With the ubiquity of mobile touchscreen devices like smartphones, two widely used text entry methods have emerged: small touch-based keyboards and speech recognition. Although speech recognition has been available on desktop computers for years, it has continued to improve at a rapid pace, and it is currently unknown how today's modern speech recognizers compare to state-of-the-art mobile touch keyboards, which also have improved considerably since their inception. To discover both methods' "upper-bound performance," we evaluated them in English and Mandarin Chinese on an Apple iPhone 6 Plus in a laboratory setting. Our experiment was carried out using Baidu's Deep Speech 2, a deep learning-based speech recognition system, and the built-in QWERTY (English) or Pinyin (Mandarin) Apple iOS keyboards. We found that with speech recognition, the English input rate was 2.93 times faster (153 vs. 52 WPM), and the Mandarin Chinese input rate was 2.87 times faster (123 vs. 43 WPM) than the keyboard for short message transcription under laboratory conditions for both methods. Furthermore, although speech made fewer errors during entry (5.30% vs. 11.22% corrected error rate), it left slightly more errors in the final transcribed text (1.30% vs. 0.79% uncorrected error rate). Our results show that comparatively, under ideal conditions for both methods, upper-bound speech recognition performance has greatly improved compared to prior systems, and might see greater uptake in the future, although further study is required to quantify performance in non-laboratory settings for both methods.