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    * Licensed to the Apache Software Foundation (ASF) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * The ASF licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *      http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  package org.apache.commons.math4.legacy.analysis.function;
  
  import org.apache.commons.math4.legacy.analysis.UnivariateFunction;
  import org.apache.commons.math4.legacy.analysis.differentiation.DerivativeStructure;
  import org.apache.commons.math4.legacy.analysis.differentiation.UnivariateDifferentiableFunction;
  import org.apache.commons.math4.core.jdkmath.JdkMath;
  import org.junit.Assert;
  import org.junit.Test;
  
  public class SqrtTest {
     @Test
     public void testComparison() {
         final Sqrt s = new Sqrt();
         final UnivariateFunction f = new UnivariateFunction() {
             @Override
             public double value(double x) {
                 return JdkMath.sqrt(x);
             }
         };
  
         for (double x = 1e-30; x < 1e10; x *= 2) {
             final double fX = f.value(x);
             final double sX = s.value(x);
             Assert.assertEquals("x=" + x, fX, sX, 0);
         }
     }
  
     @Test
     public void testDerivativeComparison() {
         final UnivariateDifferentiableFunction sPrime = new Sqrt();
         final UnivariateFunction f = new UnivariateFunction() {
                 @Override
              public double value(double x) {
                     return 1 / (2 * JdkMath.sqrt(x));
                 }
             };
  
         for (double x = 1e-30; x < 1e10; x *= 2) {
             final double fX = f.value(x);
             final double sX = sPrime.value(new DerivativeStructure(1, 1, 0, x)).getPartialDerivative(1);
             Assert.assertEquals("x=" + x, fX, sX, JdkMath.ulp(fX));
         }
     }
  
     @Test
     public void testDerivativesHighOrder() {
         DerivativeStructure s = new Sqrt().value(new DerivativeStructure(1, 5, 0, 1.2));
         Assert.assertEquals(1.0954451150103322269, s.getPartialDerivative(0), 1.0e-16);
         Assert.assertEquals(0.45643546458763842789, s.getPartialDerivative(1), 1.0e-16);
         Assert.assertEquals(-0.1901814435781826783,  s.getPartialDerivative(2), 1.0e-16);
         Assert.assertEquals(0.23772680447272834785,  s.getPartialDerivative(3), 1.0e-16);
         Assert.assertEquals(-0.49526417598485072465,   s.getPartialDerivative(4), 1.0e-16);
         Assert.assertEquals(1.4445205132891479465,  s.getPartialDerivative(5), 5.0e-16);
     }
  }