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A much more serious source of error in floating-point computation is
*significance loss* from subtraction of nearly equal values. This
means that the number of bits in the significand of the result is
fewer than the size of the value would permit. If the values being
subtracted are close enough, but still not equal, a *single
subtraction* can wipe out all correct digits, possibly contaminating
all future computations.

Floating-point calculations can sometimes be carefully designed so
that significance loss is not possible, such as summing a series where
all terms have the same sign. For example, the Taylor series
expansions of the trigonometric and hyperbolic sines have terms of
identical magnitude, of the general form

. However, those in the trigonometric sine series
alternate in sign, while those in the hyperbolic sine series are all
positive. Here is the output of two small programs that sum `x`**(2*`n` +
1) / (2*`n` + 1)!`k`
terms of the series for `sin (`

, and compare the computed
sums with known-to-be-accurate library functions:
`x`)

x = 10 k = 51 s (x) = -0.544_021_110_889_270 sin (x) = -0.544_021_110_889_370 x = 20 k = 81 s (x) = 0.912_945_250_749_573 sin (x) = 0.912_945_250_727_628 x = 30 k = 109 s (x) = -0.987_813_746_058_855 sin (x) = -0.988_031_624_092_862 x = 40 k = 137 s (x) = 0.617_400_430_980_474 sin (x) = 0.745_113_160_479_349 x = 50 k = 159 s (x) = 57_105.187_673_745_720_532 sin (x) = -0.262_374_853_703_929 // sinh(x) series summation with positive signs // with k terms needed to converge to machine precision x = 10 k = 47 t (x) = 1.101_323_287_470_340e+04 sinh (x) = 1.101_323_287_470_339e+04 x = 20 k = 69 t (x) = 2.425_825_977_048_951e+08 sinh (x) = 2.425_825_977_048_951e+08 x = 30 k = 87 t (x) = 5.343_237_290_762_229e+12 sinh (x) = 5.343_237_290_762_231e+12 x = 40 k = 105 t (x) = 1.176_926_334_185_100e+17 sinh (x) = 1.176_926_334_185_100e+17 x = 50 k = 121 t (x) = 2.592_352_764_293_534e+21 sinh (x) = 2.592_352_764_293_536e+21

We have added underscores to the numbers to enhance readability.

The `sinh (`

series with positive terms can be summed to
high accuracy. By contrast, the series for `x`)`sin (`

suffers increasing significance loss, so that when `x`)`x` = 30 only
two correct digits remain. Soon after, all digits are wrong, and the
answers are complete nonsense.

An important skill in numerical programming is to recognize when
significance loss is likely to contaminate a computation, and revise
the algorithm to reduce this problem. Sometimes, the only practical
way to do so is to compute in higher intermediate precision, which is
why the extended types like `long double`

are important.

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