Odd Angles Formula Trivia

[Albert Chan]

(08-15-2019 01:55 AM)Albert Chan Wrote:  x = 1/5³ = 0.008 // reduced angle, in radian

cos(x) ≅ 1 - x^2/2 + x^4/24 = 1 - 0.00003199982933 = 0.9999680002
cos(0.04) = 0.9992001048, error = 19 ULP
cos(0.20) = 0.9800665309, error = 469 ULP
cos(1.00) = 0.5403013120, error = 9939 ULP

A trick to improve accuracy, by defining cosm1(x) = cos(x) - 1

Polynomial to scale cosm1(x) to cosm1(5x)
= T₅(x+1) - 1
= 25x + 100x^2 + 140x^3 + 80x^4 + 16x^5
= ((4*x + 10)*x + 5)^2 * x

cosm1(x) ≅ -x^2/2 + x^4/24 = -0.00003199982933
cosm1(0.04) = -0.0007998933389, error = -1 ULP
cosm1(0.20) = -0.01993342215, error = -1 ULP
cosm1(1.00) = -0.4596976940, error = -1 ULP

cos(1) = cosm1(1) + 1 = 0.5403023060, error = -1 ULP