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Explain trig precision limits
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Michael Mclaughlin
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doc/API.html
52
doc/API.html
@ -916,6 +916,9 @@ Decimal.precision = 0
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<a href='#toNearest'><code>toNearest</code></a> and
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<a href='#trunc'><code>truncated</code></a>.
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</p>
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<p>
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See <code><a href='#Pi'>Pi</a></code> for the precision limit of the trigonometric methods.
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</p>
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<pre>Decimal.set({ precision: 5 })
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Decimal.precision // 5</pre>
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@ -1281,7 +1284,7 @@ y.cmp(NaN) // NaN</pre>
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Domain: [<code>-Infinity, Infinity</code>]<br />
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Range: [<code>-1, 1</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(0.25)
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x.cosine() // '0.96891242171064478414'
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@ -1431,7 +1434,7 @@ new Decimal(1).gte(x) // true</pre>
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Domain: [<code>-Infinity, Infinity</code>]<br />
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Range: [<code>1, Infinity</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(1)
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x.hyperbolicCosine() // '1.5430806348152437785'
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@ -1450,7 +1453,7 @@ y.cosh() // '1.1276259652063807852'</pre>
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Domain: [<code>-Infinity, Infinity</code>]<br />
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Range: [<code>-Infinity, Infinity</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(1)
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x.hyperbolicSine() // '1.1752011936438014569'
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@ -1469,7 +1472,7 @@ y.sinh() // '0.52109530549374736162'</pre>
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Domain: [<code>-Infinity, Infinity</code>]<br />
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Range: [<code>-1, 1</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(1)
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x.hyperbolicTangent() // '0.76159415595576488812'
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@ -1488,7 +1491,7 @@ y.tanh() // '0.4621171572600097585'</pre>
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Domain: [<code>-1, 1</code>]<br />
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Range: [<code>0, pi</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(0)
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x.inverseCosine() // '1.5707963267948966192'
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@ -1509,7 +1512,7 @@ y.acos() // '1.0471975511965977462'</pre>
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Domain: [<code>1, Infinity</code>]<br />
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Range: [<code>0, Infinity</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(5)
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x.inverseHyperbolicCosine() // '2.2924316695611776878'
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@ -1530,7 +1533,7 @@ y.acosh() // '4.6050701709847571595'</pre>
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Domain: [<code>-Infinity, Infinity</code>]<br />
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Range: [<code>-Infinity, Infinity</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(5)
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x.inverseHyperbolicSine() // '2.3124383412727526203'
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@ -1551,7 +1554,7 @@ y.asinh() // '4.6052701709914238266'</pre>
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Domain: [<code>-1, 1</code>]<br />
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Range: [<code>-Infinity, Infinity</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(0.5)
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x.inverseHyperbolicTangent() // '0.5493061443340548457'
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@ -1570,7 +1573,7 @@ y.atanh() // '0.97295507452765665255'</pre>
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Domain: [<code>-1, 1</code>]<br />
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Range: [<code>-pi/2, pi/2</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(0.5)
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x.inverseSine() // '0.52359877559829887308'
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@ -1589,7 +1592,7 @@ y.asin() // '0.84806207898148100805'</pre>
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Domain: [<code>-Infinity, Infinity</code>]<br />
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Range: [<code>-pi/2, pi/2</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(0.5)
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x.inverseTangent() // '0.46364760900080611621'
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@ -1741,7 +1744,8 @@ y.log(2) // '8'</pre>
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<p>Logarithms to base <code>2</code> or <code>10</code> will always be correctly rounded.</p>
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<p>
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See <a href='#pow'><code>toPower</code></a> for the circumstances in which this method may
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return an incorrectly rounded result.
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return an incorrectly rounded result, and see <a href='#ln'><code>naturalLogarithm</code></a>
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for the precision limit.
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</p>
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<p>The performance of this method degrades exponentially with increasing digits.</p>
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@ -1928,7 +1932,7 @@ x // '1234.5'</pre>
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Domain: [<code>-Infinity, Infinity</code>]<br />
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Range: [<code>-1, 1</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(0.5)
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x.sine() // '0.47942553860420300027'
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@ -1970,7 +1974,7 @@ y.sqrt().eq( y.pow(0.5) ) // true</pre>
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Domain: [<code>-Infinity, Infinity</code>]<br />
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Range: [<code>-Infinity, Infinity</code>]
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</p>
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<p>See <a href='#Pi'><code>Pi</code></a>.</p>
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<p>See <a href='#Pi'><code>Pi</code></a> for the precision limit of this method.</p>
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<pre>
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x = new Decimal(0.5)
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x.tangent() // '0.54630248984379051326'
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@ -2235,7 +2239,7 @@ x = new Decimal(1.39)
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x.toNearest(0.25) // '1.5'
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y = new Decimal(9.499)
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y.toNearest(0.5, Decimal.ROUND_UP) // '9.5'
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y.toNearest(0.5, Decimal.ROUND_UP) // '9.5'
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y.toNearest(0.5, Decimal.ROUND_DOWN) // '9'</pre>
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@ -2594,13 +2598,27 @@ try {
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</p>
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<p>
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It has a precision of <code>1025</code> digits, meaning that the trigonometric methods
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can accurately calculate up to at least <code>1000</code> digits.
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can calculate up to just over <code>1000</code> digits, but the actual figure depends on the
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precision of the argument passed to them. To calculate the actual figure use:
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</p>
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<p><b>maximum_result_precision = 1000 - argument_precision</b></p>
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For example, the following both work fine:
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<pre>
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Decimal.set({precision: 991}).tan(123456789)
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Decimal.set({precision: 9}).tan(991_digit_number)</pre>
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<p>
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as, for each, the result precision plus the argument precision, i.e. <code>991 + 9</code> and
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<code>9 + 991</code>, is less than or equal to <code>1000</code>.
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</p>
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<p>
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If greater precision is required then the value of <code>PI</code> will need to be extended to
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about <code>25</code> digits more than the precision required.
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about <code>25</code> digits more than the precision required. The time taken by the methods
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will then be the limiting factor.
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</p>
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<p>
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The value can also be shortened to reduce the size of the source file if such high precision
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is not required.
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</p>
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<p>The value can also be shortened to reduce the size of the source file.</p>
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<p>To get the value of pi:</p>
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<pre>
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pi = Decimal.acos(-1)</pre>
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