50 2.3460810354558236375089e3, // π^10/11! 51 +1.4842879303107100368487e4, // π^12/13! 52 6.9758736616563804745344e6, // π^14/15! 53 +2.5312174041370276513517e7, // π^16/17! 54 54 }; 55 55 56 /* Note: the last value should be 1.3878952462213772114468e7 (ie. 57 * π^18/18!) but we tweak it in order to get the better average precision 58 * required for tan() computations when close to π/2+kπ values. */ 56 59 static const double CC[] = 57 60 { 58 4.9348022005446793094172e0, // pi^2/2! 59 +4.0587121264167682181850e0, // pi^4/4! 60 1.3352627688545894958753e0, // pi^6/6! 61 +2.3533063035889320454188e1, // pi^8/8! 62 2.5806891390014060012598e2, // pi^10/10! 63 +1.9295743094039230479033e3, // pi^12/12! 64 1.0463810492484570711802e4, // pi^14/14! 65 +4.3030695870329470072978e6, // pi^16/16! 61 4.9348022005446793094172e0, // π^2/2! 62 +4.0587121264167682181850e0, // π^4/4! 63 1.3352627688545894958753e0, // π^6/6! 64 +2.3533063035889320454188e1, // π^8/8! 65 2.5806891390014060012598e2, // π^10/10! 66 +1.9295743094039230479033e3, // π^12/12! 67 1.0463810492484570711802e4, // π^14/14! 68 +4.3030695870329470072978e6, // π^16/16! 69 1.3777e7, 66 70 }; 67 71 68 72 /* These coefficients use Sloane’s http://oeis.org/A002430 and 69 73 * http://oeis.org/A036279 sequences for the Taylor series of tan(). 70 * Note: the last value should be 443861162*pi^18/1856156927625,ie.71 * 2.12485922978838540352881e5, but we tweak it in order to get72 * sub 1e11 precision in a larger range. */74 * Note: the last value should be 2.12485922978838540352881e5 (ie. 75 * 443861162*π^18/1856156927625), but we tweak it in order to get 76 * sub 1e11 average precision in a larger range. */ 73 77 static const double TC[] = 74 78 { 75 3.28986813369645287294483e0, // pi^2/3 76 1.29878788045336582981920e1, // 2*pi^4/15 77 5.18844961612069061254404e1, // 17*pi^6/315 78 2.07509320280908496804928e2, // 62*pi^8/2835 79 8.30024701695986756361561e2, // 1382*pi^10/155925 80 3.32009324029001216460018e3, // 21844*pi^12/6081075 81 1.32803704909665483598490e4, // 929569*pi^14/638512875 82 5.31214808666037709352112e4, // 6404582*pi^16/10854718875 83 2.373e5, // XXX: last value tweaked to improve precision 84 //2.12485922978838540352881e5, // 443861162*pi^18/1856156927625 79 3.28986813369645287294483e0, // π^2/3 80 1.29878788045336582981920e1, // 2*π^4/15 81 5.18844961612069061254404e1, // 17*π^6/315 82 2.07509320280908496804928e2, // 62*π^8/2835 83 8.30024701695986756361561e2, // 1382*π^10/155925 84 3.32009324029001216460018e3, // 21844*π^12/6081075 85 1.32803704909665483598490e4, // 929569*π^14/638512875 86 5.31214808666037709352112e4, // 6404582*π^16/10854718875 87 2.373e5, 85 88 }; 86 89 … … 344 347 double x4 = x2 * x2; 345 348 346 double subs1 = (SC[3] * x4 + SC[1]) * x4 + ONE; 349 /* Computing the Taylor series to the 11th order is enough to get 350 * x * 1e11 precision, but we push it to the 13th order so that 351 * tan() has a better precision. */ 352 double subs1 = ((SC[5] * x4 + SC[3]) * x4 + SC[1]) * x4 + ONE; 347 353 double subs2 = (SC[4] * x4 + SC[2]) * x4 + SC[0]; 348 354 double taylors = subs2 * x2 + subs1; … … 407 413 double x4 = x2 * x2; 408 414 #if defined LOL_FEATURE_VERY_CHEAP_BRANCHES 409 double subs1 = ( SC[3]* x4 + SC[1]) * x4 + ONE;415 double subs1 = ((CC[5] * x4 + SC[3]) * x4 + SC[1]) * x4 + ONE; 410 416 double subs2 = (SC[4] * x4 + SC[2]) * x4 + SC[0]; 411 417 double subc1 = ((CC[5] * x4 + CC[3]) * x4 + CC[1]) * x4 + ONE; … … 414 420 double subs1 = (((SC[7] * x4 + SC[5]) * x4 + SC[3]) * x4 + SC[1]) * x4 + ONE; 415 421 double subs2 = ((SC[6] * x4 + SC[4]) * x4 + SC[2]) * x4 + SC[0]; 422 /* Push Taylor series to the 19th order to enhance tan() accuracy. */ 416 423 double subc1 = (((CC[7] * x4 + CC[5]) * x4 + CC[3]) * x4 + CC[1]) * x4 + ONE; 417 double subc2 = (( CC[6]* x4 + CC[4]) * x4 + CC[2]) * x4 + CC[0];424 double subc2 = (((CC[8] * x4 + CC[6]) * x4 + CC[4]) * x4 + CC[2]) * x4 + CC[0]; 418 425 #endif 419 426 double taylors = subs2 * x2 + subs1;
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