core: add a code shortcut for sin() on platforms that have cheap branches.

src/core.h

@@ -16,6 +16,11 @@
 #if !defined __LOL_CORE_H__
 #define __LOL_CORE_H__
 
+// CPU features
+#if !defined __CELLOS_LV2__
+#   define LOL_FEATURE_CHEAP_BRANCHES
+#endif
+
 // Base types
 #include "trig.h"
 #include "half.h"

src/trig.cpp

@@ -23,24 +23,25 @@ using namespace std;
 namespace lol
 {
 
-static const double PI     = 3.14159265358979323846;
-static const double NEG_PI = -3.14159265358979323846;
-static const double PI_2   = PI / 2.0;
-static const double PI_4   = PI / 4.0;
-static const double INV_PI = 1.0 / PI;
-static const double ROOT3  = 1.732050807568877293527;
+static const double PI     = 3.14159265358979323846264;
+static const double NEG_PI = -3.14159265358979323846264;
+static const double PI_2   = 1.57079632679489661923132;
+static const double PI_4   = 0.785398163397448309615661;
+static const double INV_PI = 0.318309886183790671537768;
+static const double ROOT3  = 1.73205080756887729352745;
 
 static const double ZERO    = 0.0;
 static const double ONE     = 1.0;
 static const double NEG_ONE = -1.0;
 static const double HALF    = 0.5;
+static const double QUARTER = 0.25;
 static const double TWO     = 2.0;
 static const double VERY_SMALL_NUMBER = 0x1.0p-128;
 static const double TWO_EXP_52 = 4503599627370496.0;
 static const double TWO_EXP_54 = 18014398509481984.0;
 
 /** sin Taylor series coefficients. */
-static const double SC[]    =
+static const double SC[] =
 {
     -1.6449340668482264364724e-0, // pi^2/3!
     +8.1174242528335364363700e-1, // pi^4/5!
@@ -52,6 +53,18 @@ static const double SC[]    =
     +2.5312174041370276513517e-7, // pi^16/17!
 };
 
+static const double CC[] =
+{
+    -4.9348022005446793094172e-0, // pi^2/2!
+    +4.0587121264167682181850e-0, // pi^4/4!
+    -1.3352627688545894958753e-0, // pi^6/6!
+    +2.3533063035889320454188e-1, // pi^8/8!
+    -2.5806891390014060012598e-2, // pi^10/10!
+    +1.9295743094039230479033e-3, // pi^12/12!
+    -1.0463810492484570711802e-4, // pi^14/14!
+    +4.3030695870329470072978e-6, // pi^16/16!
+};
+
 /* Custom intrinsics */
 #define INLINEATTR __attribute__((always_inline))
 
@@ -167,16 +180,33 @@ static inline double lol_trunc(double x)
 double lol_sin(double x)
 {
     double absx = lol_fabs(x * INV_PI);
-    double sign = lol_fsel(x, PI, NEG_PI);
 
-    /* To compute sin(x) we build a Taylor series for |x|/pi wrapped to
-     * the range [-1, 1]. We also switch the result sign if the number
-     * of cycles is odd. */
+    /* If branches are cheap, skip the cycle count when |x| < π/4,
+     * and only do the Taylor series up to the required precision. */
+#if defined LOL_FEATURE_CHEAP_BRANCHES
+    if (absx < QUARTER)
+    {
+        /* Computing x^4 is one multiplication too many we do, but it helps
+         * interleave the Taylor series operations a lot better. */
+        double x2 = absx * absx;
+        double x4 = x2 * x2;
+        double sub1 = SC[3] * x4 + SC[1];
+        double sub2 = (SC[4] * x4 + SC[2]) * x4 + SC[0];
+        double taylor = (sub1 * x2 + sub2) * x2 + ONE;
+        return x * taylor;
+    }
+#endif
+
+    /* Wrap |x| to the range [-1, 1] and keep track of the number of
+     * cycles required. If odd, we'll need to change the sign of the
+     * result. */
 #if defined __CELLOS_LV2__
+    double sign = lol_fsel(x, PI, NEG_PI);
     double num_cycles = lol_round(absx);
     double is_even = lol_trunc(num_cycles * HALF) - (num_cycles * HALF);
     sign = lol_fsel(is_even, sign, -sign);
 #else
+    double sign = (x >= 0.0) ? PI : NEG_PI;
     double num_cycles = absx + TWO_EXP_52;
     __asm__("" : "+m" (num_cycles)); num_cycles -= TWO_EXP_52;
 
@@ -187,18 +217,36 @@ double lol_sin(double x)
     is_even -= TWO * num_cycles - ONE;
     sign *= is_even;
 #endif
-    double norm_x = absx - num_cycles;
+    absx -= num_cycles;
+
+#if defined LOL_FEATURE_VERY_CHEAP_BRANCHES
+    if (lol_fabs(absx) > QUARTER)
+    {
+        sign = (x * absx >= 0.0) ? is_even : -is_even;
+
+        double k = HALF - lol_fabs(absx);
+        double x2 = k * k;
+        double x4 = x2 * x2;
+        double sub1 = (CC[5] * x4 + CC[3]) * x4 + CC[1];
+        double sub2 = (CC[4] * x4 + CC[2]) * x4 + CC[0];
+        double taylor = (sub1 * x2 + sub2) * x2 + ONE;
+
+        return taylor * sign;
+    }
+#endif
 
-    /* Computing x^4 is one multiplication too many we do, but it helps
-     * interleave the Taylor series operations a lot better. */
-    double x2 = norm_x * norm_x;
+    double x2 = absx * absx;
     double x4 = x2 * x2;
+#if defined LOL_FEATURE_VERY_CHEAP_BRANCHES
+    double sub1 = SC[3] * x4 + SC[1];
+    double sub2 = (SC[4] * x4 + SC[2]) * x4 + SC[0];
+#else
     double sub1 = ((SC[7] * x4 + SC[5]) * x4 + SC[3]) * x4 + SC[1];
     double sub2 = ((SC[6] * x4 + SC[4]) * x4 + SC[2]) * x4 + SC[0];
+#endif
     double taylor = (sub1 * x2 + sub2) * x2 + ONE;
 
-    double result = norm_x * taylor;
-    return result * sign;
+    return absx * taylor * sign;
 }
 
 } /* namespace lol */