89 lines
19 KiB
Text
89 lines
19 KiB
Text
-->
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<span style="color: #008080;">without</span> <span style="color: #008080;">js</span> <span style="color: #000080;font-style:italic;">-- no mpfr_get_d_2exp() in mpfr.js as yet</span>
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<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.2"</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- mpfr_get_d_2exp(), mpfr_addmul_si()</span>
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<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
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<span style="color: #004080;">mpfr</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k2</span><span style="color: #0000FF;">;</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">e10</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">e2</span>
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<span style="color: #004080;">atom</span> <span style="color: #000000;">f</span>
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<span style="color: #000080;font-style:italic;">//log(x/y) with the Taylor series for atanh(x-y/x+y)</span>
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<span style="color: #008080;">procedure</span> <span style="color: #000000;">ln</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpfr</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">y</span><span style="color: #0000FF;">)</span>
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<span style="color: #004080;">mpfr</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">;</span>
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<span style="color: #7060A8;">assert</span><span style="color: #0000FF;">((</span><span style="color: #000000;">x</span><span style="color: #0000FF;">-</span><span style="color: #000000;">y</span><span style="color: #0000FF;">)==</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">mpfr_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">y</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">mpfr_si_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// s = 1 / (x + y)</span>
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<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// k2 = s * s</span>
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<span style="color: #7060A8;">mpfr_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
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<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
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<span style="color: #000000;">k</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">;</span>
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<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k2</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// d *= k2</span>
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<span style="color: #7060A8;">mpfr_div_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// q = d / k</span>
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<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// s += q</span>
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<span style="color: #0000FF;">{</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mpfr_get_d_2exp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">if</span> <span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">)>=</span><span style="color: #000000;">e2</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
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<span style="color: #7060A8;">mpfr_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">//s *= 2</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
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<span style="color: #004080;">mpfr</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">60</span><span style="color: #0000FF;">,</span> <span style="color: #000080;font-style:italic;">-- (required precision in decimal dp *6/10)</span>
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<span style="color: #000000;">n2</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">41</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">30</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">18</span><span style="color: #0000FF;">;</span>
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<span style="color: #000080;font-style:italic;">// n = 2^i * 3^j * 5^k
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// log(n) = r * log(16/15) + s * log(25/24) + t * log(81/80)
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// solve linear system for r, s, t
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// 4 -3 -4| i
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// -1 -1 4| j
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// -1 2 -1| k
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//decimal precision</span>
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<span style="color: #000000;">e10</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0.6</span><span style="color: #0000FF;">)</span>
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<span style="color: #000080;font-style:italic;">//binary precision</span>
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<span style="color: #000000;">e2</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">((</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">e10</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">0.30103</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">mpfr_set_default_precision</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e2</span><span style="color: #0000FF;">)</span>
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<span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">5</span><span style="color: #0000FF;">)</span>
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<span style="color: #000080;font-style:italic;">//Compute log terms</span>
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<span style="color: #000000;">ln</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">16</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">15</span><span style="color: #0000FF;">)</span> <span style="color: #7060A8;">mpfr_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// a = r * b</span>
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<span style="color: #000000;">ln</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">25</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">24</span><span style="color: #0000FF;">)</span> <span style="color: #000000;">mpfr_addmul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// a += s * b</span>
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<span style="color: #000000;">ln</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">81</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">80</span><span style="color: #0000FF;">)</span> <span style="color: #000000;">mpfr_addmul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// a += t * b</span>
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<span style="color: #000000;">mpfr_neg</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// a = -a</span>
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<span style="color: #7060A8;">mpfr_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// b = 1</span>
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<span style="color: #7060A8;">mpfr_set</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">mpfr_set</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">v</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
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<span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">;</span>
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<span style="color: #000000;">n2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">;</span>
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<span style="color: #7060A8;">mpfr_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// k2 = k * k (as below)</span>
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<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
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<span style="color: #000000;">mpfr_add_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">*</span><span style="color: #000000;">2</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// k2 += 2k + 1</span>
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<span style="color: #000000;">k</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">;</span>
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<span style="color: #7060A8;">mpfr_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k2</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">mpfr_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n2</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// b = b * n2 / k2</span>
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<span style="color: #7060A8;">mpfr_div_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">mpfr_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n2</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">mpfr_add</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">mpfr_div_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// a = (a * n2 / k + b) / k</span>
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<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// u += a</span>
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<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">v</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// v += b</span>
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<span style="color: #0000FF;">{</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mpfr_get_d_2exp</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">if</span> <span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">)>=</span><span style="color: #000000;">e2</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
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<span style="color: #7060A8;">mpfr_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span>
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<span style="color: #004080;">string</span> <span style="color: #000000;">su</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_get_fixed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">u</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e10</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"gamma %s (maxerr. 1e-%d)\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">su</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">e10</span><span style="color: #0000FF;">})</span>
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<!--
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