Vocoders received renewed attention recently as basic components in speech synthesis applications such as voice transformation, voice conversion and statistical parametric speech synthesis. This paper presents a new vocoder synthesizer, referred to as Vocaine, that features a novel Amplitude Modulated-Frequency Modulated (AM-FM) speech model, a new way to synthesize non-stationary sinusoids using quadratic phase splines and a super fast cosine generator. Extensive evaluations are made against several state-ofthe-art methods in Copy-Synthesis and Text-To-Speech synthesis experiments. Vocaine matches or outperforms STRAIGHT in CopySynthesis experiments and outperforms our baseline real-time optimized Mixed-Excitation vocoder with the same computational cost. We report that Vocaine considerably improves our statistical TTS synthesizers and that our new statistical parametric synthesizer  matched the quality of our mature production Unit-Selection system with uncompressed waveforms.