internal/ceres/jet_test.cc - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2015 Google Inc. All rights reserved.
 // http://ceres-solver.org/
 //
 // Redistribution and use in source and binary forms, with or without
 // modification, are permitted provided that the following conditions are met:
 //
 // * Redistributions of source code must retain the above copyright notice,
 //   this list of conditions and the following disclaimer.
 // * Redistributions in binary form must reproduce the above copyright notice,
 //   this list of conditions and the following disclaimer in the documentation
 //   and/or other materials provided with the distribution.
 // * Neither the name of Google Inc. nor the names of its contributors may be
 //   used to endorse or promote products derived from this software without
 //   specific prior written permission.
 //
 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
 // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
 // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
 // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 // POSSIBILITY OF SUCH DAMAGE.
 //
 // Author: keir@google.com (Keir Mierle)

 #include "ceres/jet.h"

 #include <algorithm>
 #include <cmath>

 #include "glog/logging.h"
 #include "gtest/gtest.h"
 #include "ceres/fpclassify.h"
 #include "ceres/stringprintf.h"
 #include "ceres/test_util.h"

 #define VL VLOG(1)

 namespace ceres {
 namespace internal {

 const double kE = 2.71828182845904523536;

 typedef Jet<double, 2> J;

 // Convenient shorthand for making a jet.
 J MakeJet(double a, double v0, double v1) {
   J z;
   z.a = a;
   z.v[0] = v0;
   z.v[1] = v1;
   return z;
 }

 // On a 32-bit optimized build, the mismatch is about 1.4e-14.
 double const kTolerance = 1e-13;

 void ExpectJetsClose(const J &x, const J &y) {
   ExpectClose(x.a, y.a, kTolerance);
   ExpectClose(x.v[0], y.v[0], kTolerance);
   ExpectClose(x.v[1], y.v[1], kTolerance);
 }

 TEST(Jet, Jet) {
   // Pick arbitrary values for x and y.
   J x = MakeJet(2.3, -2.7, 1e-3);
   J y = MakeJet(1.7,  0.5, 1e+2);

   VL << "x = " << x;
   VL << "y = " << y;

   { // Check that log(exp(x)) == x.
     J z = exp(x);
     J w = log(z);
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(w, x);
   }

   { // Check that (x * y) / x == y.
     J z = x * y;
     J w = z / x;
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(w, y);
   }

   { // Check that sqrt(x * x) == x.
     J z = x * x;
     J w = sqrt(z);
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(w, x);
   }

   { // Check that sqrt(y) * sqrt(y) == y.
     J z = sqrt(y);
     J w = z * z;
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(w, y);
   }

   { // Check that cos(2*x) = cos(x)^2 - sin(x)^2
     J z = cos(J(2.0) * x);
     J w = cos(x)*cos(x) - sin(x)*sin(x);
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(w, z);
   }

   { // Check that sin(2*x) = 2*cos(x)*sin(x)
     J z = sin(J(2.0) * x);
     J w = J(2.0)*cos(x)*sin(x);
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(w, z);
   }

   { // Check that cos(x)*cos(x) + sin(x)*sin(x) = 1
     J z = cos(x) * cos(x);
     J w = sin(x) * sin(x);
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(z + w, J(1.0));
   }

   { // Check that atan2(r*sin(t), r*cos(t)) = t.
     J t = MakeJet(0.7, -0.3, +1.5);
     J r = MakeJet(2.3, 0.13, -2.4);
     VL << "t = " << t;
     VL << "r = " << r;

     J u = atan2(r * sin(t), r * cos(t));
     VL << "u = " << u;

     ExpectJetsClose(u, t);
   }

   { // Check that tan(x) = sin(x) / cos(x).
     J z = tan(x);
     J w = sin(x) / cos(x);
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(z, w);
   }

   { // Check that tan(atan(x)) = x.
     J z = tan(atan(x));
     J w = x;
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(z, w);
   }

   { // Check that cosh(x)*cosh(x) - sinh(x)*sinh(x) = 1
     J z = cosh(x) * cosh(x);
     J w = sinh(x) * sinh(x);
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(z - w, J(1.0));
   }

   { // Check that tanh(x + y) = (tanh(x) + tanh(y)) / (1 + tanh(x) tanh(y))
     J z = tanh(x + y);
     J w = (tanh(x) + tanh(y)) / (J(1.0) + tanh(x) * tanh(y));
     VL << "z = " << z;
     VL << "w = " << w;
     ExpectJetsClose(z, w);
   }

   { // Check that pow(x, 1) == x.
     VL << "x = " << x;

     J u = pow(x, 1.);
     VL << "u = " << u;

     ExpectJetsClose(x, u);
   }

   { // Check that pow(x, 1) == x.
     J y = MakeJet(1, 0.0, 0.0);
     VL << "x = " << x;
     VL << "y = " << y;

     J u = pow(x, y);
     VL << "u = " << u;

     ExpectJetsClose(x, u);
   }

   { // Check that pow(e, log(x)) == x.
     J logx = log(x);

     VL << "x = " << x;
     VL << "y = " << y;

     J u = pow(kE, logx);
     VL << "u = " << u;

     ExpectJetsClose(x, u);
   }

   { // Check that pow(e, log(x)) == x.
     J logx = log(x);
     J e = MakeJet(kE, 0., 0.);
     VL << "x = " << x;
     VL << "log(x) = " << logx;

     J u = pow(e, logx);
     VL << "u = " << u;

     ExpectJetsClose(x, u);
   }

   { // Check that pow(e, log(x)) == x.
     J logx = log(x);
     J e = MakeJet(kE, 0., 0.);
     VL << "x = " << x;
     VL << "logx = " << logx;

     J u = pow(e, logx);
     VL << "u = " << u;

     ExpectJetsClose(x, u);
   }

   { // Check that pow(x,y) = exp(y*log(x)).
     J logx = log(x);
     J e = MakeJet(kE, 0., 0.);
     VL << "x = " << x;
     VL << "logx = " << logx;

     J u = pow(e, y*logx);
     J v = pow(x, y);
     VL << "u = " << u;
     VL << "v = " << v;

     ExpectJetsClose(v, u);
   }

   { // Check that pow(0, y) == 0 for y > 1, with both arguments Jets.
     // This tests special case handling inside pow().
     J a = MakeJet(0, 1, 2);
     J b = MakeJet(2, 3, 4);
     VL << "a = " << a;
     VL << "b = " << b;

     J c = pow(a, b);
     VL << "a^b = " << c;
     ExpectJetsClose(c, MakeJet(0, 0, 0));
   }

   { // Check that pow(0, y) == 0 for y == 1, with both arguments Jets.
     // This tests special case handling inside pow().
     J a = MakeJet(0, 1, 2);
     J b = MakeJet(1, 3, 4);
     VL << "a = " << a;
     VL << "b = " << b;

     J c = pow(a, b);
     VL << "a^b = " << c;
     ExpectJetsClose(c, MakeJet(0, 1, 2));
   }

   { // Check that pow(0, <1) is not finite, with both arguments Jets.
     for (int i = 1; i < 10; i++) {
       J a = MakeJet(0, 1, 2);
       J b = MakeJet(i*0.1, 3, 4);       // b = 0.1 ... 0.9
       VL << "a = " << a;
       VL << "b = " << b;

       J c = pow(a, b);
       VL << "a^b = " << c;
       EXPECT_EQ(c.a, 0.0);
       EXPECT_FALSE(IsFinite(c.v[0]));
       EXPECT_FALSE(IsFinite(c.v[1]));
     }
     for (int i = -10; i < 0; i++) {
       J a = MakeJet(0, 1, 2);
       J b = MakeJet(i*0.1, 3, 4);       // b = -1,-0.9 ... -0.1
       VL << "a = " << a;
       VL << "b = " << b;

       J c = pow(a, b);
       VL << "a^b = " << c;
       EXPECT_FALSE(IsFinite(c.a));
       EXPECT_FALSE(IsFinite(c.v[0]));
       EXPECT_FALSE(IsFinite(c.v[1]));
     }

     {
       // The special case of 0^0 = 1 defined by the C standard.
       J a = MakeJet(0, 1, 2);
       J b = MakeJet(0, 3, 4);
       VL << "a = " << a;
       VL << "b = " << b;

       J c = pow(a, b);
       VL << "a^b = " << c;
       EXPECT_EQ(c.a, 1.0);
       EXPECT_FALSE(IsFinite(c.v[0]));
       EXPECT_FALSE(IsFinite(c.v[1]));
     }
   }

   { // Check that pow(<0, b) is correct for integer b.
     // This tests special case handling inside pow().
     J a = MakeJet(-1.5, 3, 4);

     // b integer:
     for (int i = -10; i <= 10; i++) {
       J b = MakeJet(i, 0, 5);
       VL << "a = " << a;
       VL << "b = " << b;

       J c = pow(a, b);
       VL << "a^b = " << c;
       ExpectClose(c.a, pow(-1.5, i), kTolerance);
       EXPECT_TRUE(IsFinite(c.v[0]));
       EXPECT_FALSE(IsFinite(c.v[1]));
       ExpectClose(c.v[0], i * pow(-1.5, i - 1) * 3.0, kTolerance);
     }
   }

   { // Check that pow(<0, b) is correct for noninteger b.
     // This tests special case handling inside pow().
     J a = MakeJet(-1.5, 3, 4);
     J b = MakeJet(-2.5, 0, 5);
     VL << "a = " << a;
     VL << "b = " << b;

     J c = pow(a, b);
     VL << "a^b = " << c;
     EXPECT_FALSE(IsFinite(c.a));
     EXPECT_FALSE(IsFinite(c.v[0]));
     EXPECT_FALSE(IsFinite(c.v[1]));
   }

   {
     // Check that pow(0,y) == 0 for y == 2, with the second argument a
     // Jet.  This tests special case handling inside pow().
     double a = 0;
     J b = MakeJet(2, 3, 4);
     VL << "a = " << a;
     VL << "b = " << b;

     J c = pow(a, b);
     VL << "a^b = " << c;
     ExpectJetsClose(c, MakeJet(0, 0, 0));
   }

   {
     // Check that pow(<0,y) is correct for integer y. This tests special case
     // handling inside pow().
     double a = -1.5;
     for (int i = -10; i <= 10; i++) {
       J b = MakeJet(i, 3, 0);
       VL << "a = " << a;
       VL << "b = " << b;

       J c = pow(a, b);
       VL << "a^b = " << c;
       ExpectClose(c.a, pow(-1.5, i), kTolerance);
       EXPECT_FALSE(IsFinite(c.v[0]));
       EXPECT_TRUE(IsFinite(c.v[1]));
       ExpectClose(c.v[1], 0, kTolerance);
     }
   }

   {
     // Check that pow(<0,y) is correct for noninteger y. This tests special
     // case handling inside pow().
     double a = -1.5;
     J b = MakeJet(-3.14, 3, 0);
     VL << "a = " << a;
     VL << "b = " << b;

     J c = pow(a, b);
     VL << "a^b = " << c;
     EXPECT_FALSE(IsFinite(c.a));
     EXPECT_FALSE(IsFinite(c.v[0]));
     EXPECT_FALSE(IsFinite(c.v[1]));
   }

   { // Check that 1 + x == x + 1.
     J a = x + 1.0;
     J b = 1.0 + x;

     ExpectJetsClose(a, b);
   }

   { // Check that 1 - x == -(x - 1).
     J a = 1.0 - x;
     J b = -(x - 1.0);

     ExpectJetsClose(a, b);
   }

   { // Check that x/s == x*s.
     J a = x / 5.0;
     J b = x * 5.0;

     ExpectJetsClose(5.0 * a, b / 5.0);
   }

   { // Check that x / y == 1 / (y / x).
     J a = x / y;
     J b = 1.0 / (y / x);
     VL << "a = " << a;
     VL << "b = " << b;

     ExpectJetsClose(a, b);
   }

   { // Check that abs(-x * x) == sqrt(x * x).
     ExpectJetsClose(abs(-x), sqrt(x * x));
   }

   { // Check that cos(acos(x)) == x.
     J a = MakeJet(0.1, -2.7, 1e-3);
     ExpectJetsClose(cos(acos(a)), a);
     ExpectJetsClose(acos(cos(a)), a);

     J b = MakeJet(0.6,  0.5, 1e+2);
     ExpectJetsClose(cos(acos(b)), b);
     ExpectJetsClose(acos(cos(b)), b);
   }

   { // Check that sin(asin(x)) == x.
     J a = MakeJet(0.1, -2.7, 1e-3);
     ExpectJetsClose(sin(asin(a)), a);
     ExpectJetsClose(asin(sin(a)), a);

     J b = MakeJet(0.4,  0.5, 1e+2);
     ExpectJetsClose(sin(asin(b)), b);
     ExpectJetsClose(asin(sin(b)), b);
   }

   {
     J zero = J(0.0);

     // Check that J0(0) == 1.
     ExpectJetsClose(BesselJ0(zero), J(1.0));

     // Check that J1(0) == 0.
     ExpectJetsClose(BesselJ1(zero), zero);

     // Check that J2(0) == 0.
     ExpectJetsClose(BesselJn(2, zero), zero);

     // Check that J3(0) == 0.
     ExpectJetsClose(BesselJn(3, zero), zero);

     J z = MakeJet(0.1, -2.7, 1e-3);

     // Check that J0(z) == Jn(0,z).
     ExpectJetsClose(BesselJ0(z), BesselJn(0, z));

     // Check that J1(z) == Jn(1,z).
     ExpectJetsClose(BesselJ1(z), BesselJn(1, z));

     // Check that J0(z)+J2(z) == (2/z)*J1(z).
     // See formula http://dlmf.nist.gov/10.6.E1
     ExpectJetsClose(BesselJ0(z) + BesselJn(2, z), (2.0 / z) * BesselJ1(z));
   }

   { // Check that floor of a positive number works.
     J a = MakeJet(0.1, -2.7, 1e-3);
     J b = floor(a);
     J expected = MakeJet(floor(a.a), 0.0, 0.0);
     EXPECT_EQ(expected, b);
   }

   { // Check that floor of a negative number works.
     J a = MakeJet(-1.1, -2.7, 1e-3);
     J b = floor(a);
     J expected = MakeJet(floor(a.a), 0.0, 0.0);
     EXPECT_EQ(expected, b);
   }

   { // Check that floor of a positive number works.
     J a = MakeJet(10.123, -2.7, 1e-3);
     J b = floor(a);
     J expected = MakeJet(floor(a.a), 0.0, 0.0);
     EXPECT_EQ(expected, b);
   }

   { // Check that ceil of a positive number works.
     J a = MakeJet(0.1, -2.7, 1e-3);
     J b = ceil(a);
     J expected = MakeJet(ceil(a.a), 0.0, 0.0);
     EXPECT_EQ(expected, b);
   }

   { // Check that ceil of a negative number works.
     J a = MakeJet(-1.1, -2.7, 1e-3);
     J b = ceil(a);
     J expected = MakeJet(ceil(a.a), 0.0, 0.0);
     EXPECT_EQ(expected, b);
   }

   { // Check that ceil of a positive number works.
     J a = MakeJet(10.123, -2.7, 1e-3);
     J b = ceil(a);
     J expected = MakeJet(ceil(a.a), 0.0, 0.0);
     EXPECT_EQ(expected, b);
   }
 }

 TEST(Jet, JetsInEigenMatrices) {
   J x = MakeJet(2.3, -2.7, 1e-3);
   J y = MakeJet(1.7,  0.5, 1e+2);
   J z = MakeJet(5.3, -4.7, 1e-3);
   J w = MakeJet(9.7,  1.5, 10.1);

   Eigen::Matrix<J, 2, 2> M;
   Eigen::Matrix<J, 2, 1> v, r1, r2;

   M << x, y, z, w;
   v << x, z;

   // Check that M * v == (v^T * M^T)^T
   r1 = M * v;
   r2 = (v.transpose() * M.transpose()).transpose();

   ExpectJetsClose(r1(0), r2(0));
   ExpectJetsClose(r1(1), r2(1));
 }

 TEST(JetTraitsTest, ClassificationMixed) {
   Jet<double, 3> a(5.5, 0);
   a.v[0] = std::numeric_limits<double>::quiet_NaN();
   a.v[1] = std::numeric_limits<double>::infinity();
   a.v[2] = -std::numeric_limits<double>::infinity();
   EXPECT_FALSE(IsFinite(a));
   EXPECT_FALSE(IsNormal(a));
   EXPECT_TRUE(IsInfinite(a));
   EXPECT_TRUE(IsNaN(a));
 }

 TEST(JetTraitsTest, ClassificationNaN) {
   Jet<double, 3> a(5.5, 0);
   a.v[0] = std::numeric_limits<double>::quiet_NaN();
   a.v[1] = 0.0;
   a.v[2] = 0.0;
   EXPECT_FALSE(IsFinite(a));
   EXPECT_FALSE(IsNormal(a));
   EXPECT_FALSE(IsInfinite(a));
   EXPECT_TRUE(IsNaN(a));
 }

 TEST(JetTraitsTest, ClassificationInf) {
   Jet<double, 3> a(5.5, 0);
   a.v[0] = std::numeric_limits<double>::infinity();
   a.v[1] = 0.0;
   a.v[2] = 0.0;
   EXPECT_FALSE(IsFinite(a));
   EXPECT_FALSE(IsNormal(a));
   EXPECT_TRUE(IsInfinite(a));
   EXPECT_FALSE(IsNaN(a));
 }

 TEST(JetTraitsTest, ClassificationFinite) {
   Jet<double, 3> a(5.5, 0);
   a.v[0] = 100.0;
   a.v[1] = 1.0;
   a.v[2] = 3.14159;
   EXPECT_TRUE(IsFinite(a));
   EXPECT_TRUE(IsNormal(a));
   EXPECT_FALSE(IsInfinite(a));
   EXPECT_FALSE(IsNaN(a));
 }

 }  // namespace internal
 }  // namespace ceres
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2015 Google Inc. All rights reserved.
	// http://ceres-solver.org/
	//
	// Redistribution and use in source and binary forms, with or without
	// modification, are permitted provided that the following conditions are met:
	//
	// * Redistributions of source code must retain the above copyright notice,
	// this list of conditions and the following disclaimer.
	// * Redistributions in binary form must reproduce the above copyright notice,
	// this list of conditions and the following disclaimer in the documentation
	// and/or other materials provided with the distribution.
	// * Neither the name of Google Inc. nor the names of its contributors may be
	// used to endorse or promote products derived from this software without
	// specific prior written permission.
	//
	// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
	// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
	// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
	// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
	// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
	// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
	// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
	// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
	// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
	// POSSIBILITY OF SUCH DAMAGE.
	//
	// Author: keir@google.com (Keir Mierle)

	#include "ceres/jet.h"

	#include <algorithm>
	#include <cmath>

	#include "glog/logging.h"
	#include "gtest/gtest.h"
	#include "ceres/fpclassify.h"
	#include "ceres/stringprintf.h"
	#include "ceres/test_util.h"

	#define VL VLOG(1)

	namespace ceres {
	namespace internal {

	const double kE = 2.71828182845904523536;

	typedef Jet<double, 2> J;

	// Convenient shorthand for making a jet.
	J MakeJet(double a, double v0, double v1) {
	J z;
	z.a = a;
	z.v[0] = v0;
	z.v[1] = v1;
	return z;
	}

	// On a 32-bit optimized build, the mismatch is about 1.4e-14.
	double const kTolerance = 1e-13;

	void ExpectJetsClose(const J &x, const J &y) {
	ExpectClose(x.a, y.a, kTolerance);
	ExpectClose(x.v[0], y.v[0], kTolerance);
	ExpectClose(x.v[1], y.v[1], kTolerance);
	}

	TEST(Jet, Jet) {
	// Pick arbitrary values for x and y.
	J x = MakeJet(2.3, -2.7, 1e-3);
	J y = MakeJet(1.7, 0.5, 1e+2);

	VL << "x = " << x;
	VL << "y = " << y;

	{ // Check that log(exp(x)) == x.
	J z = exp(x);
	J w = log(z);
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(w, x);
	}

	{ // Check that (x * y) / x == y.
	J z = x * y;
	J w = z / x;
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(w, y);
	}

	{ // Check that sqrt(x * x) == x.
	J z = x * x;
	J w = sqrt(z);
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(w, x);
	}

	{ // Check that sqrt(y) * sqrt(y) == y.
	J z = sqrt(y);
	J w = z * z;
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(w, y);
	}

	{ // Check that cos(2*x) = cos(x)^2 - sin(x)^2
	J z = cos(J(2.0) * x);
	J w = cos(x)cos(x) - sin(x)sin(x);
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(w, z);
	}

	{ // Check that sin(2x) = 2cos(x)*sin(x)
	J z = sin(J(2.0) * x);
	J w = J(2.0)cos(x)sin(x);
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(w, z);
	}

	{ // Check that cos(x)cos(x) + sin(x)sin(x) = 1
	J z = cos(x) * cos(x);
	J w = sin(x) * sin(x);
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(z + w, J(1.0));
	}

	{ // Check that atan2(rsin(t), rcos(t)) = t.
	J t = MakeJet(0.7, -0.3, +1.5);
	J r = MakeJet(2.3, 0.13, -2.4);
	VL << "t = " << t;
	VL << "r = " << r;

	J u = atan2(r * sin(t), r * cos(t));
	VL << "u = " << u;

	ExpectJetsClose(u, t);
	}

	{ // Check that tan(x) = sin(x) / cos(x).
	J z = tan(x);
	J w = sin(x) / cos(x);
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(z, w);
	}

	{ // Check that tan(atan(x)) = x.
	J z = tan(atan(x));
	J w = x;
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(z, w);
	}

	{ // Check that cosh(x)cosh(x) - sinh(x)sinh(x) = 1
	J z = cosh(x) * cosh(x);
	J w = sinh(x) * sinh(x);
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(z - w, J(1.0));
	}

	{ // Check that tanh(x + y) = (tanh(x) + tanh(y)) / (1 + tanh(x) tanh(y))
	J z = tanh(x + y);
	J w = (tanh(x) + tanh(y)) / (J(1.0) + tanh(x) * tanh(y));
	VL << "z = " << z;
	VL << "w = " << w;
	ExpectJetsClose(z, w);
	}

	{ // Check that pow(x, 1) == x.
	VL << "x = " << x;

	J u = pow(x, 1.);
	VL << "u = " << u;

	ExpectJetsClose(x, u);
	}

	{ // Check that pow(x, 1) == x.
	J y = MakeJet(1, 0.0, 0.0);
	VL << "x = " << x;
	VL << "y = " << y;

	J u = pow(x, y);
	VL << "u = " << u;

	ExpectJetsClose(x, u);
	}

	{ // Check that pow(e, log(x)) == x.
	J logx = log(x);

	VL << "x = " << x;
	VL << "y = " << y;

	J u = pow(kE, logx);
	VL << "u = " << u;

	ExpectJetsClose(x, u);
	}

	{ // Check that pow(e, log(x)) == x.
	J logx = log(x);
	J e = MakeJet(kE, 0., 0.);
	VL << "x = " << x;
	VL << "log(x) = " << logx;

	J u = pow(e, logx);
	VL << "u = " << u;

	ExpectJetsClose(x, u);
	}

	{ // Check that pow(e, log(x)) == x.
	J logx = log(x);
	J e = MakeJet(kE, 0., 0.);
	VL << "x = " << x;
	VL << "logx = " << logx;

	J u = pow(e, logx);
	VL << "u = " << u;

	ExpectJetsClose(x, u);
	}

	{ // Check that pow(x,y) = exp(y*log(x)).
	J logx = log(x);
	J e = MakeJet(kE, 0., 0.);
	VL << "x = " << x;
	VL << "logx = " << logx;

	J u = pow(e, y*logx);
	J v = pow(x, y);
	VL << "u = " << u;
	VL << "v = " << v;

	ExpectJetsClose(v, u);
	}

	{ // Check that pow(0, y) == 0 for y > 1, with both arguments Jets.
	// This tests special case handling inside pow().
	J a = MakeJet(0, 1, 2);
	J b = MakeJet(2, 3, 4);
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	ExpectJetsClose(c, MakeJet(0, 0, 0));
	}

	{ // Check that pow(0, y) == 0 for y == 1, with both arguments Jets.
	// This tests special case handling inside pow().
	J a = MakeJet(0, 1, 2);
	J b = MakeJet(1, 3, 4);
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	ExpectJetsClose(c, MakeJet(0, 1, 2));
	}

	{ // Check that pow(0, <1) is not finite, with both arguments Jets.
	for (int i = 1; i < 10; i++) {
	J a = MakeJet(0, 1, 2);
	J b = MakeJet(i*0.1, 3, 4); // b = 0.1 ... 0.9
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	EXPECT_EQ(c.a, 0.0);
	EXPECT_FALSE(IsFinite(c.v[0]));
	EXPECT_FALSE(IsFinite(c.v[1]));
	}
	for (int i = -10; i < 0; i++) {
	J a = MakeJet(0, 1, 2);
	J b = MakeJet(i*0.1, 3, 4); // b = -1,-0.9 ... -0.1
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	EXPECT_FALSE(IsFinite(c.a));
	EXPECT_FALSE(IsFinite(c.v[0]));
	EXPECT_FALSE(IsFinite(c.v[1]));
	}

	{
	// The special case of 0^0 = 1 defined by the C standard.
	J a = MakeJet(0, 1, 2);
	J b = MakeJet(0, 3, 4);
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	EXPECT_EQ(c.a, 1.0);
	EXPECT_FALSE(IsFinite(c.v[0]));
	EXPECT_FALSE(IsFinite(c.v[1]));
	}
	}

	{ // Check that pow(<0, b) is correct for integer b.
	// This tests special case handling inside pow().
	J a = MakeJet(-1.5, 3, 4);

	// b integer:
	for (int i = -10; i <= 10; i++) {
	J b = MakeJet(i, 0, 5);
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	ExpectClose(c.a, pow(-1.5, i), kTolerance);
	EXPECT_TRUE(IsFinite(c.v[0]));
	EXPECT_FALSE(IsFinite(c.v[1]));
	ExpectClose(c.v[0], i * pow(-1.5, i - 1) * 3.0, kTolerance);
	}
	}

	{ // Check that pow(<0, b) is correct for noninteger b.
	// This tests special case handling inside pow().
	J a = MakeJet(-1.5, 3, 4);
	J b = MakeJet(-2.5, 0, 5);
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	EXPECT_FALSE(IsFinite(c.a));
	EXPECT_FALSE(IsFinite(c.v[0]));
	EXPECT_FALSE(IsFinite(c.v[1]));
	}

	{
	// Check that pow(0,y) == 0 for y == 2, with the second argument a
	// Jet. This tests special case handling inside pow().
	double a = 0;
	J b = MakeJet(2, 3, 4);
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	ExpectJetsClose(c, MakeJet(0, 0, 0));
	}

	{
	// Check that pow(<0,y) is correct for integer y. This tests special case
	// handling inside pow().
	double a = -1.5;
	for (int i = -10; i <= 10; i++) {
	J b = MakeJet(i, 3, 0);
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	ExpectClose(c.a, pow(-1.5, i), kTolerance);
	EXPECT_FALSE(IsFinite(c.v[0]));
	EXPECT_TRUE(IsFinite(c.v[1]));
	ExpectClose(c.v[1], 0, kTolerance);
	}
	}

	{
	// Check that pow(<0,y) is correct for noninteger y. This tests special
	// case handling inside pow().
	double a = -1.5;
	J b = MakeJet(-3.14, 3, 0);
	VL << "a = " << a;
	VL << "b = " << b;

	J c = pow(a, b);
	VL << "a^b = " << c;
	EXPECT_FALSE(IsFinite(c.a));
	EXPECT_FALSE(IsFinite(c.v[0]));
	EXPECT_FALSE(IsFinite(c.v[1]));
	}

	{ // Check that 1 + x == x + 1.
	J a = x + 1.0;
	J b = 1.0 + x;

	ExpectJetsClose(a, b);
	}

	{ // Check that 1 - x == -(x - 1).
	J a = 1.0 - x;
	J b = -(x - 1.0);

	ExpectJetsClose(a, b);
	}

	{ // Check that x/s == x*s.
	J a = x / 5.0;
	J b = x * 5.0;

	ExpectJetsClose(5.0 * a, b / 5.0);
	}

	{ // Check that x / y == 1 / (y / x).
	J a = x / y;
	J b = 1.0 / (y / x);
	VL << "a = " << a;
	VL << "b = " << b;

	ExpectJetsClose(a, b);
	}

	{ // Check that abs(-x * x) == sqrt(x * x).
	ExpectJetsClose(abs(-x), sqrt(x * x));
	}

	{ // Check that cos(acos(x)) == x.
	J a = MakeJet(0.1, -2.7, 1e-3);
	ExpectJetsClose(cos(acos(a)), a);
	ExpectJetsClose(acos(cos(a)), a);

	J b = MakeJet(0.6, 0.5, 1e+2);
	ExpectJetsClose(cos(acos(b)), b);
	ExpectJetsClose(acos(cos(b)), b);
	}

	{ // Check that sin(asin(x)) == x.
	J a = MakeJet(0.1, -2.7, 1e-3);
	ExpectJetsClose(sin(asin(a)), a);
	ExpectJetsClose(asin(sin(a)), a);

	J b = MakeJet(0.4, 0.5, 1e+2);
	ExpectJetsClose(sin(asin(b)), b);
	ExpectJetsClose(asin(sin(b)), b);
	}

	{
	J zero = J(0.0);

	// Check that J0(0) == 1.
	ExpectJetsClose(BesselJ0(zero), J(1.0));

	// Check that J1(0) == 0.
	ExpectJetsClose(BesselJ1(zero), zero);

	// Check that J2(0) == 0.
	ExpectJetsClose(BesselJn(2, zero), zero);

	// Check that J3(0) == 0.
	ExpectJetsClose(BesselJn(3, zero), zero);

	J z = MakeJet(0.1, -2.7, 1e-3);

	// Check that J0(z) == Jn(0,z).
	ExpectJetsClose(BesselJ0(z), BesselJn(0, z));

	// Check that J1(z) == Jn(1,z).
	ExpectJetsClose(BesselJ1(z), BesselJn(1, z));

	// Check that J0(z)+J2(z) == (2/z)*J1(z).
	// See formula http://dlmf.nist.gov/10.6.E1
	ExpectJetsClose(BesselJ0(z) + BesselJn(2, z), (2.0 / z) * BesselJ1(z));
	}

	{ // Check that floor of a positive number works.
	J a = MakeJet(0.1, -2.7, 1e-3);
	J b = floor(a);
	J expected = MakeJet(floor(a.a), 0.0, 0.0);
	EXPECT_EQ(expected, b);
	}

	{ // Check that floor of a negative number works.
	J a = MakeJet(-1.1, -2.7, 1e-3);
	J b = floor(a);
	J expected = MakeJet(floor(a.a), 0.0, 0.0);
	EXPECT_EQ(expected, b);
	}

	{ // Check that floor of a positive number works.
	J a = MakeJet(10.123, -2.7, 1e-3);
	J b = floor(a);
	J expected = MakeJet(floor(a.a), 0.0, 0.0);
	EXPECT_EQ(expected, b);
	}

	{ // Check that ceil of a positive number works.
	J a = MakeJet(0.1, -2.7, 1e-3);
	J b = ceil(a);
	J expected = MakeJet(ceil(a.a), 0.0, 0.0);
	EXPECT_EQ(expected, b);
	}

	{ // Check that ceil of a negative number works.
	J a = MakeJet(-1.1, -2.7, 1e-3);
	J b = ceil(a);
	J expected = MakeJet(ceil(a.a), 0.0, 0.0);
	EXPECT_EQ(expected, b);
	}

	{ // Check that ceil of a positive number works.
	J a = MakeJet(10.123, -2.7, 1e-3);
	J b = ceil(a);
	J expected = MakeJet(ceil(a.a), 0.0, 0.0);
	EXPECT_EQ(expected, b);
	}
	}

	TEST(Jet, JetsInEigenMatrices) {
	J x = MakeJet(2.3, -2.7, 1e-3);
	J y = MakeJet(1.7, 0.5, 1e+2);
	J z = MakeJet(5.3, -4.7, 1e-3);
	J w = MakeJet(9.7, 1.5, 10.1);

	Eigen::Matrix<J, 2, 2> M;
	Eigen::Matrix<J, 2, 1> v, r1, r2;

	M << x, y, z, w;
	v << x, z;

	// Check that M * v == (v^T * M^T)^T
	r1 = M * v;
	r2 = (v.transpose() * M.transpose()).transpose();

	ExpectJetsClose(r1(0), r2(0));
	ExpectJetsClose(r1(1), r2(1));
	}

	TEST(JetTraitsTest, ClassificationMixed) {
	Jet<double, 3> a(5.5, 0);
	a.v[0] = std::numeric_limits<double>::quiet_NaN();
	a.v[1] = std::numeric_limits<double>::infinity();
	a.v[2] = -std::numeric_limits<double>::infinity();
	EXPECT_FALSE(IsFinite(a));
	EXPECT_FALSE(IsNormal(a));
	EXPECT_TRUE(IsInfinite(a));
	EXPECT_TRUE(IsNaN(a));
	}

	TEST(JetTraitsTest, ClassificationNaN) {
	Jet<double, 3> a(5.5, 0);
	a.v[0] = std::numeric_limits<double>::quiet_NaN();
	a.v[1] = 0.0;
	a.v[2] = 0.0;
	EXPECT_FALSE(IsFinite(a));
	EXPECT_FALSE(IsNormal(a));
	EXPECT_FALSE(IsInfinite(a));
	EXPECT_TRUE(IsNaN(a));
	}

	TEST(JetTraitsTest, ClassificationInf) {
	Jet<double, 3> a(5.5, 0);
	a.v[0] = std::numeric_limits<double>::infinity();
	a.v[1] = 0.0;
	a.v[2] = 0.0;
	EXPECT_FALSE(IsFinite(a));
	EXPECT_FALSE(IsNormal(a));
	EXPECT_TRUE(IsInfinite(a));
	EXPECT_FALSE(IsNaN(a));
	}

	TEST(JetTraitsTest, ClassificationFinite) {
	Jet<double, 3> a(5.5, 0);
	a.v[0] = 100.0;
	a.v[1] = 1.0;
	a.v[2] = 3.14159;
	EXPECT_TRUE(IsFinite(a));
	EXPECT_TRUE(IsNormal(a));
	EXPECT_FALSE(IsInfinite(a));
	EXPECT_FALSE(IsNaN(a));
	}

	} // namespace internal
	} // namespace ceres