$builder = $this->em->createQueryBuilder();
$builder->select('a')
        ->from('EntityAppointment', 'a')
        ->andWhere('a.when = :date')
        ->setParameter('date', $date);

->andWhere('DATE(a.when) = :date')

Error: Expected known function, got 'DATE'

namespace CeradBundleCoreBundleDoctrineDQL;

use DoctrineORMQueryLexer;
use DoctrineORMQueryASTFunctionsFunctionNode;

class Date extends FunctionNode
{
    public $date;

    public function getSql(DoctrineORMQuerySqlWalker $sqlWalker)
    {
        return "DATE(" . $sqlWalker->walkArithmeticPrimary($this->date) . ")";
    }
    public function parse(DoctrineORMQueryParser $parser)
    {
        $parser->match(Lexer::T_IDENTIFIER);
        $parser->match(Lexer::T_OPEN_PARENTHESIS);

        $this->date = $parser->ArithmeticPrimary();

        $parser->match(Lexer::T_CLOSE_PARENTHESIS);
    }
}

doctrine:
  orm:
  default_entity_manager:       default
  auto_generate_proxy_classes: %kernel.debug%

  entity_managers:

    default:
      connection: default
      ...
      dql:
        datetime_functions:
          date:  CeradBundleCoreBundleDoctrineDQLDate

$config = new DoctrineORMConfiguration();
$config->addCustomDatetimeFunction('DATE', 'blahblahDate');

$builder = $this->em->createQueryBuilder();
$builder->select('a')
        ->from('EntityAppointment', 'a')
        ->andWhere('a.when >= :date_start')
        ->andWhere('a.when <= :date_end')
        ->setParameter('date_start', $date->format('Y-m-d 00:00:00'))
        ->setParameter('date_end',   $date->format('Y-m-d 23:59:59'));