# Source: quat2.js

    import * as glMatrix from "./common.js";
    import * as quat from "./quat.js";
    import * as mat4 from "./mat4.js";

    /**
     * Dual Quaternion<br>
     * Format: [real, dual]<br>
     * Quaternion format: XYZW<br>
     * Make sure to have normalized dual quaternions, otherwise the functions may not work as intended.<br>
     * @module quat2
     */

    /**
     * Creates a new identity dual quat
     *
     * @returns {quat2} a new dual quaternion [real -> rotation, dual -> translation]
     */
    export function create() {
      let dq = new glMatrix.ARRAY_TYPE(8);
      if(glMatrix.ARRAY_TYPE != Float32Array) {
        dq[0] = 0;
        dq[1] = 0;
        dq[2] = 0;
        dq[4] = 0;
        dq[5] = 0;
        dq[6] = 0;
        dq[7] = 0;
      }
      dq[3] = 1;
      return dq;
    }

    /**
     * Creates a new quat initialized with values from an existing quaternion
     *
     * @param {quat2} a dual quaternion to clone
     * @returns {quat2} new dual quaternion
     * @function
     */
    export function clone(a) {
      let dq = new glMatrix.ARRAY_TYPE(8);
      dq[0] = a[0];
      dq[1] = a[1];
      dq[2] = a[2];
      dq[3] = a[3];
      dq[4] = a[4];
      dq[5] = a[5];
      dq[6] = a[6];
      dq[7] = a[7];
      return dq;
    }

    /**
     * Creates a new dual quat initialized with the given values
     *
     * @param {Number} x1 X component
     * @param {Number} y1 Y component
     * @param {Number} z1 Z component
     * @param {Number} w1 W component
     * @param {Number} x2 X component
     * @param {Number} y2 Y component
     * @param {Number} z2 Z component
     * @param {Number} w2 W component
     * @returns {quat2} new dual quaternion
     * @function
     */
    export function fromValues(x1, y1, z1, w1, x2, y2, z2, w2) {
      let dq = new glMatrix.ARRAY_TYPE(8);
      dq[0] = x1;
      dq[1] = y1;
      dq[2] = z1;
      dq[3] = w1;
      dq[4] = x2;
      dq[5] = y2;
      dq[6] = z2;
      dq[7] = w2;
      return dq;
    }

    /**
     * Creates a new dual quat from the given values (quat and translation)
     *
     * @param {Number} x1 X component
     * @param {Number} y1 Y component
     * @param {Number} z1 Z component
     * @param {Number} w1 W component
     * @param {Number} x2 X component (translation)
     * @param {Number} y2 Y component (translation)
     * @param {Number} z2 Z component (translation)
     * @returns {quat2} new dual quaternion
     * @function
     */
    export function fromRotationTranslationValues(x1, y1, z1, w1, x2, y2, z2) {
      let dq = new glMatrix.ARRAY_TYPE(8);
      dq[0] = x1;
      dq[1] = y1;
      dq[2] = z1;
      dq[3] = w1;
      let ax = x2 * 0.5,
        ay = y2 * 0.5,
        az = z2 * 0.5;
      dq[4] = ax * w1 + ay * z1 - az * y1;
      dq[5] = ay * w1 + az * x1 - ax * z1;
      dq[6] = az * w1 + ax * y1 - ay * x1;
      dq[7] = -ax * x1 - ay * y1 - az * z1;
      return dq;
    }

    /**
     * Creates a dual quat from a quaternion and a translation
     *
     * @param {quat2} dual quaternion receiving operation result
     * @param {quat} q quaternion
     * @param {vec3} t tranlation vector
     * @returns {quat2} dual quaternion receiving operation result
     * @function
     */
    export function fromRotationTranslation(out, q, t) {
      let ax = t[0] * 0.5,
        ay = t[1] * 0.5,
        az = t[2] * 0.5,
        bx = q[0],
        by = q[1],
        bz = q[2],
        bw = q[3];
      out[0] = bx;
      out[1] = by;
      out[2] = bz;
      out[3] = bw;
      out[4] = ax * bw + ay * bz - az * by;
      out[5] = ay * bw + az * bx - ax * bz;
      out[6] = az * bw + ax * by - ay * bx;
      out[7] = -ax * bx - ay * by - az * bz;
      return out;
    }

    /**
     * Creates a dual quat from a translation
     *
     * @param {quat2} dual quaternion receiving operation result
     * @param {vec3} t translation vector
     * @returns {quat2} dual quaternion receiving operation result
     * @function
     */
    export function fromTranslation(out, t) {
      out[0] = 0;
      out[1] = 0;
      out[2] = 0;
      out[3] = 1;
      out[4] = t[0] * 0.5;
      out[5] = t[1] * 0.5;
      out[6] = t[2] * 0.5;
      out[7] = 0;
      return out;
    }

    /**
     * Creates a dual quat from a quaternion
     *
     * @param {quat2} dual quaternion receiving operation result
     * @param {quat} q the quaternion
     * @returns {quat2} dual quaternion receiving operation result
     * @function
     */
    export function fromRotation(out, q) {
      out[0] = q[0];
      out[1] = q[1];
      out[2] = q[2];
      out[3] = q[3];
      out[4] = 0;
      out[5] = 0;
      out[6] = 0;
      out[7] = 0;
      return out;
    }

    /**
     * Creates a new dual quat from a matrix (4x4)
     *
     * @param {quat2} out the dual quaternion
     * @param {mat4} a the matrix
     * @returns {quat2} dual quat receiving operation result
     * @function
     */
    export function fromMat4(out, a) {
      //TODO Optimize this
      let outer = quat.create();
      mat4.getRotation(outer, a);
      let t = new glMatrix.ARRAY_TYPE(3);
      mat4.getTranslation(t, a);
      fromRotationTranslation(out, outer, t);
      return out;
    }

    /**
     * Copy the values from one dual quat to another
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a the source dual quaternion
     * @returns {quat2} out
     * @function
     */
    export function copy(out, a) {
      out[0] = a[0];
      out[1] = a[1];
      out[2] = a[2];
      out[3] = a[3];
      out[4] = a[4];
      out[5] = a[5];
      out[6] = a[6];
      out[7] = a[7];
      return out;
    }

    /**
     * Set a dual quat to the identity dual quaternion
     *
     * @param {quat2} out the receiving quaternion
     * @returns {quat2} out
     */
    export function identity(out) {
      out[0] = 0;
      out[1] = 0;
      out[2] = 0;
      out[3] = 1;
      out[4] = 0;
      out[5] = 0;
      out[6] = 0;
      out[7] = 0;
      return out;
    }

    /**
     * Set the components of a dual quat to the given values
     *
     * @param {quat2} out the receiving quaternion
     * @param {Number} x1 X component
     * @param {Number} y1 Y component
     * @param {Number} z1 Z component
     * @param {Number} w1 W component
     * @param {Number} x2 X component
     * @param {Number} y2 Y component
     * @param {Number} z2 Z component
     * @param {Number} w2 W component
     * @returns {quat2} out
     * @function
     */
    export function set(out, x1, y1, z1, w1, x2, y2, z2, w2) {
      out[0] = x1;
      out[1] = y1;
      out[2] = z1;
      out[3] = w1;

      out[4] = x2;
      out[5] = y2;
      out[6] = z2;
      out[7] = w2;
      return out;
    }

    /**
     * Gets the real part of a dual quat
     * @param  {quat} out real part
     * @param  {quat2} a Dual Quaternion
     * @return {quat} real part
     */
    export const getReal = quat.copy;

    /**
     * Gets the dual part of a dual quat
     * @param  {quat} out dual part
     * @param  {quat2} a Dual Quaternion
     * @return {quat} dual part
     */
    export function getDual(out, a) {
      out[0] = a[4];
      out[1] = a[5];
      out[2] = a[6];
      out[3] = a[7];
      return out;
    }

    /**
     * Set the real component of a dual quat to the given quaternion
     *
     * @param {quat2} out the receiving quaternion
     * @param {quat} q a quaternion representing the real part
     * @returns {quat2} out
     * @function
     */
    export const setReal = quat.copy;

    /**
     * Set the dual component of a dual quat to the given quaternion
     *
     * @param {quat2} out the receiving quaternion
     * @param {quat} q a quaternion representing the dual part
     * @returns {quat2} out
     * @function
     */
    export function setDual(out, q) {
      out[4] = q[0];
      out[5] = q[1];
      out[6] = q[2];
      out[7] = q[3];
      return out;
    }

    /**
     * Gets the translation of a normalized dual quat
     * @param  {vec3} out translation
     * @param  {quat2} a Dual Quaternion to be decomposed
     * @return {vec3} translation
     */
    export function getTranslation(out, a) {
      let ax = a[4],
        ay = a[5],
        az = a[6],
        aw = a[7],
        bx = -a[0],
        by = -a[1],
        bz = -a[2],
        bw = a[3];
      out[0] = (ax * bw + aw * bx + ay * bz - az * by) * 2;
      out[1] = (ay * bw + aw * by + az * bx - ax * bz) * 2;
      out[2] = (az * bw + aw * bz + ax * by - ay * bx) * 2;
      return out;
    }

    /**
     * Translates a dual quat by the given vector
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a the dual quaternion to translate
     * @param {vec3} v vector to translate by
     * @returns {quat2} out
     */
    export function translate(out, a, v) {
      let ax1 = a[0],
        ay1 = a[1],
        az1 = a[2],
        aw1 = a[3],
        bx1 = v[0] * 0.5,
        by1 = v[1] * 0.5,
        bz1 = v[2] * 0.5,
        ax2 = a[4],
        ay2 = a[5],
        az2 = a[6],
        aw2 = a[7];
      out[0] = ax1;
      out[1] = ay1;
      out[2] = az1;
      out[3] = aw1;
      out[4] = aw1 * bx1 + ay1 * bz1 - az1 * by1 + ax2;
      out[5] = aw1 * by1 + az1 * bx1 - ax1 * bz1 + ay2;
      out[6] = aw1 * bz1 + ax1 * by1 - ay1 * bx1 + az2;
      out[7] = -ax1 * bx1 - ay1 * by1 - az1 * bz1 + aw2;
      return out;
    }

    /**
     * Rotates a dual quat around the X axis
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a the dual quaternion to rotate
     * @param {number} rad how far should the rotation be
     * @returns {quat2} out
     */
    export function rotateX(out, a, rad) {
      let bx = -a[0],
        by = -a[1],
        bz = -a[2],
        bw = a[3],
        ax = a[4],
        ay = a[5],
        az = a[6],
        aw = a[7],
        ax1 = ax * bw + aw * bx + ay * bz - az * by,
        ay1 = ay * bw + aw * by + az * bx - ax * bz,
        az1 = az * bw + aw * bz + ax * by - ay * bx,
        aw1 = aw * bw - ax * bx - ay * by - az * bz;
      quat.rotateX(out, a, rad);
      bx = out[0];
      by = out[1];
      bz = out[2];
      bw = out[3];
      out[4] = ax1 * bw + aw1 * bx + ay1 * bz - az1 * by;
      out[5] = ay1 * bw + aw1 * by + az1 * bx - ax1 * bz;
      out[6] = az1 * bw + aw1 * bz + ax1 * by - ay1 * bx;
      out[7] = aw1 * bw - ax1 * bx - ay1 * by - az1 * bz;
      return out;
    }

    /**
     * Rotates a dual quat around the Y axis
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a the dual quaternion to rotate
     * @param {number} rad how far should the rotation be
     * @returns {quat2} out
     */
    export function rotateY(out, a, rad) {
      let bx = -a[0],
        by = -a[1],
        bz = -a[2],
        bw = a[3],
        ax = a[4],
        ay = a[5],
        az = a[6],
        aw = a[7],
        ax1 = ax * bw + aw * bx + ay * bz - az * by,
        ay1 = ay * bw + aw * by + az * bx - ax * bz,
        az1 = az * bw + aw * bz + ax * by - ay * bx,
        aw1 = aw * bw - ax * bx - ay * by - az * bz;
      quat.rotateY(out, a, rad);
      bx = out[0];
      by = out[1];
      bz = out[2];
      bw = out[3];
      out[4] = ax1 * bw + aw1 * bx + ay1 * bz - az1 * by;
      out[5] = ay1 * bw + aw1 * by + az1 * bx - ax1 * bz;
      out[6] = az1 * bw + aw1 * bz + ax1 * by - ay1 * bx;
      out[7] = aw1 * bw - ax1 * bx - ay1 * by - az1 * bz;
      return out;
    }

    /**
     * Rotates a dual quat around the Z axis
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a the dual quaternion to rotate
     * @param {number} rad how far should the rotation be
     * @returns {quat2} out
     */
    export function rotateZ(out, a, rad) {
      let bx = -a[0],
        by = -a[1],
        bz = -a[2],
        bw = a[3],
        ax = a[4],
        ay = a[5],
        az = a[6],
        aw = a[7],
        ax1 = ax * bw + aw * bx + ay * bz - az * by,
        ay1 = ay * bw + aw * by + az * bx - ax * bz,
        az1 = az * bw + aw * bz + ax * by - ay * bx,
        aw1 = aw * bw - ax * bx - ay * by - az * bz;
      quat.rotateZ(out, a, rad);
      bx = out[0];
      by = out[1];
      bz = out[2];
      bw = out[3];
      out[4] = ax1 * bw + aw1 * bx + ay1 * bz - az1 * by;
      out[5] = ay1 * bw + aw1 * by + az1 * bx - ax1 * bz;
      out[6] = az1 * bw + aw1 * bz + ax1 * by - ay1 * bx;
      out[7] = aw1 * bw - ax1 * bx - ay1 * by - az1 * bz;
      return out;
    }

    /**
     * Rotates a dual quat by a given quaternion (a * q)
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a the dual quaternion to rotate
     * @param {quat} q quaternion to rotate by
     * @returns {quat2} out
     */
    export function rotateByQuatAppend(out, a, q) {
      let qx = q[0],
        qy = q[1],
        qz = q[2],
        qw = q[3],
        ax = a[0],
        ay = a[1],
        az = a[2],
        aw = a[3];

      out[0] = ax * qw + aw * qx + ay * qz - az * qy;
      out[1] = ay * qw + aw * qy + az * qx - ax * qz;
      out[2] = az * qw + aw * qz + ax * qy - ay * qx;
      out[3] = aw * qw - ax * qx - ay * qy - az * qz;
      ax = a[4];
      ay = a[5];
      az = a[6];
      aw = a[7];
      out[4] = ax * qw + aw * qx + ay * qz - az * qy;
      out[5] = ay * qw + aw * qy + az * qx - ax * qz;
      out[6] = az * qw + aw * qz + ax * qy - ay * qx;
      out[7] = aw * qw - ax * qx - ay * qy - az * qz;
      return out;
    }

    /**
     * Rotates a dual quat by a given quaternion (q * a)
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat} q quaternion to rotate by
     * @param {quat2} a the dual quaternion to rotate
     * @returns {quat2} out
     */
    export function rotateByQuatPrepend(out, q, a) {
      let qx = q[0],
        qy = q[1],
        qz = q[2],
        qw = q[3],
        bx = a[0],
        by = a[1],
        bz = a[2],
        bw = a[3];

      out[0] = qx * bw + qw * bx + qy * bz - qz * by;
      out[1] = qy * bw + qw * by + qz * bx - qx * bz;
      out[2] = qz * bw + qw * bz + qx * by - qy * bx;
      out[3] = qw * bw - qx * bx - qy * by - qz * bz;
      bx = a[4];
      by = a[5];
      bz = a[6];
      bw = a[7];
      out[4] = qx * bw + qw * bx + qy * bz - qz * by;
      out[5] = qy * bw + qw * by + qz * bx - qx * bz;
      out[6] = qz * bw + qw * bz + qx * by - qy * bx;
      out[7] = qw * bw - qx * bx - qy * by - qz * bz;
      return out;
    }

    /**
     * Rotates a dual quat around a given axis. Does the normalisation automatically
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a the dual quaternion to rotate
     * @param {vec3} axis the axis to rotate around
     * @param {Number} rad how far the rotation should be
     * @returns {quat2} out
     */
    export function rotateAroundAxis(out, a, axis, rad) {
      //Special case for rad = 0
      if (Math.abs(rad) < glMatrix.EPSILON) {
        return copy(out, a);
      }
      let axisLength = Math.sqrt(axis[0] * axis[0] + axis[1] * axis[1] + axis[2] * axis[2]);

      rad = rad * 0.5;
      let s = Math.sin(rad);
      let bx = s * axis[0] / axisLength;
      let by = s * axis[1] / axisLength;
      let bz = s * axis[2] / axisLength;
      let bw = Math.cos(rad);

      let ax1 = a[0],
        ay1 = a[1],
        az1 = a[2],
        aw1 = a[3];
      out[0] = ax1 * bw + aw1 * bx + ay1 * bz - az1 * by;
      out[1] = ay1 * bw + aw1 * by + az1 * bx - ax1 * bz;
      out[2] = az1 * bw + aw1 * bz + ax1 * by - ay1 * bx;
      out[3] = aw1 * bw - ax1 * bx - ay1 * by - az1 * bz;

      let ax = a[4],
        ay = a[5],
        az = a[6],
        aw = a[7];
      out[4] = ax * bw + aw * bx + ay * bz - az * by;
      out[5] = ay * bw + aw * by + az * bx - ax * bz;
      out[6] = az * bw + aw * bz + ax * by - ay * bx;
      out[7] = aw * bw - ax * bx - ay * by - az * bz;

      return out;
    }

    /**
     * Adds two dual quat's
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a the first operand
     * @param {quat2} b the second operand
     * @returns {quat2} out
     * @function
     */
    export function add(out, a, b) {
      out[0] = a[0] + b[0];
      out[1] = a[1] + b[1];
      out[2] = a[2] + b[2];
      out[3] = a[3] + b[3];
      out[4] = a[4] + b[4];
      out[5] = a[5] + b[5];
      out[6] = a[6] + b[6];
      out[7] = a[7] + b[7];
      return out;
    }

    /**
     * Multiplies two dual quat's
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a the first operand
     * @param {quat2} b the second operand
     * @returns {quat2} out
     */
    export function multiply(out, a, b) {
      let ax0 = a[0],
        ay0 = a[1],
        az0 = a[2],
        aw0 = a[3],
        bx1 = b[4],
        by1 = b[5],
        bz1 = b[6],
        bw1 = b[7],
        ax1 = a[4],
        ay1 = a[5],
        az1 = a[6],
        aw1 = a[7],
        bx0 = b[0],
        by0 = b[1],
        bz0 = b[2],
        bw0 = b[3];
      out[0] = ax0 * bw0 + aw0 * bx0 + ay0 * bz0 - az0 * by0;
      out[1] = ay0 * bw0 + aw0 * by0 + az0 * bx0 - ax0 * bz0;
      out[2] = az0 * bw0 + aw0 * bz0 + ax0 * by0 - ay0 * bx0;
      out[3] = aw0 * bw0 - ax0 * bx0 - ay0 * by0 - az0 * bz0;
      out[4] = ax0 * bw1 + aw0 * bx1 + ay0 * bz1 - az0 * by1 + ax1 * bw0 + aw1 * bx0 + ay1 * bz0 - az1 * by0;
      out[5] = ay0 * bw1 + aw0 * by1 + az0 * bx1 - ax0 * bz1 + ay1 * bw0 + aw1 * by0 + az1 * bx0 - ax1 * bz0;
      out[6] = az0 * bw1 + aw0 * bz1 + ax0 * by1 - ay0 * bx1 + az1 * bw0 + aw1 * bz0 + ax1 * by0 - ay1 * bx0;
      out[7] = aw0 * bw1 - ax0 * bx1 - ay0 * by1 - az0 * bz1 + aw1 * bw0 - ax1 * bx0 - ay1 * by0 - az1 * bz0;
      return out;
    }

    /**
     * Alias for {@link quat2.multiply}
     * @function
     */
    export const mul = multiply;

    /**
     * Scales a dual quat by a scalar number
     *
     * @param {quat2} out the receiving dual quat
     * @param {quat2} a the dual quat to scale
     * @param {Number} b amount to scale the dual quat by
     * @returns {quat2} out
     * @function
     */
    export function scale(out, a, b) {
      out[0] = a[0] * b;
      out[1] = a[1] * b;
      out[2] = a[2] * b;
      out[3] = a[3] * b;
      out[4] = a[4] * b;
      out[5] = a[5] * b;
      out[6] = a[6] * b;
      out[7] = a[7] * b;
      return out;
    }

    /**
     * Calculates the dot product of two dual quat's (The dot product of the real parts)
     *
     * @param {quat2} a the first operand
     * @param {quat2} b the second operand
     * @returns {Number} dot product of a and b
     * @function
     */
    export const dot = quat.dot;

    /**
     * Performs a linear interpolation between two dual quats's
     * NOTE: The resulting dual quaternions won't always be normalized (The error is most noticeable when t = 0.5)
     *
     * @param {quat2} out the receiving dual quat
     * @param {quat2} a the first operand
     * @param {quat2} b the second operand
     * @param {Number} t interpolation amount, in the range [0-1], between the two inputs
     * @returns {quat2} out
     */
    export function lerp(out, a, b, t) {
      let mt = 1 - t;
      if (dot(a, b) < 0) t = -t;

      out[0] = a[0] * mt + b[0] * t;
      out[1] = a[1] * mt + b[1] * t;
      out[2] = a[2] * mt + b[2] * t;
      out[3] = a[3] * mt + b[3] * t;
      out[4] = a[4] * mt + b[4] * t;
      out[5] = a[5] * mt + b[5] * t;
      out[6] = a[6] * mt + b[6] * t;
      out[7] = a[7] * mt + b[7] * t;

      return out;
    }

    /**
     * Calculates the inverse of a dual quat. If they are normalized, conjugate is cheaper
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a dual quat to calculate inverse of
     * @returns {quat2} out
     */
    export function invert(out, a) {
      let sqlen = squaredLength(a);
      out[0] = -a[0] / sqlen;
      out[1] = -a[1] / sqlen;
      out[2] = -a[2] / sqlen;
      out[3] = a[3] / sqlen;
      out[4] = -a[4] / sqlen;
      out[5] = -a[5] / sqlen;
      out[6] = -a[6] / sqlen;
      out[7] = a[7] / sqlen;
      return out;
    }

    /**
     * Calculates the conjugate of a dual quat
     * If the dual quaternion is normalized, this function is faster than quat2.inverse and produces the same result.
     *
     * @param {quat2} out the receiving quaternion
     * @param {quat2} a quat to calculate conjugate of
     * @returns {quat2} out
     */
    export function conjugate(out, a) {
      out[0] = -a[0];
      out[1] = -a[1];
      out[2] = -a[2];
      out[3] = a[3];
      out[4] = -a[4];
      out[5] = -a[5];
      out[6] = -a[6];
      out[7] = a[7];
      return out;
    }

    /**
     * Calculates the length of a dual quat
     *
     * @param {quat2} a dual quat to calculate length of
     * @returns {Number} length of a
     * @function
     */
    export const length = quat.length;

    /**
     * Alias for {@link quat2.length}
     * @function
     */
    export const len = length;

    /**
     * Calculates the squared length of a dual quat
     *
     * @param {quat2} a dual quat to calculate squared length of
     * @returns {Number} squared length of a
     * @function
     */
    export const squaredLength = quat.squaredLength;

    /**
     * Alias for {@link quat2.squaredLength}
     * @function
     */
    export const sqrLen = squaredLength;

    /**
     * Normalize a dual quat
     *
     * @param {quat2} out the receiving dual quaternion
     * @param {quat2} a dual quaternion to normalize
     * @returns {quat2} out
     * @function
     */
    export function normalize(out, a) {
      let magnitude = squaredLength(a);
      if (magnitude > 0) {
        magnitude = Math.sqrt(magnitude);

        let a0 = a[0] / magnitude;
        let a1 = a[1] / magnitude;
        let a2 = a[2] / magnitude;
        let a3 = a[3] / magnitude;

        let b0 = a[4];
        let b1 = a[5];
        let b2 = a[6];
        let b3 = a[7];

        let a_dot_b = (a0 * b0) + (a1 * b1) + (a2 * b2) + (a3 * b3);

        out[0] = a0;
        out[1] = a1;
        out[2] = a2;
        out[3] = a3;

        out[4] = (b0 - (a0 * a_dot_b)) / magnitude;
        out[5] = (b1 - (a1 * a_dot_b)) / magnitude;
        out[6] = (b2 - (a2 * a_dot_b)) / magnitude;
        out[7] = (b3 - (a3 * a_dot_b)) / magnitude;
      }
      return out;
    }

    /**
     * Returns a string representation of a dual quatenion
     *
     * @param {quat2} a dual quaternion to represent as a string
     * @returns {String} string representation of the dual quat
     */
    export function str(a) {
      return 'quat2(' + a[0] + ', ' + a[1] + ', ' + a[2] + ', ' + a[3] + ', ' +
        a[4] + ', ' + a[5] + ', ' + a[6] + ', ' + a[7] + ')';
    }

    /**
     * Returns whether or not the dual quaternions have exactly the same elements in the same position (when compared with ===)
     *
     * @param {quat2} a the first dual quaternion.
     * @param {quat2} b the second dual quaternion.
     * @returns {Boolean} true if the dual quaternions are equal, false otherwise.
     */
    export function exactEquals(a, b) {
      return a[0] === b[0] && a[1] === b[1] && a[2] === b[2] && a[3] === b[3] &&
        a[4] === b[4] && a[5] === b[5] && a[6] === b[6] && a[7] === b[7];
    }

    /**
     * Returns whether or not the dual quaternions have approximately the same elements in the same position.
     *
     * @param {quat2} a the first dual quat.
     * @param {quat2} b the second dual quat.
     * @returns {Boolean} true if the dual quats are equal, false otherwise.
     */
    export function equals(a, b) {
      let a0 = a[0],
        a1 = a[1],
        a2 = a[2],
        a3 = a[3],
        a4 = a[4],
        a5 = a[5],
        a6 = a[6],
        a7 = a[7];
      let b0 = b[0],
        b1 = b[1],
        b2 = b[2],
        b3 = b[3],
        b4 = b[4],
        b5 = b[5],
        b6 = b[6],
        b7 = b[7];
      return (Math.abs(a0 - b0) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a0), Math.abs(b0)) &&
        Math.abs(a1 - b1) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a1), Math.abs(b1)) &&
        Math.abs(a2 - b2) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a2), Math.abs(b2)) &&
        Math.abs(a3 - b3) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a3), Math.abs(b3)) &&
        Math.abs(a4 - b4) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a4), Math.abs(b4)) &&
        Math.abs(a5 - b5) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a5), Math.abs(b5)) &&
        Math.abs(a6 - b6) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a6), Math.abs(b6)) &&
        Math.abs(a7 - b7) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a7), Math.abs(b7)));
    }