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glsl: improve accuracy of atan() 

Our current atan()-approximation is pretty inaccurate at 1.0, so
let's try to improve the situation by doing a direct approximation
without going through atan.

This new implementation uses an 11th degree polynomial to approximate
atan in the [-1..1] range, and the following identitiy to reduce the
entire range to [-1..1]:

atan(x) = 0.5 * pi * sign(x) - atan(1.0 / x)

This range-reduction idea is taken from the paper "Fast computation
of Arctangent Functions for Embedded Applications: A Comparative
Analysis" (Ukil et al. 2011).

The polynomial that approximates atan(x) is:

x   * 0.9999793128310355 - x^3  * 0.3326756418091246 +
x^5 * 0.1938924977115610 - x^7  * 0.1173503194786851 +
x^9 * 0.0536813784310406 - x^11 * 0.0121323213173444

This polynomial was found with the following GNU Octave script:

x = linspace(0, 1);
y = atan(x);
n = [1, 3, 5, 7, 9, 11];
format long;
polyfitc(x, y, n)

The polyfitc function is not built-in, but too long to include here.
It can be downloaded from the following URL:

http://www.mathworks.com/matlabcentral/fileexchange/47851-constraint-polynomial-fit/content/polyfitc.m

This fixes the following piglit test:
shaders/glsl-const-folding-01

Signed-off-by: Erik Faye-Lund <kusmabite@gmail.com>
Reviewed-by: Ian Romanick <ian.d.romanick@intel.com>
This commit is contained in:
Erik Faye-Lund 2014-09-26 18:11:19 +02:00 committed by Olivier Galibert
parent 070b2c2efc
commit 326e303175
1 changed files with 55 additions and 10 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

65
src/glsl/builtin_functions.cpp
View File

@ -442,6 +442,7 @@ private:
ir_swizzle *matrix_elt(ir_variable *var, int col, int row);
ir_expression *asin_expr(ir_variable *x);
void do_atan(ir_factory &body, const glsl_type *type, ir_variable *res, operand y_over_x);
/**
* Call function \param f with parameters specified as the linked
@ -2684,11 +2685,7 @@ builtin_builder::_atan2(const glsl_type *type)
ir_factory outer_then(&outer_if->then_instructions, mem_ctx);
/* Then...call atan(y/x) */
ir_variable *y_over_x = outer_then.make_temp(glsl_type::float_type, "y_over_x");
outer_then.emit(assign(y_over_x, div(y, x)));
outer_then.emit(assign(r, mul(y_over_x, rsq(add(mul(y_over_x, y_over_x),
imm(1.0f))))));
outer_then.emit(assign(r, asin_expr(r)));
do_atan(body, glsl_type::float_type, r, div(y, x));
/* ...and fix it up: */
ir_if *inner_if = new(mem_ctx) ir_if(less(x, imm(0.0f)));
@ -2711,17 +2708,65 @@ builtin_builder::_atan2(const glsl_type *type)
return sig;
}
void
builtin_builder::do_atan(ir_factory &body, const glsl_type *type, ir_variable *res, operand y_over_x)
{
/*
* range-reduction, first step:
*
* / y_over_x if |y_over_x| <= 1.0;
* x = <
* \ 1.0 / y_over_x otherwise
*/
ir_variable *x = body.make_temp(type, "atan_x");
body.emit(assign(x, div(min2(abs(y_over_x),
imm(1.0f)),
max2(abs(y_over_x),
imm(1.0f)))));
/*
* approximate atan by evaluating polynomial:
*
* x * 0.9999793128310355 - x^3 * 0.3326756418091246 +
* x^5 * 0.1938924977115610 - x^7 * 0.1173503194786851 +
* x^9 * 0.0536813784310406 - x^11 * 0.0121323213173444
*/
ir_variable *tmp = body.make_temp(type, "atan_tmp");
body.emit(assign(tmp, mul(x, x)));
body.emit(assign(tmp, mul(add(mul(sub(mul(add(mul(sub(mul(add(mul(imm(-0.0121323213173444f),
tmp),
imm(0.0536813784310406f)),
tmp),
imm(0.1173503194786851f)),
tmp),
imm(0.1938924977115610f)),
tmp),
imm(0.3326756418091246f)),
tmp),
imm(0.9999793128310355f)),
x)));
/* range-reduction fixup */
body.emit(assign(tmp, add(tmp,
mul(b2f(greater(abs(y_over_x),
imm(1.0f, type->components()))),
add(mul(tmp,
imm(-2.0f)),
imm(M_PI_2f))))));
/* sign fixup */
body.emit(assign(res, mul(tmp, sign(y_over_x))));
}
ir_function_signature *
builtin_builder::_atan(const glsl_type *type)
{
ir_variable *y_over_x = in_var(type, "y_over_x");
MAKE_SIG(type, always_available, 1, y_over_x);
ir_variable *t = body.make_temp(type, "t");
body.emit(assign(t, mul(y_over_x, rsq(add(mul(y_over_x, y_over_x),
imm(1.0f))))));
body.emit(ret(asin_expr(t)));
ir_variable *tmp = body.make_temp(type, "tmp");
do_atan(body, type, tmp, y_over_x);
body.emit(ret(tmp));
return sig;
}