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| 22 | <title>Significance Loss (GNU C Language Manual)</title>
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| 30 | <link href="Symbol-Index.html" rel="index" title="Symbol Index">
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| 31 | <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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| 32 | <link href="Floating-Point-in-Depth.html" rel="up" title="Floating Point in Depth">
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| 55 | </head>
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| 57 | <body lang="en">
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| 58 | <span id="Significance-Loss"></span><div class="header">
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| 59 | <p>
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| 60 | Next: <a href="Fused-Multiply_002dAdd.html" accesskey="n" rel="next">Fused Multiply-Add</a>, Previous: <a href="Rounding-Issues.html" accesskey="p" rel="prev">Rounding Issues</a>, Up: <a href="Floating-Point-in-Depth.html" accesskey="u" rel="up">Floating Point in Depth</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Symbol-Index.html" title="Index" rel="index">Index</a>]</p>
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| 61 | </div>
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| 62 | <hr>
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| 63 | <span id="Significance-Loss-1"></span><h3 class="section">28.9 Significance Loss</h3>
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| 64 | <span id="index-significance-loss-_0028floating-point_0029"></span>
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| 65 | <span id="index-floating_002dpoint-significance-loss"></span>
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| 66 |
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| 67 | <p>A much more serious source of error in floating-point computation is
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| 68 | <em>significance loss</em> from subtraction of nearly equal values. This
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| 69 | means that the number of bits in the significand of the result is
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| 70 | fewer than the size of the value would permit. If the values being
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| 71 | subtracted are close enough, but still not equal, a <em>single
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| 72 | subtraction</em> can wipe out all correct digits, possibly contaminating
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| 73 | all future computations.
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| 74 | </p>
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| 75 | <p>Floating-point calculations can sometimes be carefully designed so
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| 76 | that significance loss is not possible, such as summing a series where
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| 77 | all terms have the same sign. For example, the Taylor series
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| 78 | expansions of the trigonometric and hyperbolic sines have terms of
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| 79 | identical magnitude, of the general form <code><var>x</var>**(2*<var>n</var> +
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| 80 | 1) / (2*<var>n</var> + 1)!</code>. However, those in the trigonometric sine series
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| 81 | alternate in sign, while those in the hyperbolic sine series are all
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| 82 | positive. Here is the output of two small programs that sum <var>k</var>
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| 83 | terms of the series for <code>sin (<var>x</var>)</code>, and compare the computed
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| 84 | sums with known-to-be-accurate library functions:
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| 85 | </p>
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| 86 | <div class="example">
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| 87 | <pre class="example">x = 10 k = 51
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| 88 | s (x) = -0.544_021_110_889_270
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| 89 | sin (x) = -0.544_021_110_889_370
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| 90 |
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| 91 | x = 20 k = 81
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| 92 | s (x) = 0.912_945_250_749_573
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| 93 | sin (x) = 0.912_945_250_727_628
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| 94 |
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| 95 | x = 30 k = 109
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| 96 | s (x) = -0.987_813_746_058_855
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| 97 | sin (x) = -0.988_031_624_092_862
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| 98 |
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| 99 | x = 40 k = 137
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| 100 | s (x) = 0.617_400_430_980_474
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| 101 | sin (x) = 0.745_113_160_479_349
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| 102 |
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| 103 | x = 50 k = 159
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| 104 | s (x) = 57_105.187_673_745_720_532
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| 105 | sin (x) = -0.262_374_853_703_929
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| 106 |
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| 107 | // sinh(x) series summation with positive signs
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| 108 | // with k terms needed to converge to machine precision
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| 109 |
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| 110 | x = 10 k = 47
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| 111 | t (x) = 1.101_323_287_470_340e+04
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| 112 | sinh (x) = 1.101_323_287_470_339e+04
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| 113 |
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| 114 | x = 20 k = 69
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| 115 | t (x) = 2.425_825_977_048_951e+08
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| 116 | sinh (x) = 2.425_825_977_048_951e+08
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| 117 |
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| 118 | x = 30 k = 87
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| 119 | t (x) = 5.343_237_290_762_229e+12
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| 120 | sinh (x) = 5.343_237_290_762_231e+12
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| 121 |
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| 122 | x = 40 k = 105
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| 123 | t (x) = 1.176_926_334_185_100e+17
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| 124 | sinh (x) = 1.176_926_334_185_100e+17
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| 125 |
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| 126 | x = 50 k = 121
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| 127 | t (x) = 2.592_352_764_293_534e+21
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| 128 | sinh (x) = 2.592_352_764_293_536e+21
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| 129 | </pre></div>
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| 130 |
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| 131 | <p>We have added underscores to the numbers to enhance readability.
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| 132 | </p>
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| 133 | <p>The <code>sinh (<var>x</var>)</code> series with positive terms can be summed to
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| 134 | high accuracy. By contrast, the series for <code>sin (<var>x</var>)</code>
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| 135 | suffers increasing significance loss, so that when <var>x</var> = 30 only
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| 136 | two correct digits remain. Soon after, all digits are wrong, and the
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| 137 | answers are complete nonsense.
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| 138 | </p>
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| 139 | <p>An important skill in numerical programming is to recognize when
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| 140 | significance loss is likely to contaminate a computation, and revise
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| 141 | the algorithm to reduce this problem. Sometimes, the only practical
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| 142 | way to do so is to compute in higher intermediate precision, which is
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| 143 | why the extended types like <code>long double</code> are important.
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| 144 | </p>
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| 145 |
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| 146 |
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| 147 | <hr>
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| 148 | <div class="header">
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| 149 | <p>
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| 150 | Next: <a href="Fused-Multiply_002dAdd.html" accesskey="n" rel="next">Fused Multiply-Add</a>, Previous: <a href="Rounding-Issues.html" accesskey="p" rel="prev">Rounding Issues</a>, Up: <a href="Floating-Point-in-Depth.html" accesskey="u" rel="up">Floating Point in Depth</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Symbol-Index.html" title="Index" rel="index">Index</a>]</p>
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