[Mesa-dev] [PATCH 5/8] glsl: Rewrite atan2 implementation to fix accuracy and handling of zero/infinity.

[Ian Romanick]

It's a real bummer that we have two implementations of this function
that are basically written in assembly... I'm not sure what else you'd
call generating IR by hand.  The code review and maintenance costs are
of the same magnitude for sure.

We could move this to GLSL and let the standalone compiler generate the
builder code.  I don't think that is currently helpful.  However, for
future "soft" int64 and fp64 work the standalone compiler will need to
be extended to also generate NIR builder.  Once that is done, I think
the cost-benefit analysis changes.

On 01/24/2017 03:26 PM, Francisco Jerez wrote:
> This addresses several issues of the current atan2 implementation:
> 
>  - Negative zero (and negative denorms which end up getting flushed to
>    zero) isn't handled correctly by the current implementation.  The
>    reason is that it does 'y >= 0' and 'x < 0' comparisons to decide
>    on which side of the branch cut the argument is, which causes us to
>    return incorrect results (off by up to 2π) for very small negative
>    values.
> 
>  - There is a serious precision problem for x values of large enough
>    magnitude introduced by the floating point division operation being
>    implemented as a mul+rcp sequence.  This can lead to the quotient
>    getting flushed to zero in some cases introducing an error of over
>    8e6 ULP in the result -- Or in the most catastrophic case will
>    cause us to return NaN instead of the correct value ±π/2 for y=±∞
>    and x very large.  We can fix this easily by scaling down both
>    arguments when the absolute value of the denominator goes above
>    certain threshold.  The error of this atan2 implementation remains
>    below 25 ULP in most of its domain except for a neighborhood of y=0
>    where it reaches a maximum error of about 180 ULP.
> 
>  - It emits a bunch of instructions including no less than three
>    if-else branches per scalar component that don't seem to get
>    optimized out later on.  This implementation uses about 13% less
>    instructions on Intel SKL hardware and doesn't emit any control
>    flow instructions.
> ---
>  src/compiler/glsl/builtin_functions.cpp | 82 ++++++++++++++++++---------------
>  1 file changed, 46 insertions(+), 36 deletions(-)
> 
> diff --git a/src/compiler/glsl/builtin_functions.cpp b/src/compiler/glsl/builtin_functions.cpp
> index 4a6c5af..fd59381 100644
> --- a/src/compiler/glsl/builtin_functions.cpp
> +++ b/src/compiler/glsl/builtin_functions.cpp
> @@ -3560,44 +3560,54 @@ builtin_builder::_acos(const glsl_type *type)
>  ir_function_signature *
>  builtin_builder::_atan2(const glsl_type *type)
>  {
> -   ir_variable *vec_y = in_var(type, "vec_y");
> -   ir_variable *vec_x = in_var(type, "vec_x");
> -   MAKE_SIG(type, always_available, 2, vec_y, vec_x);
> -
> -   ir_variable *vec_result = body.make_temp(type, "vec_result");
> -   ir_variable *r = body.make_temp(glsl_type::float_type, "r");
> -   for (int i = 0; i < type->vector_elements; i++) {
> -      ir_variable *y = body.make_temp(glsl_type::float_type, "y");
> -      ir_variable *x = body.make_temp(glsl_type::float_type, "x");
> -      body.emit(assign(y, swizzle(vec_y, i, 1)));
> -      body.emit(assign(x, swizzle(vec_x, i, 1)));
> -
> -      /* If |x| >= 1.0e-8 * |y|: */
> -      ir_if *outer_if =
> -         new(mem_ctx) ir_if(greater(abs(x), mul(imm(1.0e-8f), abs(y))));
> -
> -      ir_factory outer_then(&outer_if->then_instructions, mem_ctx);
> -
> -      /* Then...call atan(y/x) */
> -      do_atan(outer_then, 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)));
> -      inner_if->then_instructions.push_tail(
> -         if_tree(gequal(y, imm(0.0f)),
> -                 assign(r, add(r, imm(M_PIf))),
> -                 assign(r, sub(r, imm(M_PIf)))));
> -      outer_then.emit(inner_if);
> -
> -      /* Else... */
> -      outer_if->else_instructions.push_tail(
> -         assign(r, mul(sign(y), imm(M_PI_2f))));
> +   const unsigned n = type->vector_elements;
> +   ir_variable *y = in_var(type, "y");
> +   ir_variable *x = in_var(type, "x");
> +   MAKE_SIG(type, always_available, 2, y, x);
>  
> -      body.emit(outer_if);
> +   /* If we're on the left half-plane rotate the coordinates π/2 clock-wise
> +    * for the y=0 discontinuity to end up aligned with the vertical
> +    * discontinuity of atan(s/t) along t=0.
> +    */
> +   ir_variable *flip = body.make_temp(glsl_type::bvec(n), "flip");
> +   body.emit(assign(flip, less(x, imm(0.0f, n))));
> +   ir_variable *s = body.make_temp(type, "s");
> +   body.emit(assign(s, csel(flip, abs(x), y)));
> +   ir_variable *t = body.make_temp(type, "t");
> +   body.emit(assign(t, csel(flip, y, abs(x))));
>  
> -      body.emit(assign(vec_result, r, 1 << i));
> -   }
> -   body.emit(ret(vec_result));
> +   /* If the magnitude of the denominator exceeds some huge value, scale down
> +    * the arguments in order to prevent the reciprocal operation from flushing
> +    * its result to zero, which would cause precision problems, and for s
> +    * infinite would cause us to return a NaN instead of the correct finite
> +    * value.
> +    */
> +   ir_constant *huge = imm(1e37f, n);
> +   ir_variable *scale = body.make_temp(type, "scale");
> +   body.emit(assign(scale, csel(gequal(abs(t), huge),
> +                                imm(0.0625f, n), imm(1.0f, n))));

Out of curiosity, how did you arrive at 1/16 for the scale?  I wonder if
r300-era GPUs would want a different huge-val and scale.  I'm pretty
sure r300 still only used 24-bit floats.  I think all the old NVIDIA and
Intel GPUs used 32-bit... I don't think there are any Mesa drivers for
other weird GPUs of that era.

> +   ir_variable *rcp_scaled_t = body.make_temp(type, "rcp_scaled_t");
> +   body.emit(assign(rcp_scaled_t, rcp(mul(t, scale))));
> +   ir_expression *s_over_t = mul(mul(s, scale), rcp_scaled_t);

If I'm reading this right,

    s_over_t = (s * scale) * rcp(t * scale);

I want to make sure we have a specific test case that will fail if
someone tries to add an algebraic optimization that tries to remove the
'* scale'.

> +
> +   /* Calculate the arctangent and fix up the result if we had flipped the
> +    * coordinate system.
> +    */
> +   ir_variable *arc = body.make_temp(type, "arc");
> +   do_atan(body, type, arc, abs(s_over_t));
> +   body.emit(assign(arc, add(arc, mul(b2f(flip), imm(M_PI_2f)))));

I'll spare the details of the deep rat hole I went down with algebraic
optimizations related to things like  b ? x : x + y vs x + float(b) * y.
 In this particular case, I think x + csel(flip, M_PI_2, 0) is
marginally better because the csel can be scheduled early and flip
killed.  Eh... but a csel with two immediate values usually results in
dumb code.  I'm already heading back down the rat hole... never mind.

> +
> +   /* Rather convoluted calculation of the sign of the result.  When x < 0 we
> +    * cannot use fsign because we need to be able to distinguish between
> +    * negative and positive zero.  Unfortunately we cannot use bitwise
> +    * arithmetic tricks either because of back-ends without integer support.
> +    * When x >= 0 rcp_scaled_t will always be non-negative so this won't be
> +    * able to distinguish between negative and positive zero, but we don't
> +    * care because atan2 is continuous along the whole positive y = 0
> +    * half-line, so it won't affect the result.
> +    */
> +   body.emit(ret(csel(less(min2(y, rcp_scaled_t), imm(0.0f, n)),
> +                      neg(arc), arc)));

FWIW, b ? -x : x vs. -float(b) * x was part of the previous rat hole.

>  
>     return sig;
>  }
>