Change _cairo_fixed_from_double to use the "magic number" technique

See long thread here:
http://lists.freedesktop.org/archives/cairo/2006-October/008285.html

This patch provides a 3x performance improvement (on x86) for the
conversion of floating-point to fixed-point values as measured by
the recent pattern_create_radial performance test:

image-rgba      pattern_create_radial-16     8.98 3.36% ->   2.97 1.03%:  3.38x speedup
██▍
image-rgb       pattern_create_radial-16     8.94 3.21% ->   2.97 0.18%:  3.36x speedup
██▍
 xlib-rgb       pattern_create_radial-16     9.55 3.17% ->   3.64 0.51%:  2.93x speedup
█▉
 xlib-rgba      pattern_create_radial-16     9.63 3.53% ->   3.69 0.66%:  2.91x speedup
█▉

1 files changed, 47 insertions, 1 deletions

diff --git a/src/cairo-fixed.c b/src/cairo-fixed.c
index 604c9e729..fe6c2dc3e 100644
--- a/src/cairo-fixed.c
+++ b/src/cairo-fixed.c
@@ -42,10 +42,56 @@ _cairo_fixed_from_int (int i)
     return i << 16;
 }
 
+/* This is the "magic number" approach to converting a double into fixed
+ * point as described here:
+ *
+ * http://www.stereopsis.com/sree/fpu2006.html (an overview)
+ * http://www.d6.com/users/checker/pdfs/gdmfp.pdf (in detail)
+ *
+ * The basic idea is to add a large enough number to the double that the
+ * literal floating point is moved up to the extent that it forces the
+ * double's value to be shifted down to the bottom of the mantissa (to make
+ * room for the large number being added in). Since the mantissa is, at a
+ * given moment in time, a fixed point integer itself, one can convert a
+ * float to various fixed point representations by moving around the point
+ * of a floating point number through arithmetic operations. This behavior
+ * is reliable on most modern platforms as it is mandated by the IEEE-754
+ * standard for floating point arithmetic.
+ *
+ * For our purposes, a "magic number" must be carefully selected that is
+ * both large enough to produce the desired point-shifting effect, and also
+ * has no lower bits in its representation that would interfere with our
+ * value at the bottom of the mantissa. The magic number is calculated as
+ * follows:
+ *
+ *          (2 ^ (MANTISSA_SIZE - FRACTIONAL_SIZE)) * 1.5
+ *
+ * where in our case:
+ *  - MANTISSA_SIZE for 64-bit doubles is 52
+ *  - FRACTIONAL_SIZE for 16.16 fixed point is 16
+ *
+ * Although this approach provides a very large speedup of this function
+ * on a wide-array of systems, it does come with two caveats:
+ *
+ * 1) It uses banker's rounding as opposed to arithmetic rounding.
+ * 2) It doesn't function properly if the FPU is in single-precision
+ *    mode.
+ */
+#define CAIRO_MAGIC_NUMBER_FIXED_16_16 (103079215104.0)
 cairo_fixed_t
 _cairo_fixed_from_double (double d)
 {
-    return (cairo_fixed_t) floor (d * 65536 + 0.5);
+    union {
+        double d;
+        int32_t i[2];
+    } u;
+
+    u.d = d + CAIRO_MAGIC_NUMBER_FIXED_16_16;
+#ifdef FLOAT_WORDS_BIGENDIAN
+    return u.i[1];
+#else
+    return u.i[0];
+#endif
 }
 
 cairo_fixed_t