babeljs.io version of arrow function fails in safari

The fourth line of the following code (produces the error SyntaxError: Unexpected keyword 'function'. Expected ')' to end a argument list. is produced from babeljs.io. Can you fix the error? Here is a working jsfiddle of the original unbabeled code to use.

"use strict";

var recordAudio = function recordAudio() {
  return new Promise(async function (resolve) {         
    var stream = await navigator.mediaDevices.getUserMedia({
      audio: true
    });
    var mediaRecorder = new MediaRecorder(stream);
    var audioChunks = [];
    mediaRecorder.addEventListener("dataavailable", function (event) {
      audioChunks.push(event.data);
    });

    var start = function start() {
      return mediaRecorder.start();
    };

    var stop = function stop() {
      return new Promise(function (resolve) {
        mediaRecorder.addEventListener("stop", function () {
          var audioBlob = new Blob(audioChunks);
          var audioUrl = URL.createObjectURL(audioBlob);
          var audio = new Audio(audioUrl);

          var play = function play() {
            return audio.play();
          };

          resolve({
            audioBlob: audioBlob,
            audioUrl: audioUrl,
            play: play
          });
        });
        mediaRecorder.stop();
      });
    };

    resolve({
      start: start,
      stop: stop
    });
  });
};

var sleep = function sleep(time) {
  return new Promise(function (resolve) {
    return setTimeout(resolve, time);
  });
};

var handleAction = async function handleAction() {
  var recorder = await recordAudio();
  var actionButton = document.getElementById('action');
  actionButton.disabled = true;
  recorder.start();
  await sleep(3000);
  var audio = await recorder.stop();
  audio.play();
  await sleep(3000);
  actionButton.disabled = false;
};

The above code is produced by the following code "arrow function" version taken from Bryan Jennings

const recordAudio = () =>
  new Promise(async resolve => {
    const stream = await navigator.mediaDevices.getUserMedia({ audio: true });
    const mediaRecorder = new MediaRecorder(stream);
    const audioChunks = [];

    mediaRecorder.addEventListener("dataavailable", event => {
      audioChunks.push(event.data);
    });

    const start = () => mediaRecorder.start();

    const stop = () =>
      new Promise(resolve => {
        mediaRecorder.addEventListener("stop", () => {
          const audioBlob = new Blob(audioChunks);
          const audioUrl = URL.createObjectURL(audioBlob);
          const audio = new Audio(audioUrl);
          const play = () => audio.play();
          resolve({ audioBlob, audioUrl, play });
        });

        mediaRecorder.stop();
      });

    resolve({ start, stop });
  });

const sleep = time => new Promise(resolve => setTimeout(resolve, time));

const handleAction = async () => {
  const recorder = await recordAudio();
  const actionButton = document.getElementById('action');
  actionButton.disabled = true;
  recorder.start();
  await sleep(3000);
  const audio = await recorder.stop();
  audio.play();
  await sleep(3000);
  actionButton.disabled = false;
}