doubles_floats_impl.h

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/* Copyright (C) 2013-2016, The Regents of The University of Michigan.
All rights reserved.

This software was developed in the APRIL Robotics Lab under the
direction of Edwin Olson, ebolson@umich.edu. This software may be
available under alternative licensing terms; contact the address above.

This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.

This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see <http://www.gnu.org/licenses/>.
*/

#include <stdio.h>
#include <math.h>
#include <string.h>
#include <float.h>

#include "matd.h"
#include "math_util.h"

// XXX Write unit tests for me!
// XXX Rewrite matd_coords in terms of this.

/*
  This file provides conversions between the following formats:

  quaternion (TNAME[4], { w, x, y, z})

  xyt  (translation in x, y, and rotation in radians.)

  xytcov  (xyt as a TNAME[3] followed by covariance TNAME[9])

  xy, xyz  (translation in x, y, and z)

  mat44 (4x4 rigid-body transformation matrix, row-major
  order. Conventions: We assume points are projected via right
  multiplication. E.g., p' = Mp.) Note: some functions really do rely
  on it being a RIGID, scale=1 transform.

  angleaxis (TNAME[4], { angle-rads, x, y, z }

  xyzrpy (translation x, y, z, euler angles)

  Roll Pitch Yaw are evaluated in the order: roll, pitch, then yaw. I.e.,
  rollPitchYawToMatrix(rpy) = rotateZ(rpy[2]) * rotateY(rpy[1]) * rotateX(rpy[0])
*/

#define TRRFN(root, suffix) root ## suffix
#define TRFN(root, suffix) TRRFN(root, suffix)
#define TFN(suffix) TRFN(TNAME, suffix)

// if V is null, returns null.
static inline TNAME *TFN(s_dup)(const TNAME *v, int len)
{
    if (!v)
        return NULL;

    TNAME *r = malloc(len * sizeof(TNAME));
    memcpy(r, v, len * sizeof(TNAME));
    return r;
}

static inline void TFN(s_print)(const TNAME *a, int len, const char *fmt)
{
    for (int i = 0; i < len; i++)
        printf(fmt, a[i]);
    printf("\n");
}

static inline void TFN(s_print_mat)(const TNAME *a, int nrows, int ncols, const char *fmt)
{
    for (int i = 0; i < nrows * ncols; i++) {
        printf(fmt, a[i]);
        if ((i % ncols) == (ncols - 1))
            printf("\n");
    }
}

static inline void TFN(s_print_mat44)(const TNAME *a, const char *fmt)
{
    for (int i = 0; i < 4 * 4; i++) {
        printf(fmt, a[i]);
        if ((i % 4) == 3)
            printf("\n");
    }
}

static inline void TFN(s_add)(const TNAME *a, const TNAME *b, int len, TNAME *r)
{
    for (int i = 0; i < len; i++)
        r[i] = a[i] + b[i];
}

static inline void TFN(s_subtract)(const TNAME *a, const TNAME *b, int len, TNAME *r)
{
    for (int i = 0; i < len; i++)
        r[i] = a[i] - b[i];
}

static inline void TFN(s_scale)(TNAME s, const TNAME *v, int len, TNAME *r)
{
    for (int i = 0; i < len; i++)
        r[i] = s * v[i];
}

static inline TNAME TFN(s_dot)(const TNAME *a, const TNAME *b, int len)
{
    TNAME acc = 0;
    for (int i = 0; i < len; i++)
        acc += a[i] * b[i];
    return acc;
}

static inline TNAME TFN(s_distance)(const TNAME *a, const TNAME *b, int len)
{
    TNAME acc = 0;
    for (int i = 0; i < len; i++)
        acc += (a[i] - b[i])*(a[i] - b[i]);
    return sqrt(acc);
}

static inline TNAME TFN(s_squared_distance)(const TNAME *a, const TNAME *b, int len)
{
    TNAME acc = 0;
    for (int i = 0; i < len; i++)
        acc += (a[i] - b[i])*(a[i] - b[i]);
    return acc;
}

static inline TNAME TFN(s_squared_magnitude)(const TNAME *v, int len)
{
    TNAME acc = 0;
    for (int i = 0; i < len; i++)
        acc += v[i]*v[i];
    return acc;
}

static inline TNAME TFN(s_magnitude)(const TNAME *v, int len)
{
    TNAME acc = 0;
    for (int i = 0; i < len; i++)
        acc += v[i]*v[i];
    return sqrt(acc);
}

static inline void TFN(s_normalize)(const TNAME *v, int len, TNAME *r)
{
    TNAME mag = TFN(s_magnitude)(v, len);
    for (int i = 0; i < len; i++)
        r[i] = v[i] / mag;
}

static inline void TFN(s_normalize_self)(TNAME *v, int len)
{
    TNAME mag = TFN(s_magnitude)(v, len);
    for (int i = 0; i < len; i++)
        v[i] /= mag;
}

static inline void TFN(s_scale_self)(TNAME *v, int len, double scale)
{
    for (int i = 0; i < len; i++)
        v[i] *= scale;
}

static inline void TFN(s_quat_rotate)(const TNAME q[4], const TNAME v[3], TNAME r[3])
{
    TNAME t2, t3, t4, t5, t6, t7, t8, t9, t10;

    t2 = q[0]*q[1];
    t3 = q[0]*q[2];
    t4 = q[0]*q[3];
    t5 = -q[1]*q[1];
    t6 = q[1]*q[2];
    t7 = q[1]*q[3];
    t8 = -q[2]*q[2];
    t9 = q[2]*q[3];
    t10 = -q[3]*q[3];

    r[0] = 2*((t8+t10)*v[0] + (t6-t4)*v[1]  + (t3+t7)*v[2]) + v[0];
    r[1] = 2*((t4+t6)*v[0]  + (t5+t10)*v[1] + (t9-t2)*v[2]) + v[1];
    r[2] = 2*((t7-t3)*v[0]  + (t2+t9)*v[1]  + (t5+t8)*v[2]) + v[2];
}

static inline void TFN(s_quat_multiply)(const TNAME a[4], const TNAME b[4], TNAME r[4])
{
    r[0] = a[0]*b[0] - a[1]*b[1] - a[2]*b[2] - a[3]*b[3];
    r[1] = a[0]*b[1] + a[1]*b[0] + a[2]*b[3] - a[3]*b[2];
    r[2] = a[0]*b[2] - a[1]*b[3] + a[2]*b[0] + a[3]*b[1];
    r[3] = a[0]*b[3] + a[1]*b[2] - a[2]*b[1] + a[3]*b[0];
}

static inline void TFN(s_quat_inverse)(const TNAME q[4], TNAME r[4])
{
    TNAME mag = TFN(s_magnitude)(q, 4);
    r[0] = q[0]/mag;
    r[1] = -q[1]/mag;
    r[2] = -q[2]/mag;
    r[3] = -q[3]/mag;
}

static inline void TFN(s_copy)(const TNAME *src, TNAME *dst, int n)
{
    memcpy(dst, src, n * sizeof(TNAME));
}

static inline void TFN(s_xyt_copy)(const TNAME xyt[3], TNAME r[3])
{
    TFN(s_copy)(xyt, r, 3);
}

static inline void TFN(s_xyt_to_mat44)(const TNAME xyt[3], TNAME r[16])
{
    TNAME s = sin(xyt[2]), c = cos(xyt[2]);
    memset(r, 0, sizeof(TNAME)*16);
    r[0] = c;
    r[1] = -s;
    r[3] = xyt[0];
    r[4] = s;
    r[5] = c;
    r[7] = xyt[1];
    r[10] = 1;
    r[15] = 1;
}

static inline void TFN(s_xyt_transform_xy)(const TNAME xyt[3], const TNAME xy[2], TNAME r[2])
{
    TNAME s = sin(xyt[2]), c = cos(xyt[2]);
    r[0] = c*xy[0] - s*xy[1] + xyt[0];
    r[1] = s*xy[0] + c*xy[1] + xyt[1];
}

static inline void TFN(s_mat_transform_xyz)(const TNAME M[16], const TNAME xyz[3], TNAME r[3])
{
    r[0] = M[0]*xyz[0] + M[1]*xyz[1] + M[2]*xyz[2]  + M[3];
    r[1] = M[4]*xyz[0] + M[5]*xyz[1] + M[6]*xyz[2]  + M[7];
    r[2] = M[8]*xyz[0] + M[9]*xyz[1] + M[10]*xyz[2] + M[11];
}

static inline void TFN(s_quat_to_angleaxis)(const TNAME _q[4], TNAME r[4])
{
    TNAME q[4];
    TFN(s_normalize)(_q, 4, q);

    // be polite: return an angle from [-pi, pi]
    // use atan2 to be 4-quadrant safe
    TNAME mag = TFN(s_magnitude)(&q[1], 3);
    r[0] = mod2pi(2 * atan2(mag, q[0]));
    if (mag != 0) {
        r[1] = q[1] / mag;
        r[2] = q[2] / mag;
        r[3] = q[3] / mag;
    } else {
        r[1] = 1;
        r[2] = 0;
        r[3] = 0;
    }
}

static inline void TFN(s_angleaxis_to_quat)(const TNAME aa[4], TNAME q[4])
{
    TNAME rad = aa[0];
    q[0] = cos(rad / 2.0);
    TNAME s = sin(rad / 2.0);

    TNAME v[3] = { aa[1], aa[2], aa[3] };
    TFN(s_normalize)(v, 3, v);

    q[1] = s * v[0];
    q[2] = s * v[1];
    q[3] = s * v[2];
}

static inline void TFN(s_quat_to_mat44)(const TNAME q[4], TNAME r[16])
{
    TNAME w = q[0], x = q[1], y = q[2], z = q[3];

    r[0] = w*w + x*x - y*y - z*z;
    r[1] = 2*x*y - 2*w*z;
    r[2] = 2*x*z + 2*w*y;
    r[3] = 0;

    r[4] = 2*x*y + 2*w*z;
    r[5] = w*w - x*x + y*y - z*z;
    r[6] = 2*y*z - 2*w*x;
    r[7] = 0;

    r[8] = 2*x*z - 2*w*y;
    r[9] = 2*y*z + 2*w*x;
    r[10] = w*w - x*x - y*y + z*z;
    r[11] = 0;

    r[12] = 0;
    r[13] = 0;
    r[14] = 0;
    r[15] = 1;
}

static inline void TFN(s_angleaxis_to_mat44)(const TNAME aa[4], TNAME r[4])
{
    TNAME q[4];

    TFN(s_angleaxis_to_quat)(aa, q);
    TFN(s_quat_to_mat44)(q, r);
}

static inline void TFN(s_quat_xyz_to_mat44)(const TNAME q[4], const TNAME xyz[3], TNAME r[16])
{
    TFN(s_quat_to_mat44)(q, r);

    if (xyz != NULL) {
        r[3] = xyz[0];
        r[7] = xyz[1];
        r[11] = xyz[2];
    }
}

static inline void TFN(s_rpy_to_quat)(const TNAME rpy[3], TNAME quat[4])
{
    TNAME roll = rpy[0], pitch = rpy[1], yaw = rpy[2];

    TNAME halfroll = roll / 2;
    TNAME halfpitch = pitch / 2;
    TNAME halfyaw = yaw / 2;

    TNAME sin_r2 = sin(halfroll);
    TNAME sin_p2 = sin(halfpitch);
    TNAME sin_y2 = sin(halfyaw);

    TNAME cos_r2 = cos(halfroll);
    TNAME cos_p2 = cos(halfpitch);
    TNAME cos_y2 = cos(halfyaw);

    quat[0] = cos_r2 * cos_p2 * cos_y2 + sin_r2 * sin_p2 * sin_y2;
    quat[1] = sin_r2 * cos_p2 * cos_y2 - cos_r2 * sin_p2 * sin_y2;
    quat[2] = cos_r2 * sin_p2 * cos_y2 + sin_r2 * cos_p2 * sin_y2;
    quat[3] = cos_r2 * cos_p2 * sin_y2 - sin_r2 * sin_p2 * cos_y2;
}

// Reference: "A tutorial on SE(3) transformation parameterizations and
// on-manifold optimization" by Jose-Luis Blanco
static inline void TFN(s_quat_to_rpy)(const TNAME q[4], TNAME rpy[3])
{
    const TNAME qr = q[0];
    const TNAME qx = q[1];
    const TNAME qy = q[2];
    const TNAME qz = q[3];

    TNAME disc = qr*qy - qx*qz;

    if (fabs(disc+0.5) < DBL_EPSILON) {         // near -1/2
        rpy[0] = 0;
        rpy[1] = -M_PI/2;
        rpy[2] = 2 * atan2(qx, qr);
    }
    else if (fabs(disc-0.5) < DBL_EPSILON) {    // near  1/2
        rpy[0] = 0;
        rpy[1] = M_PI/2;
        rpy[2] = -2 * atan2(qx, qr);
    }
    else {
        // roll
        TNAME roll_a = 2 * (qr*qx + qy*qz);
        TNAME roll_b = 1 - 2 * (qx*qx + qy*qy);
        rpy[0] = atan2(roll_a, roll_b);

        // pitch
        rpy[1] = asin(2*disc);

        // yaw
        TNAME yaw_a = 2 * (qr*qz + qx*qy);
        TNAME yaw_b = 1 - 2 * (qy*qy + qz*qz);
        rpy[2] = atan2(yaw_a, yaw_b);
    }
}

static inline void TFN(s_rpy_to_mat44)(const TNAME rpy[3], TNAME M[16])
{
    TNAME q[4];
    TFN(s_rpy_to_quat)(rpy, q);
    TFN(s_quat_to_mat44)(q, M);
}


static inline void TFN(s_xyzrpy_to_mat44)(const TNAME xyzrpy[6], TNAME M[16])
{
    TFN(s_rpy_to_mat44)(&xyzrpy[3], M);
    M[3] = xyzrpy[0];
    M[7] = xyzrpy[1];
    M[11] = xyzrpy[2];
}

static inline void TFN(s_mat44_transform_xyz)(const TNAME M[16], const TNAME in[3], TNAME out[3])
{
    for (int i = 0; i < 3; i++)
        out[i] = M[4*i + 0]*in[0] + M[4*i + 1]*in[1] + M[4*i + 2]*in[2] + M[4*i + 3];
}

// out = (upper 3x3 of M) * in
static inline void TFN(s_mat44_rotate_vector)(const TNAME M[16], const TNAME in[3], TNAME out[3])
{
    for (int i = 0; i < 3; i++)
        out[i] = M[4*i + 0]*in[0] + M[4*i + 1]*in[1] + M[4*i + 2]*in[2];
}

static inline void TFN(s_mat44_to_xyt)(const TNAME M[16], TNAME xyt[3])
{
    // c -s
    // s  c
    xyt[0] = M[3];
    xyt[1] = M[7];
    xyt[2] = atan2(M[4], M[0]);
}

static inline void TFN(s_mat_to_xyz)(const TNAME M[16], TNAME xyz[3])
{
    xyz[0] = M[3];
    xyz[1] = M[7];
    xyz[2] = M[11];
}

static inline void TFN(s_mat_to_quat)(const TNAME M[16], TNAME q[4])
{
    double T = M[0] + M[5] + M[10] + 1.0;
    double S;

    if (T > 0.0000001) {
        S = sqrt(T) * 2;
        q[0] = 0.25 * S;
        q[1] = (M[9] - M[6]) / S;
        q[2] = (M[2] - M[8]) / S;
        q[3] = (M[4] - M[1]) / S;
    } else if (M[0] > M[5] && M[0] > M[10]) {   // Column 0:
        S = sqrt(1.0 + M[0] - M[5] - M[10]) * 2;
        q[0] = (M[9] - M[6]) / S;
        q[1] = 0.25 * S;
        q[2] = (M[4] + M[1]) / S;
        q[3] = (M[2] + M[8]) / S;
    } else if (M[5] > M[10]) {                  // Column 1:
        S = sqrt(1.0 + M[5] - M[0] - M[10]) * 2;
        q[0] = (M[2] - M[8]) / S;
        q[1] = (M[4] + M[1]) / S;
        q[2] = 0.25 * S;
        q[3] = (M[9] + M[6]) / S;
    } else {                                    // Column 2:
        S = sqrt(1.0 + M[10] - M[0] - M[5]);
        q[0] = (M[4] - M[1]) / S;
        q[1] = (M[2] + M[8]) / S;
        q[2] = (M[9] + M[6]) / S;
        q[3] = 0.25 * S;
    }

    TFN(s_normalize)(q, 4, q);
}

static inline void TFN(s_quat_xyz_to_xyt)(const TNAME q[4], const TNAME xyz[3], TNAME xyt[3])
{
    TNAME M[16];
    TFN(s_quat_xyz_to_mat44)(q, xyz, M);
    TFN(s_mat44_to_xyt)(M, xyt);
}

// xytr = xyta * xytb;
static inline void TFN(s_xyt_mul)(const TNAME xyta[3], const TNAME xytb[3], TNAME xytr[3])
{
    TNAME xa = xyta[0], ya = xyta[1], ta = xyta[2];
    TNAME s = sin(ta), c = cos(ta);

    xytr[0] = c*xytb[0] - s*xytb[1] + xa;
    xytr[1] = s*xytb[0] + c*xytb[1] + ya;
    xytr[2] = ta + xytb[2];
}

static inline void TFN(s_xytcov_copy)(const TNAME xyta[3], const TNAME Ca[9],
                                      TNAME xytr[3], TNAME Cr[9])
{
    memcpy(xytr, xyta, 3 * sizeof(TNAME));
    memcpy(Cr, Ca, 9 * sizeof(TNAME));
}

static inline void TFN(s_xytcov_mul)(const TNAME xyta[3], const TNAME Ca[9],
                                      const TNAME xytb[3], const TNAME Cb[9],
                                      TNAME xytr[3], TNAME Cr[9])
{
    TNAME xa = xyta[0], ya = xyta[1], ta = xyta[2];
    TNAME xb = xytb[0], yb = xytb[1];

    TNAME sa = sin(ta), ca = cos(ta);

    TNAME P11 = Ca[0], P12 = Ca[1], P13 = Ca[2];
    TNAME              P22 = Ca[4], P23 = Ca[5];
    TNAME                           P33 = Ca[8];

    TNAME Q11 = Cb[0], Q12 = Cb[1], Q13 = Cb[2];
    TNAME              Q22 = Cb[4], Q23 = Cb[5];
    TNAME                           Q33 = Cb[8];

    TNAME JA13 = -sa*xb - ca*yb;
    TNAME JA23 = ca*xb - sa*yb;
    TNAME JB11 = ca;
    TNAME JB12 = -sa;
    TNAME JB21 = sa;
    TNAME JB22 = ca;

    Cr[0] = P33*JA13*JA13 + 2*P13*JA13 + Q11*JB11*JB11 + 2*Q12*JB11*JB12 + Q22*JB12*JB12 + P11;
    Cr[1] = P12 + JA23*(P13 + JA13*P33) + JA13*P23 + JB21*(JB11*Q11 + JB12*Q12) + JB22*(JB11*Q12 + JB12*Q22);
    Cr[2] = P13 + JA13*P33 + JB11*Q13 + JB12*Q23;
    Cr[3] = Cr[1];
    Cr[4] = P33*JA23*JA23 + 2*P23*JA23 + Q11*JB21*JB21 + 2*Q12*JB21*JB22 + Q22*JB22*JB22 + P22;
    Cr[5] = P23 + JA23*P33 + JB21*Q13 + JB22*Q23;
    Cr[6] = Cr[2];
    Cr[7] = Cr[5];
    Cr[8] = P33 + Q33;

    xytr[0] = ca*xb - sa*yb + xa;
    xytr[1] = sa*xb + ca*yb + ya;
    xytr[2] = xyta[2] + xytb[2];

/*
  // the code above is just an unrolling of the following:

        TNAME JA[][] = new TNAME[][] { { 1, 0, -sa*xb - ca*yb },
                                         { 0, 1, ca*xb - sa*yb },
                                         { 0, 0, 1 } };
        TNAME JB[][] = new TNAME[][] { { ca, -sa, 0 },
                                         { sa, ca, 0 },
                                         { 0,  0,  1 } };

        newge.P = LinAlg.add(LinAlg.matrixABCt(JA, P, JA),
                             LinAlg.matrixABCt(JB, ge.P, JB));
*/
}


static inline void TFN(s_xyt_inv)(const TNAME xyta[3], TNAME xytr[3])
{
    TNAME s = sin(xyta[2]), c = cos(xyta[2]);
    xytr[0] = -s*xyta[1] - c*xyta[0];
    xytr[1] = -c*xyta[1] + s*xyta[0];
    xytr[2] = -xyta[2];
}

static inline void TFN(s_xytcov_inv)(const TNAME xyta[3], const TNAME Ca[9],
                                      TNAME xytr[3], TNAME Cr[9])
{
    TNAME x = xyta[0], y = xyta[1], theta = xyta[2];
    TNAME s = sin(theta), c = cos(theta);

    TNAME J11 = -c, J12 = -s, J13 = -c*y + s*x;
    TNAME J21 = s,  J22 = -c, J23 = s*y + c*x;

    TNAME P11 = Ca[0], P12 = Ca[1], P13 = Ca[2];
    TNAME              P22 = Ca[4], P23 = Ca[5];
    TNAME                           P33 = Ca[8];

    Cr[0] = P11*J11*J11 + 2*P12*J11*J12 + 2*P13*J11*J13 +
        P22*J12*J12 + 2*P23*J12*J13 + P33*J13*J13;
    Cr[1] = J21*(J11*P11 + J12*P12 + J13*P13) +
        J22*(J11*P12 + J12*P22 + J13*P23) +
        J23*(J11*P13 + J12*P23 + J13*P33);
    Cr[2] = - J11*P13 - J12*P23 - J13*P33;
    Cr[3] = Cr[1];
    Cr[4] = P11*J21*J21 + 2*P12*J21*J22 + 2*P13*J21*J23 +
        P22*J22*J22 + 2*P23*J22*J23 + P33*J23*J23;
    Cr[5] = - J21*P13 - J22*P23 - J23*P33;
    Cr[6] = Cr[2];
    Cr[7] = Cr[5];
    Cr[8] = P33;

    /*
    // the code above is just an unrolling of the following:

    TNAME J[][] = new TNAME[][] { { -c, -s, -c*y + s*x },
                                    { s,  -c,  s*y + c*x },
                                    { 0,   0,     -1     } };
    ge.P = LinAlg.matrixABCt(J, P, J);
    */

    xytr[0] = -s*y - c*x;
    xytr[1] = -c*y + s*x;
    xytr[2] = -xyta[2];
}

// xytr = inv(xyta) * xytb
static inline void TFN(s_xyt_inv_mul)(const TNAME xyta[3], const TNAME xytb[3], TNAME xytr[3])
{
    TNAME theta = xyta[2];
    TNAME ca = cos(theta);
    TNAME sa = sin(theta);
    TNAME dx = xytb[0] - xyta[0];
    TNAME dy = xytb[1] - xyta[1];

    xytr[0] = ca*dx + sa*dy;
    xytr[1] = -sa*dx + ca*dy;
    xytr[2]= xytb[2] - xyta[2];
}

static inline void TFN(s_mat_add)(const TNAME *A, int Arows, int Acols,
                                   const TNAME *B, int Brows, int Bcols,
                                   TNAME *R, int Rrows, int Rcols)
{
    assert(Arows == Brows);
    assert(Arows == Rrows);
    assert(Bcols == Bcols);
    assert(Bcols == Rcols);

    for (int i = 0; i < Arows; i++)
        for (int j = 0; j < Bcols; j++)
            R[i*Acols + j] = A[i*Acols + j] + B[i*Acols + j];
}

// matrix should be in row-major order, allocated in a single packed
// array. (This is compatible with matd.)
static inline void TFN(s_mat_AB)(const TNAME *A, int Arows, int Acols,
                                  const TNAME *B, int Brows, int Bcols,
                                  TNAME *R, int Rrows, int Rcols)
{
    assert(Acols == Brows);
    assert(Rrows == Arows);
    assert(Bcols == Rcols);

    for (int Rrow = 0; Rrow < Rrows; Rrow++) {
        for (int Rcol = 0; Rcol < Rcols; Rcol++) {
            TNAME acc = 0;
            for (int i = 0; i < Acols; i++)
                acc += A[Rrow*Acols + i] * B[i*Bcols + Rcol];
            R[Rrow*Rcols + Rcol] = acc;
        }
    }
}

// matrix should be in row-major order, allocated in a single packed
// array. (This is compatible with matd.)
static inline void TFN(s_mat_ABt)(const TNAME *A, int Arows, int Acols,
                                  const TNAME *B, int Brows, int Bcols,
                                  TNAME *R, int Rrows, int Rcols)
{
    assert(Acols == Bcols);
    assert(Rrows == Arows);
    assert(Brows == Rcols);

    for (int Rrow = 0; Rrow < Rrows; Rrow++) {
        for (int Rcol = 0; Rcol < Rcols; Rcol++) {
            TNAME acc = 0;
            for (int i = 0; i < Acols; i++)
                acc += A[Rrow*Acols + i] * B[Rcol*Bcols + i];
            R[Rrow*Rcols + Rcol] = acc;
        }
    }
}

static inline void TFN(s_mat_ABC)(const TNAME *A, int Arows, int Acols,
                                  const TNAME *B, int Brows, int Bcols,
                                  const TNAME *C, int Crows, int Ccols,
                                  TNAME *R, int Rrows, int Rcols)
{
    TNAME tmp[Arows*Bcols];

    TFN(s_mat_AB)(A, Arows, Acols, B, Brows, Bcols, tmp, Arows, Bcols);
    TFN(s_mat_AB)(tmp, Arows, Bcols, C, Crows, Ccols, R, Rrows, Rcols);
}

static inline void TFN(s_mat_Ab)(const TNAME *A, int Arows, int Acols,
                                  const TNAME *B, int Blength,
                                  TNAME *R, int Rlength)
{
    assert(Acols == Blength);
    assert(Arows == Rlength);

    for (int Ridx = 0; Ridx < Rlength; Ridx++) {
        TNAME acc = 0;
        for (int i = 0; i < Blength; i++)
            acc += A[Ridx*Acols + i] * B[i];
        R[Ridx] = acc;
    }
}

static inline void TFN(s_mat_AtB)(const TNAME *A, int Arows, int Acols,
                                   const TNAME *B, int Brows, int Bcols,
                                   TNAME *R, int Rrows, int Rcols)
{
    assert(Arows == Brows);
    assert(Rrows == Acols);
    assert(Bcols == Rcols);

    for (int Rrow = 0; Rrow < Rrows; Rrow++) {
        for (int Rcol = 0; Rcol < Rcols; Rcol++) {
            TNAME acc = 0;
            for (int i = 0; i < Acols; i++)
                acc += A[i*Acols + Rrow] * B[i*Bcols + Rcol];
            R[Rrow*Rcols + Rcol] = acc;
        }
    }
}

static inline void TFN(s_quat_slerp)(const TNAME q0[4], const TNAME _q1[4], TNAME r[4], TNAME w)
{
    TNAME dot = TFN(s_dot)(q0, _q1, 4);

    TNAME q1[4];
    memcpy(q1, _q1, sizeof(TNAME) * 4);

    if (dot < 0) {
        // flip sign on one of them so we don't spin the "wrong
        // way" around. This doesn't change the rotation that the
        // quaternion represents.
        dot = -dot;
        for (int i = 0; i < 4; i++)
            q1[i] *= -1;
    }

    // if large dot product (1), slerp will scale both q0 and q1
    // by 0, and normalization will blow up.
    if (dot > 0.95) {

        for (int i = 0; i < 4; i++)
            r[i] = q0[i]*(1-w) + q1[i]*w;

    } else {
        TNAME angle = acos(dot);

        TNAME w0 = sin(angle*(1-w)), w1 = sin(angle*w);

        for (int i = 0; i < 4; i++)
            r[i] = q0[i]*w0 + q1[i]*w1;

        TFN(s_normalize)(r, 4, r);
    }
}

static inline void TFN(s_cross_product)(const TNAME v1[3], const TNAME v2[3], TNAME r[3])
{
    r[0] = v1[1]*v2[2] - v1[2]*v2[1];
    r[1] = v1[2]*v2[0] - v1[0]*v2[2];
    r[2] = v1[0]*v2[1] - v1[1]*v2[0];
}

static inline void TFN(s_mat44_identity)(TNAME out[16])
{
    memset(out, 0, 16 * sizeof(TNAME));
    out[0] = 1;
    out[5] = 1;
    out[10] = 1;
    out[15] = 1;
}

static inline void TFN(s_mat44_translate)(const TNAME txyz[3], TNAME out[16])
{
    TFN(s_mat44_identity)(out);

    for (int i = 0; i < 3; i++)
        out[4*i + 3] += txyz[i];
}

static inline void TFN(s_mat44_scale)(const TNAME sxyz[3], TNAME out[16])
{
    TFN(s_mat44_identity)(out);

    for (int i = 0; i < 3; i++)
        out[4*i + i] = sxyz[i];
}

static inline void TFN(s_mat44_rotate_z)(TNAME rad, TNAME out[16])
{
    TFN(s_mat44_identity)(out);
    double s = sin(rad), c = cos(rad);
    out[0*4 + 0] = c;
    out[0*4 + 1] = -s;
    out[1*4 + 0] = s;
    out[1*4 + 1] = c;
}

static inline void TFN(s_mat44_rotate_y)(TNAME rad, TNAME out[16])
{
    TFN(s_mat44_identity)(out);
    double s = sin(rad), c = cos(rad);
    out[0*4 + 0] = c;
    out[0*4 + 2] = s;
    out[2*4 + 0] = -s;
    out[2*4 + 2] = c;
}

static inline void TFN(s_mat44_rotate_x)(TNAME rad, TNAME out[16])
{
    TFN(s_mat44_identity)(out);
    double s = sin(rad), c = cos(rad);
    out[1*4 + 1] = c;
    out[1*4 + 2] = -s;
    out[2*4 + 1] = s;
    out[2*4 + 2] = c;
}

// out = out * translate(txyz)
static inline void TFN(s_mat44_translate_self)(const TNAME txyz[3], TNAME out[16])
{
    TNAME tmp[16], prod[16];
    TFN(s_mat44_translate(txyz, tmp));
    TFN(s_mat_AB)(out, 4, 4, tmp, 4, 4, prod, 4, 4);
    memcpy(out, prod, sizeof(TNAME)*16);
}

static inline void TFN(s_mat44_scale_self)(const TNAME sxyz[3], TNAME out[16])
{
    TNAME tmp[16], prod[16];
    TFN(s_mat44_scale(sxyz, tmp));
    TFN(s_mat_AB)(out, 4, 4, tmp, 4, 4, prod, 4, 4);
    memcpy(out, prod, sizeof(TNAME)*16);
}

static inline void TFN(s_mat44_rotate_z_self)(TNAME rad, TNAME out[16])
{
    TNAME tmp[16], prod[16];
    TFN(s_mat44_rotate_z(rad, tmp));
    TFN(s_mat_AB)(out, 4, 4, tmp, 4, 4, prod, 4, 4);
    memcpy(out, prod, sizeof(TNAME)*16);
}

// out = inv(M)*in. Note: this assumes that mat44 is a rigid-body transformation.
static inline void TFN(s_mat44_inv)(const TNAME M[16], TNAME out[16])
{
// NB: M = T*R,  inv(M) = inv(R) * inv(T)

    // transpose of upper-left corner
    for (int i = 0; i < 3; i++)
        for (int j = 0; j < 3; j++)
            out[4*i + j] = M[4*j + i];

    out[4*0 + 3] = 0;
    out[4*1 + 3] = 0;
    out[4*2 + 3] = 0;

    for (int i = 0; i < 3; i++)
        for (int j = 0; j < 3; j++)
            out[4*i + 3] -= out[4*i + j] * M[4*j + 3];

    out[4*3 + 0] = 0;
    out[4*3 + 1] = 0;
    out[4*3 + 2] = 0;
    out[4*3 + 3] = 1;

/*    TNAME tmp[16];
    TFN(s_mat_AB)(M, 4, 4, out, 4, 4, tmp, 4, 4);
    printf("identity: ");
    TFN(s_print_mat)(tmp, 4, 4, "%15f"); */
}

// out = inv(M)*in
static inline void TFN(s_mat44_inv_transform_xyz)(const TNAME M[16], const TNAME in[3], TNAME out[3])
{
    TNAME T[16];
    TFN(s_mat44_inv)(M, T);

    TFN(s_mat44_transform_xyz)(T, in, out);
}

// out = (upper 3x3 of inv(M)) * in
static inline void TFN(s_mat44_inv_rotate_vector)(const TNAME M[16], const TNAME in[3], TNAME out[3])
{
    TNAME T[16];
    TFN(s_mat44_inv)(M, T);

    TFN(s_mat44_rotate_vector)(T, in, out);
}

static inline void TFN(s_elu_to_mat44)(const TNAME eye[3], const TNAME lookat[3], const TNAME _up[3],
                                       TNAME M[16])
{
    TNAME f[3];
    TFN(s_subtract)(lookat, eye, 3, f);
    TFN(s_normalize)(f, 3, f);

    TNAME up[3];

    // remove any component of 'up' that isn't perpendicular to the look direction.
    TFN(s_normalize)(_up, 3, up);

    TNAME up_dot = TFN(s_dot)(f, up, 3);
    for (int i = 0; i < 3; i++)
        up[i] -= up_dot*f[i];

    TFN(s_normalize_self)(up, 3);

    TNAME s[3], u[3];
    TFN(s_cross_product)(f, up, s);
    TFN(s_cross_product)(s, f, u);

    TNAME R[16] = {  s[0],  s[1],  s[2], 0,
                     u[0],  u[1],  u[2], 0,
                    -f[0], -f[1], -f[2], 0,
                     0,     0,     0,    1};

    TNAME T[16] = {1, 0, 0,  -eye[0],
                    0, 1, 0, -eye[1],
                    0, 0, 1, -eye[2],
                    0, 0, 0, 1};

    // M is the extrinsics matrix [R | t] where t = -R*c
    TNAME tmp[16];
    TFN(s_mat_AB)(R, 4, 4, T, 4, 4, tmp, 4, 4);
    TFN(s_mat44_inv)(tmp, M);
}

// Computes the cholesky factorization of A, putting the lower
// triangular matrix into R.
static inline void TFN(s_mat33_chol)(const TNAME *A, int Arows, int Acols,
                                     TNAME *R, int Brows, int Bcols)
{
    assert(Arows == Brows);
    assert(Bcols == Bcols);

    // A[0] = R[0]*R[0]
    R[0] = sqrt(A[0]);

    // A[1] = R[0]*R[3];
    R[3] = A[1] / R[0];

    // A[2] = R[0]*R[6];
    R[6] = A[2] / R[0];

    // A[4] = R[3]*R[3] + R[4]*R[4]
    R[4] = sqrt(A[4] - R[3]*R[3]);

    // A[5] = R[3]*R[6] + R[4]*R[7]
    R[7] = (A[5] - R[3]*R[6]) / R[4];

    // A[8] = R[6]*R[6] + R[7]*R[7] + R[8]*R[8]
    R[8] = sqrt(A[8] - R[6]*R[6] - R[7]*R[7]);

    R[1] = 0;
    R[2] = 0;
    R[5] = 0;
}

static inline void TFN(s_mat33_lower_tri_inv)(const TNAME *A, int Arows, int Acols,
                                              TNAME *R, int Rrows, int Rcols)
{
    // A[0]*R[0] = 1
    R[0] = 1 / A[0];

    // A[3]*R[0] + A[4]*R[3] = 0
    R[3] = -A[3]*R[0] / A[4];

    // A[4]*R[4] = 1
    R[4] = 1 / A[4];

    // A[6]*R[0] + A[7]*R[3] + A[8]*R[6] = 0
    R[6] = (-A[6]*R[0] - A[7]*R[3]) / A[8];

    // A[7]*R[4] + A[8]*R[7] = 0
    R[7] = -A[7]*R[4] / A[8];

    // A[8]*R[8] = 1
    R[8] = 1 / A[8];
}


static inline void TFN(s_mat33_sym_solve)(const TNAME *A, int Arows, int Acols,
                                          const TNAME *B, int Brows, int Bcols,
                                          TNAME *R, int Rrows, int Rcols)
{
    assert(Arows == Acols);
    assert(Acols == 3);
    assert(Brows == 3);
    assert(Bcols == 1);
    assert(Rrows == 3);
    assert(Rcols == 1);

    TNAME L[9];
    TFN(s_mat33_chol)(A, 3, 3, L, 3, 3);

    TNAME M[9];
    TFN(s_mat33_lower_tri_inv)(L, 3, 3, M, 3, 3);

    double tmp[3];
    tmp[0] = M[0]*B[0];
    tmp[1] = M[3]*B[0] + M[4]*B[1];
    tmp[2] = M[6]*B[0] + M[7]*B[1] + M[8]*B[2];

    R[0] = M[0]*tmp[0] + M[3]*tmp[1] + M[6]*tmp[2];
    R[1] = M[4]*tmp[1] + M[7]*tmp[2];
    R[2] = M[8]*tmp[2];
}

/*
// solve Ax = B. Assumes A is symmetric; uses cholesky factorization
static inline void TFN(s_mat_solve_chol)(const TNAME *A, int Arows, int Acols,
                                         const TNAME *B, int Brows, int Bcols,
                                         TNAME *R, int Rrows, int Rcols)
{
    assert(Arows == Acols);
    assert(Arows == Brows);
    assert(Acols == Rrows);
    assert(Bcols == Rcols);

    //
}
*/
#undef TRRFN
#undef TRFN
#undef TFN