[Mesa-dev] [PATCH] glsl: improve the accuracy of the asin() builtin function.

[Paul Berry]

The previous formula for asin(x) was algebraically equivalent to:

sign(x)*(pi/2 - sqrt(1-|x|)*(A + B|x| + C|x|^2))

where A, B, and C were arbitrary constants determined by a curve fit.

This formula had a worst case absolute error of 0.00448, an unbounded
worst case relative error, and a discontinuity near x=0.

Changed the formula to:

sign(x)*(pi/2 - sqrt(1-|x|)*(pi/2 + (pi/4-1)|x| + A|x|^2 + B|x|^3))

where A and B are arbitrary constants determined by a curve fit.  This
has a worst case absolute error of 0.00039, a worst case relative
error of 0.000405, and no discontinuities.

I don't expect a significant performance degradation, since the extra
multiply-accumulate should be fast compared to the sqrt() computation.
---
 src/glsl/builtins/ir/asin |   68 ++++++++++++++++++++++++++------------------
 1 files changed, 40 insertions(+), 28 deletions(-)

diff --git a/src/glsl/builtins/ir/asin b/src/glsl/builtins/ir/asin
index e230ad6..45d9e67 100644
--- a/src/glsl/builtins/ir/asin
+++ b/src/glsl/builtins/ir/asin
@@ -5,23 +5,26 @@
      ((return (expression float *
 	       (expression float sign (var_ref x))
 	       (expression float -
-		(expression float *
-		 (constant float (3.1415926))
-		 (constant float (0.5)))
+		(constant float (1.5707964))
 		(expression float *
 		 (expression float sqrt
 		  (expression float -
 		   (constant float (1.0))
 		   (expression float abs (var_ref x))))
 		 (expression float +
-		  (constant float (1.5707288))
+		  (constant float (1.5707964))
 		  (expression float *
 		   (expression float abs (var_ref x))
 		   (expression float +
-		    (constant float (-0.2121144))
+		    (constant float (-0.21460183))
 		    (expression float *
-		     (constant float (0.0742610))
-		     (expression float abs (var_ref x))))))))))))
+		     (expression float abs (var_ref x))
+                     (expression float +
+                      (constant float (0.086566724))
+                      (expression float *
+                       (expression float abs (var_ref x))
+                       (constant float (-0.03102955))
+		     ))))))))))))
 
    (signature vec2
      (parameters
@@ -29,23 +32,26 @@
      ((return (expression vec2 *
 	       (expression vec2 sign (var_ref x))
 	       (expression vec2 -
-		(expression float *
-		 (constant float (3.1415926))
-		 (constant float (0.5)))
+		(constant float (1.5707964))
 		(expression vec2 *
 		 (expression vec2 sqrt
 		  (expression vec2 -
 		   (constant float (1.0))
 		   (expression vec2 abs (var_ref x))))
 		 (expression vec2 +
-		  (constant float (1.5707288))
+		  (constant float (1.5707964))
 		  (expression vec2 *
 		   (expression vec2 abs (var_ref x))
 		   (expression vec2 +
-		    (constant float (-0.2121144))
+		    (constant float (-0.21460183))
 		    (expression vec2 *
-		     (constant float (0.0742610))
-		     (expression vec2 abs (var_ref x))))))))))))
+		     (expression vec2 abs (var_ref x))
+                     (expression vec2 +
+                      (constant float (0.086566724))
+                      (expression vec2 *
+                       (expression vec2 abs (var_ref x))
+                       (constant float (-0.03102955))
+		     ))))))))))))
 
    (signature vec3
      (parameters
@@ -53,23 +59,26 @@
      ((return (expression vec3 *
 	       (expression vec3 sign (var_ref x))
 	       (expression vec3 -
-		(expression float *
-		 (constant float (3.1415926))
-		 (constant float (0.5)))
+		(constant float (1.5707964))
 		(expression vec3 *
 		 (expression vec3 sqrt
 		  (expression vec3 -
 		   (constant float (1.0))
 		   (expression vec3 abs (var_ref x))))
 		 (expression vec3 +
-		  (constant float (1.5707288))
+		  (constant float (1.5707964))
 		  (expression vec3 *
 		   (expression vec3 abs (var_ref x))
 		   (expression vec3 +
-		    (constant float (-0.2121144))
+		    (constant float (-0.21460183))
 		    (expression vec3 *
-		     (constant float (0.0742610))
-		     (expression vec3 abs (var_ref x))))))))))))
+		     (expression vec3 abs (var_ref x))
+                     (expression vec3 +
+                      (constant float (0.086566724))
+                      (expression vec3 *
+                       (expression vec3 abs (var_ref x))
+                       (constant float (-0.03102955))
+		     ))))))))))))
 
    (signature vec4
      (parameters
@@ -77,21 +86,24 @@
      ((return (expression vec4 *
 	       (expression vec4 sign (var_ref x))
 	       (expression vec4 -
-		(expression float *
-		 (constant float (3.1415926))
-		 (constant float (0.5)))
+		(constant float (1.5707964))
 		(expression vec4 *
 		 (expression vec4 sqrt
 		  (expression vec4 -
 		   (constant float (1.0))
 		   (expression vec4 abs (var_ref x))))
 		 (expression vec4 +
-		  (constant float (1.5707288))
+		  (constant float (1.5707964))
 		  (expression vec4 *
 		   (expression vec4 abs (var_ref x))
 		   (expression vec4 +
-		    (constant float (-0.2121144))
+		    (constant float (-0.21460183))
 		    (expression vec4 *
-		     (constant float (0.0742610))
-		     (expression vec4 abs (var_ref x))))))))))))
+		     (expression vec4 abs (var_ref x))
+                     (expression vec4 +
+                      (constant float (0.086566724))
+                      (expression vec4 *
+                       (expression vec4 abs (var_ref x))
+                       (constant float (-0.03102955))
+		     ))))))))))))
 ))
-- 
1.7.6