14.1. Fast Math Functions Test

14.1.1. Overview

This example demonstrates the usage and performance of the fast math functions implemented in the HPM Math middleware. The implementation provides optimized trigonometric and inverse trigonometric functions with configurable trade-offs between speed and accuracy.

14.1.2. Supported Functions

1. Sin/Cos Functions (lookup table based)

   • hpm_math_fast_sinf / hpm_math_fast_cosf: No interpolation (fastest)

   • hpm_math_fast_sinf_linear / hpm_math_fast_cosf_linear: Linear interpolation

   • hpm_math_fast_sinf_quadratic / hpm_math_fast_cosf_quadratic: Quadratic interpolation (most accurate)

2. SinCos Functions (optimized for simultaneous sin+cos calculation)

   • hpm_math_fast_sincosf_linear: Compute sin and cos together with linear interpolation

   • hpm_math_fast_sincosf_quadratic: Compute sin and cos together with quadratic interpolation

   • Much faster than calling sinf and cosf separately (shares angle normalization and index calculation)

3. Tan Functions (using optimized sincosf)

   • hpm_math_fast_tanf_linear: Linear interpolation

   • hpm_math_fast_tanf_quadratic: Quadratic interpolation

   • hpm_math_fast_tanf_hybrid: Hybrid method (highest accuracy, ~2.5x improvement)

     • Uses polynomial approximation near singularities to avoid division error amplification

     • Uses sin/cos division elsewhere for best overall accuracy

4. Atan/Atan2 Functions (polynomial with range reduction)

   • hpm_math_fast_atanf: Fast arctangent with range reduction

   • hpm_math_fast_atan2f: Fast two-argument arctangent

14.1.3. Test Items

The test includes the following items:

Performance Test

Measures execution time (in CPU ticks) for each function over 1000 iterations.

1. Sin Functions

   • sinf (stdlib): Standard library sine function

   • hpm_math_fast_sinf: Fast sine without interpolation

   • hpm_math_fast_sinf_linear: Fast sine with linear interpolation

   • hpm_math_fast_sinf_quadratic: Fast sine with quadratic interpolation

2. Cos Functions

   • cosf (stdlib): Standard library cosine function

   • hpm_math_fast_cosf: Fast cosine without interpolation

   • hpm_math_fast_cosf_linear: Fast cosine with linear interpolation

   • hpm_math_fast_cosf_quadratic: Fast cosine with quadratic interpolation

3. SinCos Functions

   • hpm_math_fast_sincosf_linear: Simultaneous sin+cos with linear interpolation

   • hpm_math_fast_sincosf_quadratic: Simultaneous sin+cos with quadratic interpolation

4. Tan Functions

   • tanf (stdlib): Standard library tangent function

   • hpm_math_fast_tanf_linear: Fast tangent with linear interpolation

   • hpm_math_fast_tanf_quadratic: Fast tangent with quadratic interpolation

   • hpm_math_fast_tanf_hybrid: Fast tangent with hybrid method

5. Atan Functions

   • atanf (stdlib): Standard library arctangent function

   • hpm_math_fast_atanf: Fast arctangent with range reduction

6. Atan2 Functions

   • atan2f (stdlib): Standard library two-argument arctangent

   • hpm_math_fast_atan2f: Fast two-argument arctangent

Accuracy Test

Measures maximum and average error compared to standard library over millions of test points.

1. Sin Accuracy: Tests 6,530,344 points in [0, 2π] with 1e-6 rad step

2. Cos Accuracy: Tests 6,530,344 points in [0, 2π] with 1e-6 rad step

3. Tan Accuracy: Tests 6,110,914 points in [0, 2π] avoiding singularities (|x - π/2| > 0.1)

4. Atan Accuracy: Tests 199,874 points in [-10, 10] with 0.0001 step

5. Atan2 Accuracy: Tests 6,284 points using unit circle coordinates

14.1.4. Performance Results

The following results were obtained on HPM6E00EVK at 600MHz (Table size: 4096).

Note

Performance may vary depending on the chip model, compiler optimization level, cache configuration, and other system settings. These results are for reference only.

Sin Functions (ticks per 1000 operations)

Function	Total	Per Op
sinf (stdlib)	104053	104
hpm_math_fast_sinf	14118	14
hpm_math_fast_sinf_linear	26442	26
hpm_math_fast_sinf_quadratic	39815	40

Cos Functions (ticks per 1000 operations)

Function	Total	Per Op
cosf (stdlib)	106108	106
hpm_math_fast_cosf	18059	18
hpm_math_fast_cosf_linear	29426	29
hpm_math_fast_cosf_quadratic	43418	43

SinCos Functions (ticks per 1000 operations)

Function	Total	Per Op
hpm_math_fast_sincosf_linear	33480	33
hpm_math_fast_sincosf_quadratic	47801	48

Tan Functions (ticks per 1000 operations)

Function	Total	Per Op
tanf (stdlib)	144019	144
hpm_math_fast_tanf_linear	51416	51
hpm_math_fast_tanf_quadratic	65778	66
hpm_math_fast_tanf_hybrid	105441	105

Atan/Atan2 Functions (ticks per 1000 operations)

Function	Total	Per Op
atanf (stdlib)	178337	178
hpm_math_fast_atanf	58052	58
atan2f (stdlib)	218747	218
hpm_math_fast_atan2f	57391	57

14.1.5. Accuracy Results

Sin Accuracy (6,530,344 test points)

Method	Max Error	Avg Error
No interpolation	1.53e-03	4.84e-04
Linear interpolation	5.36e-07	1.46e-07
Quadratic interpolation	5.59e-07	6.41e-08

Cos Accuracy (6,530,344 test points)

Method	Max Error	Avg Error
No interpolation	1.53e-03	4.87e-04
Linear interpolation	6.56e-07	1.60e-07
Quadratic interpolation	5.36e-07	7.61e-08

Tan Accuracy (6,110,914 test points, avoiding singularities)

Method	Max Error	Avg Error
Linear interpolation	3.91e-05	7.53e-07
Quadratic interpolation	3.91e-05	7.56e-07
Hybrid method	1.53e-05	6.17e-07

Atan Accuracy (199,874 test points, range [-10, 10])

Method	Max Error	Avg Error
Fast (range reduction)	5.01e-06	2.49e-07

Atan2 Accuracy (6,284 test points)

Method	Max Error	Avg Error
Fast (range reduction)	5.01e-06	3.88e-07

14.1.6. Test Configuration

Error thresholds (configurable via macros):

• Sin/Cos no interpolation: 1e-2

• Sin/Cos linear interpolation: 1e-4

• Sin/Cos quadratic interpolation: 1e-5

• Tan linear: 1e-3

• Tan quadratic: 1e-4

• Tan hybrid: 2e-5

• Atan/Atan2: 1e-4

14.1.7. Usage

The example automatically runs comprehensive tests and displays results on the debug console. No user interaction is required.

14.1.8. License

BSD-3-Clause