system/embdrv/lc3/src/fastmath.h

/******************************************************************************
 *
 *  Copyright 2022 Google LLC
 *
 *  Licensed under the Apache License, Version 2.0 (the "License");
 *  you may not use this file except in compliance with the License.
 *  You may obtain a copy of the License at:
 *
 *  http://www.apache.org/licenses/LICENSE-2.0
 *
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS,
 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
 *  limitations under the License.
 *
 ******************************************************************************/

/**
 * LC3 - Mathematics function approximation
 */

#ifndef __LC3_FASTMATH_H
#define __LC3_FASTMATH_H

#include <stdint.h>
#include <math.h>


/**
 * Fast 2^n approximation
 * x               Operand, range -8 to 8
 * return          2^x approximation (max relative error ~ 7e-6)
 */
static inline float fast_exp2f(float x)
{
    float y;

    /* --- Polynomial approx in range -0.5 to 0.5 --- */

    static const float c[] = { 1.27191277e-09, 1.47415221e-07,
        1.35510312e-05, 9.38375815e-04, 4.33216946e-02 };

    y = (    c[0]) * x;
    y = (y + c[1]) * x;
    y = (y + c[2]) * x;
    y = (y + c[3]) * x;
    y = (y + c[4]) * x;
    y = (y + 1.f);

    /* --- Raise to the power of 16  --- */

    y = y*y;
    y = y*y;
    y = y*y;
    y = y*y;

    return y;
}

/**
 * Fast log2(x) approximation
 * x               Operand, greater than 0
 * return          log2(x) approximation (max absolute error ~ 1e-4)
 */
static inline float fast_log2f(float x)
{
    float y;
    int e;

    /* --- Polynomial approx in range 0.5 to 1 --- */

    static const float c[] = {
        -1.29479677, 5.11769018, -8.42295281, 8.10557963, -3.50567360 };

    x = frexpf(x, &e);

    y = (    c[0]) * x;
    y = (y + c[1]) * x;
    y = (y + c[2]) * x;
    y = (y + c[3]) * x;
    y = (y + c[4]);

    /* --- Add log2f(2^e) and return --- */

    return e + y;
}

/**
 * Fast log10(x) approximation
 * x               Operand, greater than 0
 * return          log10(x) approximation (max absolute error ~ 1e-4)
 */
static inline float fast_log10f(float x)
{
    return log10f(2) * fast_log2f(x);
}

/**
 * Fast `10 * log10(x)` (or dB) approximation in fixed Q16
 * x               Operand, in range 2^-63 to 2^63 (1e-19 to 1e19)
 * return          10 * log10(x) in fixed Q16 (-190 to 192 dB)
 *
 * - The 0 value is accepted and return the minimum value ~ -191dB
 * - This function assumed that float 32 bits is coded IEEE 754
 */
static inline int32_t fast_db_q16(float x)
{
    /* --- Table in Q15 --- */

    static const uint16_t t[][2] = {

        /* [n][0] = 10 * log10(2) * log2(1 + n/32), with n = [0..15]     */
        /* [n][1] = [n+1][0] - [n][0] (while defining [16][0])           */

        {     0, 4379 }, {  4379, 4248 }, {  8627, 4125 }, { 12753, 4009 },
        { 16762, 3899 }, { 20661, 3795 }, { 24456, 3697 }, { 28153, 3603 },
        { 31755, 3514 }, { 35269, 3429 }, { 38699, 3349 }, { 42047, 3272 },
        { 45319, 3198 }, { 48517, 3128 }, { 51645, 3061 }, { 54705, 2996 },

        /* [n][0] = 10 * log10(2) * log2(1 + n/32) - 10 * log10(2) / 2,  */
        /*     with n = [16..31]                                         */
        /* [n][1] = [n+1][0] - [n][0] (while defining [32][0])           */

        {  8381, 2934 }, { 11315, 2875 }, { 14190, 2818 }, { 17008, 2763 },
        { 19772, 2711 }, { 22482, 2660 }, { 25142, 2611 }, { 27754, 2564 },
        { 30318, 2519 }, { 32837, 2475 }, { 35312, 2433 }, { 37744, 2392 },
        { 40136, 2352 }, { 42489, 2314 }, { 44803, 2277 }, { 47080, 2241 },

    };

    /* --- Approximation ---
     *
     *   10 * log10(x^2) = 10 * log10(2) * log2(x^2)
     *
     *   And log2(x^2) = 2 * log2( (1 + m) * 2^e )
     *                 = 2 * (e + log2(1 + m)) , with m in range [0..1]
     *
     * Split the float values in :
     *   e2  Double value of the exponent (2 * e + k)
     *   hi  High 5 bits of mantissa, for precalculated result `t[hi][0]`
     *   lo  Low 16 bits of mantissa, for linear interpolation `t[hi][1]`
     *
     * Two cases, from the range of the mantissa :
     *   0 to 0.5   `k = 0`, use 1st part of the table
     *   0.5 to 1   `k = 1`, use 2nd part of the table  */

    union { float f; uint32_t u; } x2 = { .f = x*x };

    int e2 = (int)(x2.u >> 22) - 2*127;
    int hi = (x2.u >> 18) & 0x1f;
    int lo = (x2.u >>  2) & 0xffff;

    return e2 * 49321 + t[hi][0] + ((t[hi][1] * lo) >> 16);
}


#endif /* __LC3_FASTMATH_H */