Smoothly accelerates and decelerates to land exactly on target

Approacher2d.cpp

void updateApproacher(v2& currentPosition, v2& currentVelocity, v2 target, float delta, float acceleration, float deceleration, float maxSpeed){
	if(delta == 0) return;
	v2 dto = target - currentPosition;
	float dtm = dto.magnitude();
	float vm = currentVelocity.magnitude();
	auto distanceToStop = [&](float initialVelocity, float deceleration){
		float deltdec = deceleration*delta;
		float numberOfFramesToSlowDown = max(0.0f, floor((initialVelocity - deltdec)/(deltdec)));
		return numberOfFramesToSlowDown*delta*(initialVelocity - (deltdec*(numberOfFramesToSlowDown + 1))/2);
	};
	if(vm < deceleration*delta && dtm < (vm + deceleration*delta)*delta){
		currentPosition = target;
		currentVelocity = v2(0,0);
	}else{
		float deltacc = delta*acceleration;
		float deltdec = delta*deceleration;
		v2 initialProjection;
		v2 vn;
		if(vm != 0){
			vn = currentVelocity/vm;
			initialProjection = currentPosition + distanceToStop(vm, deceleration)*vn;
		}else{
			initialProjection = currentPosition;
		}
		v2 targetRelativeToStoppingPoint = target - initialProjection;
		float trm = targetRelativeToStoppingPoint.magnitude();
		if(trm != 0){
			v2 trn = targetRelativeToStoppingPoint/trm;
			if(vm != 0){
				if(vn.dot(trn) < 0){
					currentVelocity = (currentVelocity - deltdec*vn).capLengthAt(maxSpeed);
				}else{ //then we have some room for cleverness
					//look over the range of reasonable alterations we could make. If one goes too far and the other goes not far enough, interpolate between them
					v2 prospectiveFullAheadVelocity = (currentVelocity + deltacc*trn).capLengthAt(maxSpeed);
					float pfavm = prospectiveFullAheadVelocity.magnitude();
					if(pfavm){
						v2 pfavn = prospectiveFullAheadVelocity/pfavm;
						float frontDistanceToGo = pfavn.dot(dto);
						float frontDistanceToStop = distanceToStop(pfavm, deceleration) + pfavm*delta;
						if(frontDistanceToGo < frontDistanceToStop){
							//have to do interpolation
							v2 prospectiveFullBehindVelocity = (currentVelocity - deltdec*vn).capLengthAt(maxSpeed);
							float fbm = prospectiveFullBehindVelocity.magnitude();
							float behindDistanceToStop;
							if(fbm){
								behindDistanceToStop = distanceToStop(fbm, deceleration) + fbm*delta;
							}else{
								behindDistanceToStop = 0;
							}
							currentVelocity = lerp(
								prospectiveFullBehindVelocity,
								prospectiveFullAheadVelocity,
								unlerpUnit(behindDistanceToStop, frontDistanceToStop, frontDistanceToGo)
							);
						}else{
							//don't have to do interpolation, prospectiveFullAheadVelocity is fine
							currentVelocity = prospectiveFullAheadVelocity;
						}
					}else{
						currentVelocity = prospectiveFullAheadVelocity;
					}
				}
			}else{
				v2 dtn = dto/dtm;
				currentVelocity = currentVelocity + deltacc*dtn;
			}
		}else{
			currentVelocity = currentVelocity.shortenBy(deltdec);
		}
		currentPosition += currentVelocity*delta;
	}
}