A simple code snippet for authenticating a user using email address

authenticate_with_email.php

public function authenticate_with_email( $user, $email, $password ) {
    if ( is_wp_error( $user ) ) {
        // Already failed
        return $user;
    }

    if( empty( $email ) || empty ( $password ) ) {
        $error = new WP_Error();
        if ( empty( $email ) ) {
            $error->add( 'empty_username', __( '<strong>ERROR</strong>: Email field is empty.' ) );
        }

        if ( empty( $password ) ) {
            $error->add( 'empty_password', __( '<strong>ERROR</strong>: Password field is empty.' ) );
        }

        return $error;
    }

    // Check if user exists in WordPress database and password is valid
    // Notice that we look the user up using the email
    $user = get_user_by( 'email', $email );
    if ( ! $user ) {
        // Optional: if no user with this email exists, you can check the username if you like...
        $user = get_user_by( 'login', $email );
    }

    // Check password
    if ( ! $user || ! wp_check_password( $password, $user->user_pass, $user->ID ) ) {
        $error = new WP_Error();
        $error->add( 'invalid', __('<strong>ERROR</strong>: Either the email or password you entered is invalid.' ) );
        return $error;
    }

    // User logged in OK
    return $user;
}