HP HP-15C - When No Root Is Found

184 Appendix: Accuracy of Numerical Calculations
Level 2: Correctly Rounded for Possibly
Perturbed Input
Trigonometric Functions of Real Radian Angles
Recall example 3, which noted that the calculator's (jr] key delivers
an approximation to TT correct to 10 significant digits but still
slightly different from rr, so 0 = sin(Tr) ¥= sin (0) f°r which the
calculator delivers
fSINl (H) = -4.100000000 X ID'10.
This computed value is not quite the same as the true value
sin (H) = -4.10206761537356... X 10"10.
Whether the discrepancy looks small (absolute error less than 2.1
X 10"13) or relatively large (wrong in the fourth significant digit) for
a 10-significant-digit calculator, the discrepancy deserves to be
understood because it foreshadows other errors that look, at first
sight, much more serious.
Consider
10147r = 314159265358979.3238462643...
with sin (10147r) = 0 and
1014 X 0 = 314159265400000
with [SIN] (1014 0) = 0.7990550814, although the true
sin (1014H) = -0.78387....
The wrong sign is an error too serious to ignore; it seems to suggest
a defect in the calculator. To understand the error in trigonometric
functions we must pay attention to small differences among TT and
two approximations to TT:
true TT = 3.1415926535897932384626433...
key H = 3.141592654 (matches rr to 10 digits)
internal p = 3.141592653590 (matches TT to 13 digits)
Then all is explained by the following formula for the calculated
value: |SIN|(;c) — sm(xir/p) to within ±0.6 units in its last (10th)
significant digit.
More generally, if trig(^c) is any of the functions sin(jc), cos(x), or
tan(x), evaluated in real Radians mode, the HP-15C produces