From 326e303175ab700d59602f0e344e8d84e15f1aa8 Mon Sep 17 00:00:00 2001 From: Erik Faye-Lund Date: Fri, 26 Sep 2014 18:11:19 +0200 Subject: [PATCH] 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 Reviewed-by: Ian Romanick --- src/glsl/builtin_functions.cpp | 65 +++++++++++++++++++++++++++++++++++------- 1 file changed, 55 insertions(+), 10 deletions(-) diff --git a/src/glsl/builtin_functions.cpp b/src/glsl/builtin_functions.cpp index 7d61fcc..bb7fbcd 100644 --- a/src/glsl/builtin_functions.cpp +++ b/src/glsl/builtin_functions.cpp @@ -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; } -- 2.7.4