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sin
Circular sine.SYNOPSIS:
double x, y, sin();
y = sin(x);
DESCRIPTION:
Range reduction is into intervals of pi/4. The reduction error is nearly eliminated by contriving an extended precision modular arithmetic.
Two polynomial approximating functions are employed.
Between 0 and pi/4 the sine is approximated by:
x + x**3 P(x**2).
Between pi/4 and pi/2 the cosine is represented as:
1 - x**2 Q(x**2).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0, 10 150000 3.0e-17 7.8e-18
IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
ERROR MESSAGES:
message condition value returned
total loss x > 1.073741824e9 0.0
Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9. The loss is not gradual, but jumps suddenly to about 1 part in 10e7. Results might be meaningless for x > 2**49 = 5.6e14. The routine as implemented flags a TLOSS error for x > 2**30 and returns 0.0.