Methods and apparatuses for computing trigonometric functions with high precision

1. A method implemented by a GPU (Graphics Processing Unit) for computing trigonometric functions in three-dimensional graphics Application Programming Interfaces (APIs), comprising:

computing, by a range reduction unit of an ALU (Arithmetic Logic Unit), a remainder r and a reduction value x* corresponding to an input parameter x;

communicating the remainder r to an SFU (Special Function Unit);

computing, by the SFU, an intermediate function f(x*) corresponding to the reduction value x*;

communicating the intermediate function f(x*) to the ALU; and

computing, by a post-computation unit of the ALU, a multiplication of the reduction value x* by the intermediate function f(x*) as a computation result of a trigonometric function,

wherein the reduction value x* is computed by an Equation:

x* =π/2* frc ( x* 2/π).

2. The method of claim 1 , wherein the step for computing, by a range reduction unit of an ALU (Arithmetic Logic Unit), a remainder r and a reduction value x* corresponding to an input parameter x, further comprises:

computing the remainder r corresponding to the input parameter x using an Equation:

r=frc ( x *2/π);

determining whether a host requests that cos(x) be computed;

when the host requests that cos(x) be computed, adjusting the remainder r using an Equation:

r= 1.0 −r ; and

computing the reduction value x* using an Equation:

x*=π /2 *r.

3. The method of claim 1 , wherein the step for computing, by an SFU (Special Function Unit), an intermediate function f(x*) corresponding to the reduction value x*, further comprises:

obtaining coefficients C 0 , C 1 and C 2 of a second-degree polynomial from a LUT (Look-Up Table) according to a first magnitude X 1 of the remainder r as an index;

obtaining a second magnitude X 2 of the remainder r; and

computing the intermediate function f(x*) using an Equation:

f ( x* )= C 0 +C 1 *X 2 +C 2 *X 2 2 .

4. The method of claim 3 , wherein the intermediate function f(x*) is a result of Sin(x*)/x* or Cos(x*)/x*.

5. The method of claim 3 , wherein the first magnitude X 1 is 7 bits and the second magnitude X 2 is 24 bits.

6. The method of claim 5 , wherein the LUT comprises 128 records and each record stores the coefficients C 0 , C 1 and C 2 of the second-degree polynomial associated with one of binary values from “0b0000000” to “0b1111111”.

7. An apparatus in a GPU (Graphics Processing Unit) for computing trigonometric functions in three-dimensional graphics Application Programming Interfaces (APIs), comprising:

an ALU (Arithmetic Logic Unit); and

an SFU (Special Function Unit), coupled to the ALU,

wherein the ALU includes range reduction circuitry configured to compute a remainder r and a reduction value x* corresponding to an input parameter x; the SFU includes second-degree-polynomial computation circuitry configured to compute an intermediate function f(x*) corresponding to the reduction value x*; and the ALU includes post-computation circuitry configured to compute a multiplication of the reduction value x* by the intermediate function f(x*) as a computation result of a trigonometric function,

wherein the reduction value x* is computed by an Equation:

x* =π/2* frc ( x* 2/π).

8. The apparatus of claim 7 , wherein the ALU is configured to compute the remainder r corresponding to the input parameter x using an Equation: r=frc(x*2/π); determines whether a host requests that cos(x) be computed; when the host requests that cos(x) be computed, adjusts the remainder r using an Equation: r=1.0−r; and computes the reduction value x* using an Equation: x*=π/2*r.

9. The apparatus of claim 8 , wherein the SFU further comprises:

a volatile memory device, coupled to the ALU, storing the remainder r;

a non-volatile memory device, storing a LUT (Look-Up Table); and

wherein the SFU obtains coefficients C 0 , C 1 and C 2 of a second-degree polynomial from the LUT according to a first magnitude X 1 of the remainder r as an index; obtains the coefficients C 0 , C 1 and C 2 of the second-degree polynomial; obtains a second magnitude X 2 of the remainder r from the volatile memory device; obtains a square of the second magnitude X 2 from the square computation unit; and computes the intermediate function f(x*) using an Equation: f(x*)=C 0 +C 1 *X 2 +C 2 *X 2 2 .

10. The apparatus of claim 9 , wherein the intermediate function f(x*) is a result of Sin(x*)/x* or Cos(x*)/x*.

11. The apparatus of claim 9 , wherein the first magnitude X 1 is 7 bits and the second magnitude X 2 is 24 bits.

12. The apparatus of claim 11 , wherein the LUT comprises 128 records and each record stores the coefficients C 0 , C 1 and C 2 of the second-degree polynomial associated with one of binary values from “0b0000000” to “0b1111111”.