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    <p>I wrote small test. It calcutates expected value with double precision. Then it calculates an error with your options. The first option is better:</p> <pre><code>Algorithm: FloatTest$1 option 1 error = 3.802792362162126 option 2 error = 4.333273185303996 Algorithm: FloatTest$2 option 1 error = 3.802792362167937 option 2 error = 4.333273185305868 </code></pre> <p>The Java code: </p> <pre><code>import org.junit.Test; public class FloatTest { @Test public void test() { testImpl(new ExpectedAlgorithm() { public double te(double cos_theta) { return Math.sqrt(1.0f - cos_theta * cos_theta); } }); testImpl(new ExpectedAlgorithm() { public double te(double cos_theta) { return Math.sqrt((1.0f + cos_theta) * (1.0f - cos_theta)); } }); } public void testImpl(ExpectedAlgorithm ea) { double delta1 = 0; double delta2 = 0; for (double cos_theta = -1; cos_theta &lt;= 1; cos_theta += 1e-8) { double[] delta = delta(cos_theta, ea); delta1 += delta[0]; delta2 += delta[1]; } System.out.println("Algorithm: " + ea.getClass().getName()); System.out.println("option 1 error = " + delta1); System.out.println("option 2 error = " + delta2); } private double[] delta(double cos_theta, ExpectedAlgorithm ea) { double expected = ea.te(cos_theta); double delta1 = Math.abs(expected - t1((float) cos_theta)); double delta2 = Math.abs(expected - t2((float) cos_theta)); return new double[]{delta1, delta2}; } private double t1(float cos_theta) { return Math.sqrt(1.0f - cos_theta * cos_theta); } private double t2(float cos_theta) { return Math.sqrt((1.0f + cos_theta) * (1.0f - cos_theta)); } interface ExpectedAlgorithm { double te(double cos_theta); } } </code></pre>
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    1. CONice! But I don't understand the difference between `FloatTest$1` and `FloatTest$2`. Why do you have 4 different numbers (instead of two)? It's been a while since I wrote Java myself.
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    2. COI use same options to calculate expected value :) So FloatTest$1 tests with option 1 algorithm. And FloatTest$2 tests with option 2 algorithm.
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    3. COAh, the *expected* value has to be computed as well, right - hello combinatorics ;) !
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