HP -15C Advanced Functions Handbook page 193

Appendix: Accuracy of Numerical Calculations 191
Example 6: The Smaller Root of a Quadratic. The two roots x
and y of the quadratic equation c — 2bz + az
2
= 0 are real whenever
d = b
2
— ac is nonnegative. Then the root y of smaller magnitude
can be regarded as a function y = f(a,b,c) of the quadratic's
coefficients
I (6-x/dsgn(6))/a i f a ^ O
f(a,b,c)— <
I (c/6)/2 otherwise.
Were this formula translated directly in a program F(a, b, c)
intended to calculate f ( a , b, c), then whenever ac is so small
compared with b
2
that the computed value of d rounds to b
2
, that
program could deliver F = 0 even though f^ 0. So drastic an error
cannot be explained by backward error analysis because no
relatively small perturbations to each coefficient a, b, and c could
drive c to zero, as would be necessary to change the smaller root y
into 0. On the other hand, the algebraically equivalent formula
(
c /(b + \fd sgn( b)) if divisor is nonzero
0 otherwise
translates into a much more accurate program F whose errors do
no more damage than would a perturbation in the last (10th)
significant digit of c. Such a program will be listed later (page 205)
and must be used in those instances, common in engineering, when
the smaller root y is needed accurately despite the fact that the
quadratic's other unwanted root is relatively large.
Almost all the functions built into the HP-15C have been designed
so that backward error analysis will account for their errors
satisfactorily. The exceptions are [SOLVE], [7T|, and the statistics
keys [si, | L.R. |, and |y,r| which can malfunction in certain
pathological cases. Otherwise, every calculator function F
intended to produce f ( x ) produces instead a value F(x) no farther
from/(;t) than if first x had been perturbed to x + 8x with|<5;c|^ TI\X\,
thenf(x + dx) were perturbed to (f+df)(x + 5x) with \8f\ e|/|. The
tolerances r\d e vary a little from function to function; roughly
speaking,
rj — 0 and e < 10~
9
for all functions in Level 1,
77 < 10~
12
and e < 6 X 10~
10
for other real and complex functions.