Abstract
We developed a new method to compute the cosine amplitude function, $$c\equiv \mathrm{cn}(u|m)$$ , by using its double argument formula. The accumulation of round-off errors is effectively suppressed by the introduction of a complementary variable, $$b\equiv 1-c$$ , and a conditional switch between the duplication of $$b$$ and $$c$$ . The sine and delta amplitude functions, $$s \equiv \mathrm{sn}(u|m)$$ and $$d \equiv \mathrm{dn}(u|m)$$ , are evaluated from thus computed $$b$$ or $$c$$ by means of their identity relations. The new method is sufficiently precise as its errors are less than a few machine epsilons. Also, it is significantly faster than the existing procedures. In case of single precision computation, it runs more than 50 times faster than Bulirsch's sncndn based on the Gauss transformation and 2.7 times faster than our previous method based on the simultaneous duplication of $$s,c$$ and $$d$$ . The ratios change to 7.6 and 3.5 respectively in case of the double precision environment.
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