added opencl hex-double generator #4

libmandel/include/ClGenerators.h

@@ -27,6 +27,7 @@ namespace mnd
     class ClGeneratorDoubleDouble;
     class ClGeneratorTripleDouble;
     class ClGeneratorQuadDouble;
+    class ClGeneratorHexDouble;
     class ClGenerator128;
     class ClGenerator64;
 }
@@ -125,6 +126,19 @@ protected:
 };
 
 
+class mnd::ClGeneratorHexDouble : public ClGenerator
+{
+    bool smooth;
+public:
+    ClGeneratorHexDouble(mnd::MandelDevice& device);
+    virtual ~ClGeneratorHexDouble(void) = default;
+
+    virtual void generate(const MandelInfo& info, float* data) override;
+protected:
+    virtual std::string getKernelCode(bool smooth) const;
+};
+
+
 class mnd::ClGenerator128 : public ClGenerator
 {
 public:

libmandel/include/Generators.h

@@ -23,13 +23,14 @@ namespace mnd
         DOUBLE,
         DOUBLE_DOUBLE,
         TRIPLE_DOUBLE,
+        QUAD_DOUBLE,
+        HEX_DOUBLE,
         FLOAT128,
         FLOAT256,
         FLOAT512,
         FIXED64,
         FIXED128,
         FIXED512,
-        QUAD_DOUBLE,
         INF_PREC,
     };
 
@@ -57,6 +58,7 @@ namespace mnd
     template<> Real getPrecision<DoubleDouble>();
     template<> Real getPrecision<TripleDouble>();
     template<> Real getPrecision<QuadDouble>();
+    template<> Real getPrecision<HexDouble>();
     template<> Real getPrecision<Fixed64>();
     template<> Real getPrecision<Fixed128>();
     template<> Real getPrecision<Fixed512>();
@@ -71,6 +73,7 @@ namespace mnd
     template<> inline Precision getType<DoubleDouble>() { return Precision::DOUBLE_DOUBLE; }
     template<> inline Precision getType<TripleDouble>() { return Precision::TRIPLE_DOUBLE; }
     template<> inline Precision getType<QuadDouble>() { return Precision::QUAD_DOUBLE; }
+    template<> inline Precision getType<HexDouble>() { return Precision::HEX_DOUBLE; }
     template<> inline Precision getType<Fixed64>() { return Precision::FIXED64; }
     template<> inline Precision getType<Fixed128>() { return Precision::FIXED128; }
     template<> inline Precision getType<Fixed512>() { return Precision::FIXED512; }

libmandel/include/OpenClCode.h

@@ -11,6 +11,7 @@ namespace mnd
     std::string getDoubleDouble_cl();
     std::string getTripleDouble_cl();
     std::string getQuadDouble_cl();
+    std::string getHexDouble_cl();
     std::string getFixed64_cl();
     std::string getFixed128_cl();
     std::string getFixed512_cl();

libmandel/include/PolyfloatUtil.h

@@ -12,6 +12,23 @@ namespace mnd
             T b;
         };
 
+        template<typename T>
+        struct Triple
+        {
+            T a;
+            T b;
+            T c;
+        };
+
+        template<typename T>
+        struct Quadruple
+        {
+            T a;
+            T b;
+            T c;
+            T d;
+        };
+
         using DoublePair = Pair<double>;
         using FloatPair = Pair<float>;
 
@@ -33,6 +50,15 @@ namespace mnd
         }
 
         template<typename T>
+        inline Triple<T> threeSum(T a, T b, T c)
+        {
+            auto[t1, t2] = twoSum(a, b);
+            auto[r0, t3] = twoSum(t1, c);
+            auto[r1, r2] = twoSum(t2, t3);
+            return { r0, r1, r2 };
+        }
+
+        template<typename T>
         inline Pair<T> threeTwoSum(T a, T b, T c)
         {
             auto[t1, t2] = twoSum(a, b);
@@ -40,6 +66,54 @@ namespace mnd
             return { r0, t2 + t3 };
         }
 
+        template<typename T>
+        inline Quadruple<T> fourSum(T a, T b, T c, T d)
+        {
+            auto[t1, t2] = twoSum(a, b);
+            auto[t3, t4] = twoSum(t1, c);
+            auto[r0, t5] = twoSum(t3, d);
+            auto[r1, r2, r3] = threeSum(t2, t4, t5);
+            return { r0, r1, r2, r3 };
+        }
+
+        template<typename T>
+        inline Pair<T> fiveTwoSum(T a, T b, T c, T d, T e)
+        {
+            auto[t1, t2] = twoSum(a, b);
+            auto[t3, t4] = twoSum(t1, c);
+            auto[t5, t6] = twoSum(t3, d);
+            auto[r0, t7] = twoSum(t5, e);
+            return { r0, t2 + t4 + t6 + t7 };
+        }
+
+        template<typename T>
+        inline Triple<T> sixThreeSum(T a, T b, T c, T d, T e, T f)
+        {
+            auto[a1, a2, a3] = threeSum(a, b, c);
+            auto[b1, b2, b3] = threeSum(d, e, f);
+            
+            auto[r1, t1] = twoSum(a1, b1);
+            auto[t2, t3] = twoSum(a2, b2);
+            auto t4 = a3 + b3;
+
+            auto[r2, t5] = twoSum(t1, t2);
+            auto r3 = t4 + t3 + t5;
+            return { r1, r2, r3 };
+        }
+
+        template<typename T>
+        inline Triple<T> nineThreeSum(T a, T b, T c, T d, T e, T f, T g, T h, T i)
+        {
+            auto[a1, a2, a3] = threeSum(a, b, c);
+            auto[b1, b2, b3] = threeSum(d, e, f);
+            auto[c1, c2, c3] = threeSum(g, h, i);
+            
+            auto[r1, t1, t2] = threeSum(a1, b1, c1);
+            auto[r2, t3, t4, t5] = fourSum(a2, b2, c2, t1);
+            auto r3 = a3 + b3 + c3 + t2 + t3 + t4 + t5;
+            return { r1, r2, r3 };
+        }
+
         inline Pair<double> split(double a)
         {
             static const double splitter = double((1ULL << 27) + 1);

libmandel/include/Types.h

@@ -24,6 +24,7 @@
 
 #include "LightDoubleDouble.h"
 #include "TripleDouble.h"
+#include "HexDouble.h"
 
 namespace mnd
 {
@@ -235,7 +236,36 @@ namespace mnd
     template<>
     inline float convert<float, QuadDouble>(const QuadDouble& x)
     {
-        return float(x.x[0] + x.x[1] + x.x[2] + x.x[3]);
+        return float(x.x[0] + x.x[1]);
+    }
+
+    template<>
+    inline Real convert<Real, HexDouble>(const HexDouble& x)
+    {
+        return Real{ x[0] } + x[1] + x[2] + x[3] + x[4] + x[5];
+    }
+
+    template<>
+    inline HexDouble convert<HexDouble, Real>(const Real& x)
+    {
+        double s = static_cast<double>(x);
+        Real tmp = x - s;
+        double e1 = static_cast<double>(tmp);
+        tmp = tmp - e1;
+        double e2 = static_cast<double>(tmp);
+        tmp = tmp - e2;
+        double e3 = static_cast<double>(tmp);
+        tmp = tmp - e3;
+        double e4 = static_cast<double>(tmp);
+        tmp = tmp - e4;
+        double e5 = static_cast<double>(tmp);
+        return HexDouble { s, e1, e2, e3, e4, e5 };
+    }
+
+    template<>
+    inline float convert<float, HexDouble>(const HexDouble& x)
+    {
+        return float(x.x[0] + x.x[1]);
     }
 
     template<>

libmandel/src/ClGenerators.cpp

@@ -19,6 +19,7 @@ using mnd::ClGeneratorDouble;
 using mnd::ClGeneratorDoubleDouble;
 using mnd::ClGeneratorTripleDouble;
 using mnd::ClGeneratorQuadDouble;
+using mnd::ClGeneratorHexDouble;
 using mnd::ClGenerator128;
 using mnd::ClGenerator64;
 
@@ -504,6 +505,61 @@ std::string ClGeneratorQuadDouble::getKernelCode(bool smooth) const
 }
 
 
+ClGeneratorHexDouble::ClGeneratorHexDouble(mnd::MandelDevice& device) :
+    ClGenerator{ device, getHexDouble_cl(), mnd::Precision::HEX_DOUBLE }
+{
+    kernel = Kernel(program, "iterate");
+}
+
+
+void ClGeneratorHexDouble::generate(const mnd::MandelInfo& info, float* data)
+{
+    ::size_t bufferSize = info.bWidth * info.bHeight * sizeof(float);
+
+    Buffer buffer_A(context, CL_MEM_WRITE_ONLY, bufferSize);
+
+    mnd::HexDouble x = mnd::convert<mnd::HexDouble>(info.view.x);
+    mnd::HexDouble y = mnd::convert<mnd::HexDouble>(info.view.y);
+
+    mnd::HexDouble psx = mnd::convert<mnd::HexDouble>(info.view.width / info.bWidth);
+    mnd::HexDouble psy = mnd::convert<mnd::HexDouble>(info.view.height / info.bHeight);
+
+    mnd::HexDouble jx = mnd::convert<mnd::HexDouble>(info.juliaX);
+    mnd::HexDouble jy = mnd::convert<mnd::HexDouble>(info.juliaY);
+
+    double vals[] = {250, 250, 250, 250, 250, 250 };
+    const size_t argBufSize = 6 * sizeof(double);
+    Buffer xbuf(context, CL_MEM_READ_ONLY | CL_MEM_COPY_HOST_PTR, argBufSize, x.x);
+    Buffer ybuf(context, CL_MEM_READ_ONLY | CL_MEM_COPY_HOST_PTR, argBufSize, y.x);
+    Buffer psxbuf(context, CL_MEM_READ_ONLY | CL_MEM_COPY_HOST_PTR, argBufSize, psx.x);
+    Buffer psybuf(context, CL_MEM_READ_ONLY | CL_MEM_COPY_HOST_PTR, argBufSize, psy.x);
+    Buffer jxbuf(context, CL_MEM_READ_ONLY | CL_MEM_COPY_HOST_PTR, argBufSize, jx.x);
+    Buffer jybuf(context, CL_MEM_READ_ONLY | CL_MEM_COPY_HOST_PTR, argBufSize);
+
+    kernel.setArg(0, buffer_A);
+    kernel.setArg(1, int(info.bWidth));
+    kernel.setArg(2, xbuf);
+    kernel.setArg(3, ybuf);
+    kernel.setArg(4, psxbuf);
+    kernel.setArg(5, psybuf);
+    kernel.setArg(6, int(info.maxIter));
+    kernel.setArg(7, int(info.smooth ? 1 : 0));
+    kernel.setArg(8, int(info.julia ? 1 : 0));
+    kernel.setArg(9, jxbuf);
+    kernel.setArg(10, jybuf);
+
+    cl_int result = queue.enqueueNDRangeKernel(kernel, 0, NDRange(info.bWidth * info.bHeight));
+    queue.enqueueReadBuffer(buffer_A, CL_TRUE, 0, bufferSize, data);
+
+}
+
+
+std::string ClGeneratorHexDouble::getKernelCode(bool smooth) const
+{
+    return getHexDouble_cl();
+}
+
+
 ClGenerator128::ClGenerator128(mnd::MandelDevice& device) :
     ClGenerator{ device, getFixed512_cl(), mnd::Precision::FIXED128 }
 {

libmandel/src/CpuGenerators.cpp

@@ -53,6 +53,9 @@ namespace mnd
 
     template class CpuGenerator<mnd::QuadDouble, mnd::NONE, false>;
     template class CpuGenerator<mnd::QuadDouble, mnd::NONE, true>;
+
+    template class CpuGenerator<mnd::HexDouble, mnd::NONE, false>;
+    template class CpuGenerator<mnd::HexDouble, mnd::NONE, true>;
 #endif
 }
 

libmandel/src/CpuGeneratorsAVX.cpp

@@ -279,6 +279,9 @@ void generateDoubleAvx(long width, long height, float* data, bool parallel,
 }
 
 
+namespace avx_private
+{
+
 struct VecPair
 {
     __m256d a;
@@ -402,6 +405,73 @@ struct AvxDoubleDouble
 };
 
 
+struct AvxTripleDouble
+{
+    __m256d x[3];
+
+    inline AvxTripleDouble(__m256d a, __m256d b, __m256d c) :
+        x{ a, b, c }
+    {}
+
+    inline AvxTripleDouble(double a, double b, double c) :
+        x{ _mm256_set1_pd(a), _mm256_set1_pd(b), _mm256_set1_pd(c) }
+    {}
+
+    inline AvxTripleDouble operator + (const AvxTripleDouble& b) const
+    {
+        const auto& a = *this;
+        auto[r0, t0] = twoSum(a.x[0], b.x[0]);
+        auto[t1, t2] = twoSum(a.x[1], b.x[1]);
+        auto[r1, t3] = twoSum(t0, t1);
+        auto r2 = _mm256_add_pd(_mm256_add_pd(t2, _mm256_add_pd(a.x[2], b.x[2])), t3);
+
+        auto[re1, t4] = quickTwoSum(r0, r1);
+        auto[re2, re3] = quickTwoSum(t4, r2);
+        return { re1, re2, re3 };
+    }
+
+    inline AvxTripleDouble operator - (const AvxTripleDouble& b) const
+    {
+        const auto& a = *this;
+        auto[r0, t0] = twoDiff(a.x[0], b.x[0]);
+        auto[t1, t2] = twoDiff(a.x[1], b.x[1]);
+        auto[r1, t3] = twoSum(t0, t1);
+        auto r2 = _mm256_add_pd(_mm256_add_pd(t2, _mm256_sub_pd(a.x[2], b.x[2])), t3);
+
+        auto[re1, t4] = quickTwoSum(r0, r1);
+        auto[re2, re3] = quickTwoSum(t4, r2);
+        return { re1, re2, re3 };
+    }
+
+    inline AvxTripleDouble operator * (const AvxTripleDouble& b) const
+    {
+        const auto& a = *this;
+        auto[p1_0, p2_0] = twoProd(a.x[0], b.x[0]);
+        auto[p2_1, p3_0] = twoProd(a.x[0], b.x[1]);
+        auto[p2_2, p3_1] = twoProd(a.x[1], b.x[0]);
+
+        auto[t2, tl3] = threeTwoSum(p2_0, p2_1, p2_2);
+        auto t3 = _mm256_add_pd(tl3,
+            _mm256_add_pd(
+                _mm256_add_pd(p3_0, p3_1),
+                _mm256_add_pd(
+                    _mm256_mul_pd(a.x[1], b.x[1]),
+                    _mm256_add_pd(
+                        _mm256_mul_pd(a.x[2], b.x[0]),
+                        _mm256_mul_pd(a.x[0], b.x[2])
+                    )
+                )
+            )
+            );
+        auto[re0, q2] = quickTwoSum(p1_0, t2);
+        auto[re1, re2] = quickTwoSum(q2, t3);
+        return { re0, re1, re2 };
+    }
+};
+
+} // namespace avx_private
+
+
 void generateDoubleDoubleAvx(long width, long height, float* data, bool parallel,
     double vx1, double vx2, double vy1, double vy2, double vw1, double vw2, double vh1, double vh2, int maxIter, bool smooth,
     bool julia, double jX1, double jX2, double jY1, double jY2)
@@ -495,71 +565,6 @@ void generateDoubleDoubleAvx(long width, long height, float* data, bool parallel
     }
 }
 
-struct AvxTripleDouble
-{
-    __m256d x[3];
-
-    inline AvxTripleDouble(__m256d a, __m256d b, __m256d c) :
-        x{ a, b, c }
-    {}
-
-    inline AvxTripleDouble(double a, double b, double c) :
-        x{ _mm256_set1_pd(a), _mm256_set1_pd(b), _mm256_set1_pd(c) }
-    {}
-
-    inline AvxTripleDouble operator + (const AvxTripleDouble& b) const
-    {
-        const auto& a = *this;
-        auto[r0, t0] = twoSum(a.x[0], b.x[0]);
-        auto[t1, t2] = twoSum(a.x[1], b.x[1]);
-        auto[r1, t3] = twoSum(t0, t1);
-        auto r2 = _mm256_add_pd(_mm256_add_pd(t2, _mm256_add_pd(a.x[2], b.x[2])), t3);
-
-        auto[re1, t4] = quickTwoSum(r0, r1);
-        auto[re2, re3] = quickTwoSum(t4, r2);
-        return { re1, re2, re3 };
-    }
-
-    inline AvxTripleDouble operator - (const AvxTripleDouble& b) const
-    {
-        const auto& a = *this;
-        auto[r0, t0] = twoDiff(a.x[0], b.x[0]);
-        auto[t1, t2] = twoDiff(a.x[1], b.x[1]);
-        auto[r1, t3] = twoSum(t0, t1);
-        auto r2 = _mm256_add_pd(_mm256_add_pd(t2, _mm256_sub_pd(a.x[2], b.x[2])), t3);
-
-        auto[re1, t4] = quickTwoSum(r0, r1);
-        auto[re2, re3] = quickTwoSum(t4, r2);
-        return { re1, re2, re3 };
-    }
-
-    inline AvxTripleDouble operator * (const AvxTripleDouble& b) const
-    {
-        const auto& a = *this;
-        auto[p1_0, p2_0] = twoProd(a.x[0], b.x[0]);
-        auto[p2_1, p3_0] = twoProd(a.x[0], b.x[1]);
-        auto[p2_2, p3_1] = twoProd(a.x[1], b.x[0]);
-
-        auto[t2, tl3] = threeTwoSum(p2_0, p2_1, p2_2);
-        auto t3 = _mm256_add_pd(tl3,
-            _mm256_add_pd(
-                _mm256_add_pd(p3_0, p3_1),
-                _mm256_add_pd(
-                    _mm256_mul_pd(a.x[1], b.x[1]),
-                    _mm256_add_pd(
-                        _mm256_mul_pd(a.x[2], b.x[0]),
-                        _mm256_mul_pd(a.x[0], b.x[2])
-                    )
-                )
-            )
-            );
-        auto[re0, q2] = quickTwoSum(p1_0, t2);
-        auto[re1, re2] = quickTwoSum(q2, t3);
-        return { re0, re1, re2 };
-    }
-};
-
-
 void generateTripleDoubleAvx(long width, long height, float* data, bool parallel,
     double vx1, double vx2, double vx3, double vy1, double vy2, double vy3,
     double vw1, double vw2, double vw3, double vh1, double vh2, double vh3,

libmandel/src/Generators.cpp

@@ -132,6 +132,10 @@ namespace mnd
             return "double-double";
         case Precision::TRIPLE_DOUBLE:
             return "triple-double";
+        case Precision::QUAD_DOUBLE:
+            return "quad-double";
+        case Precision::HEX_DOUBLE:
+            return "hex-double";
         case Precision::FLOAT128:
             return "float128";
         case Precision::FLOAT256:
@@ -144,8 +148,6 @@ namespace mnd
             return "fixed128";
         case Precision::FIXED512:
             return "fixed512";
-        case Precision::QUAD_DOUBLE:
-            return "quad-double";
         case Precision::INF_PREC:
             return "real";
         }
@@ -182,6 +184,7 @@ namespace mnd
             { Precision::DOUBLE_DOUBLE, Real("1.0e-29") },
             { Precision::TRIPLE_DOUBLE, Real("1.0e-47") },
             { Precision::QUAD_DOUBLE, Real("1.0e-56") },
+            { Precision::HEX_DOUBLE, Real("1.0e-94") },
             { Precision::FIXED64, Real("3.5e-15") },
             { Precision::FIXED128, Real("1.317e-29") },
             { Precision::FIXED512, Real("1.5e-130") },
@@ -215,6 +218,10 @@ namespace mnd
         return Real("3.0e-64");
     }
     template<>
+    Real getPrecision<HexDouble>() {
+        return Real("3.0e-94");
+    }
+    template<>
     Real getPrecision<Fixed64>() {
         return Real("3.5e-15");
     }

libmandel/src/Mandel.cpp

@@ -132,12 +132,14 @@ MandelContext::MandelContext(void)
 #endif // WITH_BOOST
 
     auto dd = std::make_unique<CpuGenerator<DoubleDouble, mnd::NONE, true>>();
+    auto td = std::make_unique<CpuGenerator<TripleDouble, mnd::NONE, true>>();
     auto qd = std::make_unique<CpuGenerator<QuadDouble, mnd::NONE, true>>();
+    auto hd = std::make_unique<CpuGenerator<HexDouble, mnd::NONE, true>>();
     cpuGenerators.insert({ { Precision::DOUBLE_DOUBLE, CpuExtension::NONE }, std::move(dd) });
+    cpuGenerators.insert({ { Precision::TRIPLE_DOUBLE, CpuExtension::NONE }, std::move(td) });
     cpuGenerators.insert({ { Precision::QUAD_DOUBLE, CpuExtension::NONE }, std::move(qd) });
+    cpuGenerators.insert({ { Precision::HEX_DOUBLE, CpuExtension::NONE }, std::move(hd) });
 
-    auto td = std::make_unique<CpuGenerator<TripleDouble, mnd::NONE, true>>();
-    cpuGenerators.insert({ { Precision::TRIPLE_DOUBLE, CpuExtension::NONE }, std::move(td) });
 
     auto fix512 = std::make_unique<CpuGenerator<Fixed512, mnd::NONE, true>>();
     cpuGenerators.insert({ { Precision::FIXED512, CpuExtension::NONE }, std::move(fix512) });
@@ -158,8 +160,8 @@ std::unique_ptr<mnd::AdaptiveGenerator> MandelContext::createAdaptiveGenerator(v
         Precision::DOUBLE_DOUBLE,
         Precision::TRIPLE_DOUBLE,
         Precision::QUAD_DOUBLE,
-        Precision::FLOAT256,
-        Precision::FLOAT512
+        Precision::HEX_DOUBLE,
+        Precision::FIXED512
     };
 
     auto ag = std::make_unique<AdaptiveGenerator>();
@@ -253,6 +255,7 @@ std::vector<std::unique_ptr<MandelDevice>> MandelContext::createDevices(void)
                     md.mandelGenerators.insert({ Precision::DOUBLE_DOUBLE, std::make_unique<ClGeneratorDoubleDouble>(md) });
                     md.mandelGenerators.insert({ Precision::TRIPLE_DOUBLE, std::make_unique<ClGeneratorTripleDouble>(md) });
                     md.mandelGenerators.insert({ Precision::QUAD_DOUBLE, std::make_unique<ClGeneratorQuadDouble>(md) });
+                    md.mandelGenerators.insert({ Precision::HEX_DOUBLE, std::make_unique<ClGeneratorHexDouble>(md) });
                 }
                 catch (const std::string& err) {
                     printf("err: %s", err.c_str());

libmandel/src/OpenClCode.cpp

@@ -6,6 +6,7 @@
 #include "opencl/doubledouble.h"
 #include "opencl/tripledouble.h"
 #include "opencl/quaddouble.h"
+#include "opencl/hexdouble.h"
 
 #include "opencl/fixed64.h"
 #include "opencl/fixed128.h"
@@ -37,6 +38,10 @@ namespace mnd
         return std::string{ (char*) quaddouble_cl, quaddouble_cl_len };
     }
 
+    std::string getHexDouble_cl() {
+        return std::string{ (char*) hexdouble_cl, hexdouble_cl_len };
+    }
+
     std::string getFixed64_cl() {
         return std::string{ (char*) fixed64_cl, fixed64_cl_len };
     }

libmandel/src/opencl/float.h

@@ -1,6 +1,77 @@
 unsigned char float_cl[] = {
-  0x0d, 0x0a, 0x5f, 0x5f, 0x6b, 0x65, 0x72, 0x6e, 0x65, 0x6c, 0x20, 0x76,
-  0x6f, 0x69, 0x64, 0x20, 0x69, 0x74, 0x65, 0x72, 0x61, 0x74, 0x65, 0x28,
+  0x0a, 0x5f, 0x5f, 0x6b, 0x65, 0x72, 0x6e, 0x65, 0x6c, 0x20, 0x76, 0x6f,
+  0x69, 0x64, 0x20, 0x69, 0x74, 0x65, 0x72, 0x61, 0x74, 0x65, 0x28, 0x5f,
+  0x5f, 0x67, 0x6c, 0x6f, 0x62, 0x61, 0x6c, 0x20, 0x66, 0x6c, 0x6f, 0x61,
+  0x74, 0x2a, 0x20, 0x41, 0x2c, 0x20, 0x63, 0x6f, 0x6e, 0x73, 0x74, 0x20,
+  0x69, 0x6e, 0x74, 0x20, 0x77, 0x69, 0x64, 0x74, 0x68, 0x2c, 0x20, 0x66,
+  0x6c, 0x6f, 0x61, 0x74, 0x20, 0x78, 0x6c, 0x2c, 0x20, 0x66, 0x6c, 0x6f,
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@@ -243,12 +236,12 @@ unsigned char float_cl[] = {
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-  0x2e, 0x30, 0x66, 0x29, 0x3b, 0x0d, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x7d,
-  0x0d, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x65, 0x6c, 0x73, 0x65, 0x0d, 0x0a,
-  0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x41, 0x5b, 0x69, 0x6e,
-  0x64, 0x65, 0x78, 0x20, 0x2b, 0x20, 0x69, 0x5d, 0x20, 0x3d, 0x20, 0x28,
-  0x28, 0x66, 0x6c, 0x6f, 0x61, 0x74, 0x29, 0x20, 0x63, 0x6f, 0x75, 0x6e,
-  0x74, 0x5b, 0x69, 0x5d, 0x29, 0x3b, 0x0d, 0x0a, 0x20, 0x20, 0x20, 0x7d,
-  0x0d, 0x0a, 0x7d, 0x0d, 0x0a, 0x2a, 0x2f, 0x0d, 0x0a
+  0x2e, 0x30, 0x66, 0x29, 0x3b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x7d, 0x0a,
+  0x20, 0x20, 0x20, 0x20, 0x65, 0x6c, 0x73, 0x65, 0x0a, 0x20, 0x20, 0x20,
+  0x20, 0x20, 0x20, 0x20, 0x20, 0x41, 0x5b, 0x69, 0x6e, 0x64, 0x65, 0x78,
+  0x20, 0x2b, 0x20, 0x69, 0x5d, 0x20, 0x3d, 0x20, 0x28, 0x28, 0x66, 0x6c,
+  0x6f, 0x61, 0x74, 0x29, 0x20, 0x63, 0x6f, 0x75, 0x6e, 0x74, 0x5b, 0x69,
+  0x5d, 0x29, 0x3b, 0x0a, 0x20, 0x20, 0x20, 0x7d, 0x0a, 0x7d, 0x0a, 0x2a,
+  0x2f, 0x0a
 };
-unsigned int float_cl_len = 3009;
+unsigned int float_cl_len = 2918;

libmandel/src/opencl/hexdouble.cl

@@ -0,0 +1,225 @@
+#pragma OPENCL EXTENSION cl_khr_fp64 : enable
+
+
+typedef struct hex_double {
+    double x[6];
+} HexDouble;
+
+inline double2 twoSum(double a, double b) {
+    double s = a + b;
+    double bb = s - a;
+    double e = (a - (s - bb)) + (b - bb);
+    return (double2)(s, e);
+}
+
+inline double two_sum(double a, double b, double* err) {
+    double s = a + b;
+    double bb = s - a;
+    *err = (a - (s - bb)) + (b - bb);
+    return s;
+}
+
+inline double2 quickTwoSum(double a, double b) {
+    double s = a + b;
+    double e = b - (s - a);
+    return (double2)(s, e);
+}
+
+
+inline double2 twoProd(double a, double b) {
+    double p = a * b;
+    double e = fma(a, b, -p);
+    return (double2)(p, e);
+}
+
+
+inline double3 threeSum(double a, double b, double c)
+{
+    double2 x = twoSum(a, b);
+    double2 y = twoSum(x.s0, c);
+    double2 z = twoSum(x.s1, y.s1);
+    return (double3)(y.s0, z.s0, z.s1);
+}
+
+
+inline double4 fourSum(double a, double b, double c, double d)
+{
+    double2 x = twoSum(a, b);
+    double2 y = twoSum(x.s0, c);
+    double2 z = twoSum(y.s0, d);
+    double3 r = threeSum(x.s1, y.s1, z.s1);
+    return (double4)(z.s0, r.s0, r.s1, r.s2);
+}
+
+
+inline double2 fiveTwoSum(double a, double b, double c, double d, double e)
+{
+    double2 x = twoSum(a, b);
+    double2 y = twoSum(x.s0, c);
+    double2 z = twoSum(y.s0, d);
+    double2 w = twoSum(z.s0, e);
+    return (double2)(w.s0, x.s1 + y.s1 + z.s1 + w.s1);
+}
+
+
+inline double3 sixThreeSum(double a, double b, double c, double d, double e, double f)
+{
+    double3 av = threeSum(a, b, c);
+    double3 bv = threeSum(d, e, f);
+    
+    double2 r1t1 = twoSum(av.s0, bv.s0);
+    double2 t2t3 = twoSum(av.s1, bv.s1);
+    double t4 = av.s2 + bv.s2;
+
+    double2 r2t5 = twoSum(r1t1.s1, t2t3.s0);
+    double r3 = t4 + t2t3.s1 + r2t5.s1;
+    return (double3)(r1t1.s0, r2t5.s0, r3);
+}
+
+
+inline double3 nineThreeSum(double a, double b, double c, double d, double e,
+                            double f, double g, double h, double i)
+{
+    double3 av = threeSum(a, b, c);
+    double3 bv = threeSum(d, e, f);
+    double3 cv = threeSum(g, h, i);
+    
+    double3 r1t1t2 = threeSum(av.s0, bv.s0, cv.s0);
+    double4 r2t3t4t5 = fourSum(av.s1, bv.s1, cv.s1, r1t1t2.s1);
+    double r3 = av.s2 + bv.s2 + cv.s2 + r1t1t2.s2 + r2t3t4t5.s1 + r2t3t4t5.s2 + r2t3t4t5.s3;
+    return (double3)(r1t1t2.s0, r2t3t4t5.s0, r3);
+}
+
+
+inline HexDouble add(const __private HexDouble* a, const __private HexDouble* b)
+{
+    double2 ap = twoSum(a->x[0], b->x[0]);
+    double2 bp = twoSum(a->x[1], b->x[1]);
+    double2 cp = twoSum(a->x[2], b->x[2]);
+    double2 dp = twoSum(a->x[3], b->x[3]);
+    double2 ep = twoSum(a->x[4], b->x[4]);
+
+    double t0 = ap.s0;
+    double2 t1p1 = twoSum(ap.s1, bp.s0);
+    double3 t2p2p3 = threeSum(bp.s1, cp.s0, t1p1.s1);
+    double4 t3p456 = fourSum(cp.s1, dp.s0, t2p2p3.s1, t2p2p3.s2);
+    double2 t4p7 = fiveTwoSum(dp.s1, ep.s0, t3p456.s1, t3p456.s2, t3p456.s3);
+    double t5 = a->x[5] + b->x[5] + ep.s1 + t4p7.s1;
+
+    double2 r0 = quickTwoSum(t0, t1p1.s0);
+    double2 r1 = quickTwoSum(r0.s1, t2p2p3.s0);
+    double2 r2 = quickTwoSum(r1.s1, t3p456.s0);
+    double2 r3 = quickTwoSum(r2.s1, t4p7.s0);
+    double2 r45 = quickTwoSum(r3.s1, t5);
+
+    return (HexDouble){ r0.s0, r1.s0, r2.s0, r3.s0, r45.s0, r45.s1 };
+}
+
+
+inline HexDouble mul(const __private HexDouble* a, const __private HexDouble* b)
+{
+    double2 p0 = twoProd(a->x[0], b->x[0]);
+
+    double2 q0 = twoProd(a->x[0], b->x[1]);
+    double2 q1 = twoProd(a->x[1], b->x[0]);
+
+    double2 r0 = twoProd(a->x[2], b->x[0]);
+    double2 r1 = twoProd(a->x[1], b->x[1]);
+    double2 r2 = twoProd(a->x[0], b->x[2]);
+
+    double2 s0 = twoProd(a->x[3], b->x[0]);
+    double2 s1 = twoProd(a->x[2], b->x[1]);
+    double2 s2 = twoProd(a->x[1], b->x[2]);
+    double2 s3 = twoProd(a->x[0], b->x[3]);
+
+    double2 t0 = twoProd(a->x[4], b->x[0]);
+    double2 t1 = twoProd(a->x[3], b->x[1]);
+    double2 t2 = twoProd(a->x[2], b->x[2]);
+    double2 t3 = twoProd(a->x[1], b->x[3]);
+    double2 t4 = twoProd(a->x[0], b->x[4]);
+
+    double v1 = p0.s0;
+    double3 t2L34 = threeSum(p0.s1, q0.s0, q1.s0);
+    double3 t3L45 = sixThreeSum(q0.s1, q1.s1, r0.s0, r1.s0, r2.s0, t2L34.s1);
+    double3 t4L56 = nineThreeSum(r0.s1, r1.s1, r2.s1, s0.s0, s1.s0, s2.s0, s3.s0, t2L34.s2, t3L45.s1);
+    double3 x123 = nineThreeSum(s0.s1, s1.s1, s2.s1, s3.s1, t0.s0, t1.s0, t2.s0, t3.s0, t4.s0);
+    double3 t5L67 = sixThreeSum(x123.s0, x123.s1, x123.s2, t3L45.s2, t4L56.s1, 0.0);
+
+    double v6 = t4L56.s2 + t5L67.s1 + t5L67.s2 + t0.s1 + t1.s1 + t2.s1 + t3.s1 + t4.s1 +
+        a->x[5] * b->x[0] + a->x[4] * b->x[1] + a->x[3] * b->x[2] + a->x[2] * b->x[3] + a->x[1] * b->x[4] + a->x[0] * b->x[5];
+
+    double2 re0 = quickTwoSum(v1, t2L34.s0);
+    double2 re1 = quickTwoSum(re0.s1, t3L45.s0);
+    double2 re2 = quickTwoSum(re1.s1, t4L56.s0);
+    double2 re3 = quickTwoSum(re2.s1, t5L67.s0);
+    double2 re45 = quickTwoSum(re3.s1, v6);
+    return (HexDouble){ re0.s0, re1.s0, re2.s0, re3.s0, re45.s0, re45.s1 };
+}
+
+
+HexDouble loadHd(const __global double* v) {
+    return (HexDouble) {
+        v[0],
+        v[1],
+        v[2],
+        v[3],
+        v[4],
+        v[5],
+    };
+}
+
+
+__kernel void iterate(__global float* A, const int width,
+                      __global double* x, __global double* y,
+                      __global double* pw, __global double* ph, int max, int smooth, int julia,
+                      __global double* jx, __global double* jy) {
+    int index = get_global_id(0);
+    int px = index % width;
+    int py = index / width;
+
+    HexDouble xl = loadHd(x);
+    HexDouble yt = loadHd(y);
+    HexDouble pixelScaleX = loadHd(pw);
+    HexDouble pixelScaleY = loadHd(ph);
+
+    HexDouble pxh = { px, 0, 0, 0, 0, 0 };
+    HexDouble pyh = { py, 0, 0, 0, 0, 0 };
+
+    HexDouble caOff = mul(&pixelScaleX, &pxh);
+    HexDouble cbOff = mul(&pixelScaleY, &pyh);
+    HexDouble ca = add(&caOff, &xl); // pixelScaleX * px + xl
+    HexDouble cb = add(&cbOff, &yt); // pixelScaleY * py + yt
+    HexDouble a = ca;
+    HexDouble b = cb;
+    if (julia != 0) {
+        ca = loadHd(jx);
+        cb = loadHd(jy);
+    }
+
+    int n = 0;
+    while (n < max - 1) {
+        HexDouble aa = mul(&a, &a); /* TODO add squaring function */
+        HexDouble bb = mul(&b, &b);
+        HexDouble ab = mul(&a, &b);
+        HexDouble minusbb = { -bb.x[0], -bb.x[1], -bb.x[2], -bb.x[3], -bb.x[4], -bb.x[5] };
+        HexDouble aambb = add(&aa, &minusbb);
+        HexDouble abab = add(&ab, &ab);
+        a = add(&aambb, &ca);
+        b = add(&abab, &cb);
+        if (aa.x[0] + bb.x[0] > 16) break;
+        n++;
+    }
+
+    // N + 1 - log (log  |Z(N)|) / log 2
+    if (n >= max - 1)
+        A[index] = max;
+    else {
+        if (smooth != 0)
+            A[index] = ((float) n) + 1 - log2(log(a.x[0] * a.x[0] + b.x[0] * b.x[0]) * 0.5);
+        else
+            A[index] = ((float)n);
+    }
+    //               A[index] = ((float)n) + 1 - (a * a + b * b - 16) / (256 - 16);
+    //           A[get_global_id(0)] = 5;
+}
+

libmandel/src/opencl/hexdouble.h

@@ -0,0 +1,584 @@
+unsigned char hexdouble_cl[] = {
+  0x23, 0x70, 0x72, 0x61, 0x67, 0x6d, 0x61, 0x20, 0x4f, 0x50, 0x45, 0x4e,
+  0x43, 0x4c, 0x20, 0x45, 0x58, 0x54, 0x45, 0x4e, 0x53, 0x49, 0x4f, 0x4e,
+  0x20, 0x63, 0x6c, 0x5f, 0x6b, 0x68, 0x72, 0x5f, 0x66, 0x70, 0x36, 0x34,
+  0x20, 0x3a, 0x20, 0x65, 0x6e, 0x61, 0x62, 0x6c, 0x65, 0x0a, 0x0a, 0x0a,
+  0x74, 0x79, 0x70, 0x65, 0x64, 0x65, 0x66, 0x20, 0x73, 0x74, 0x72, 0x75,
+  0x63, 0x74, 0x20, 0x68, 0x65, 0x78, 0x5f, 0x64, 0x6f, 0x75, 0x62, 0x6c,
+  0x65, 0x20, 0x7b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x64, 0x6f, 0x75, 0x62,
+  0x6c, 0x65, 0x20, 0x78, 0x5b, 0x36, 0x5d, 0x3b, 0x0a, 0x7d, 0x20, 0x48,
+  0x65, 0x78, 0x44, 0x6f, 0x75, 0x62, 0x6c, 0x65, 0x3b, 0x0a, 0x0a, 0x69,
+  0x6e, 0x6c, 0x69, 0x6e, 0x65, 0x20, 0x64, 0x6f, 0x75, 0x62, 0x6c, 0x65,
+  0x32, 0x20, 0x74, 0x77, 0x6f, 0x53, 0x75, 0x6d, 0x28, 0x64, 0x6f, 0x75,
+  0x62, 0x6c, 0x65, 0x20, 0x61, 0x2c, 0x20, 0x64, 0x6f, 0x75, 0x62, 0x6c,
+  0x65, 0x20, 0x62, 0x29, 0x20, 0x7b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x64,
+  0x6f, 0x75, 0x62, 0x6c, 0x65, 0x20, 0x73, 0x20, 0x3d, 0x20, 0x61, 0x20,
+  0x2b, 0x20, 0x62, 0x3b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x64, 0x6f, 0x75,
+  0x62, 0x6c, 0x65, 0x20, 0x62, 0x62, 0x20, 0x3d, 0x20, 0x73, 0x20, 0x2d,
+  0x20, 0x61, 0x3b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x64, 0x6f, 0x75, 0x62,
+  0x6c, 0x65, 0x20, 0x65, 0x20, 0x3d, 0x20, 0x28, 0x61, 0x20, 0x2d, 0x20,
+  0x28, 0x73, 0x20, 0x2d, 0x20, 0x62, 0x62, 0x29, 0x29, 0x20, 0x2b, 0x20,
+  0x28, 0x62, 0x20, 0x2d, 0x20, 0x62, 0x62, 0x29, 0x3b, 0x0a, 0x20, 0x20,
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+  0x20, 0x63, 0x62, 0x20, 0x3d, 0x20, 0x6c, 0x6f, 0x61, 0x64, 0x48, 0x64,
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+  0x0a, 0x20, 0x20, 0x20, 0x20, 0x69, 0x6e, 0x74, 0x20, 0x6e, 0x20, 0x3d,
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+  0x65, 0x20, 0x28, 0x6e, 0x20, 0x3c, 0x20, 0x6d, 0x61, 0x78, 0x20, 0x2d,
+  0x20, 0x31, 0x29, 0x20, 0x7b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20,
+  0x20, 0x20, 0x48, 0x65, 0x78, 0x44, 0x6f, 0x75, 0x62, 0x6c, 0x65, 0x20,
+  0x61, 0x61, 0x20, 0x3d, 0x20, 0x6d, 0x75, 0x6c, 0x28, 0x26, 0x61, 0x2c,
+  0x20, 0x26, 0x61, 0x29, 0x3b, 0x20, 0x2f, 0x2a, 0x20, 0x54, 0x4f, 0x44,
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+  0x61, 0x61, 0x2e, 0x78, 0x5b, 0x30, 0x5d, 0x20, 0x2b, 0x20, 0x62, 0x62,
+  0x2e, 0x78, 0x5b, 0x30, 0x5d, 0x20, 0x3e, 0x20, 0x31, 0x36, 0x29, 0x20,
+  0x62, 0x72, 0x65, 0x61, 0x6b, 0x3b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x20,
+  0x20, 0x20, 0x20, 0x6e, 0x2b, 0x2b, 0x3b, 0x0a, 0x20, 0x20, 0x20, 0x20,
+  0x7d, 0x0a, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x2f, 0x2f, 0x20, 0x4e, 0x20,
+  0x2b, 0x20, 0x31, 0x20, 0x2d, 0x20, 0x6c, 0x6f, 0x67, 0x20, 0x28, 0x6c,
+  0x6f, 0x67, 0x20, 0x20, 0x7c, 0x5a, 0x28, 0x4e, 0x29, 0x7c, 0x29, 0x20,
+  0x2f, 0x20, 0x6c, 0x6f, 0x67, 0x20, 0x32, 0x0a, 0x20, 0x20, 0x20, 0x20,
+  0x69, 0x66, 0x20, 0x28, 0x6e, 0x20, 0x3e, 0x3d, 0x20, 0x6d, 0x61, 0x78,
+  0x20, 0x2d, 0x20, 0x31, 0x29, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20,
+  0x20, 0x20, 0x41, 0x5b, 0x69, 0x6e, 0x64, 0x65, 0x78, 0x5d, 0x20, 0x3d,
+  0x20, 0x6d, 0x61, 0x78, 0x3b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x65, 0x6c,
+  0x73, 0x65, 0x20, 0x7b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20,
+  0x20, 0x69, 0x66, 0x20, 0x28, 0x73, 0x6d, 0x6f, 0x6f, 0x74, 0x68, 0x20,
+  0x21, 0x3d, 0x20, 0x30, 0x29, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20,
+  0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x41, 0x5b, 0x69, 0x6e, 0x64, 0x65,
+  0x78, 0x5d, 0x20, 0x3d, 0x20, 0x28, 0x28, 0x66, 0x6c, 0x6f, 0x61, 0x74,
+  0x29, 0x20, 0x6e, 0x29, 0x20, 0x2b, 0x20, 0x31, 0x20, 0x2d, 0x20, 0x6c,
+  0x6f, 0x67, 0x32, 0x28, 0x6c, 0x6f, 0x67, 0x28, 0x61, 0x2e, 0x78, 0x5b,
+  0x30, 0x5d, 0x20, 0x2a, 0x20, 0x61, 0x2e, 0x78, 0x5b, 0x30, 0x5d, 0x20,
+  0x2b, 0x20, 0x62, 0x2e, 0x78, 0x5b, 0x30, 0x5d, 0x20, 0x2a, 0x20, 0x62,
+  0x2e, 0x78, 0x5b, 0x30, 0x5d, 0x29, 0x20, 0x2a, 0x20, 0x30, 0x2e, 0x35,
+  0x29, 0x3b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x65,
+  0x6c, 0x73, 0x65, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20,
+  0x20, 0x20, 0x20, 0x20, 0x41, 0x5b, 0x69, 0x6e, 0x64, 0x65, 0x78, 0x5d,
+  0x20, 0x3d, 0x20, 0x28, 0x28, 0x66, 0x6c, 0x6f, 0x61, 0x74, 0x29, 0x6e,
+  0x29, 0x3b, 0x0a, 0x20, 0x20, 0x20, 0x20, 0x7d, 0x0a, 0x20, 0x20, 0x20,
+  0x20, 0x2f, 0x2f, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20,
+  0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x41, 0x5b, 0x69, 0x6e, 0x64, 0x65,
+  0x78, 0x5d, 0x20, 0x3d, 0x20, 0x28, 0x28, 0x66, 0x6c, 0x6f, 0x61, 0x74,
+  0x29, 0x6e, 0x29, 0x20, 0x2b, 0x20, 0x31, 0x20, 0x2d, 0x20, 0x28, 0x61,
+  0x20, 0x2a, 0x20, 0x61, 0x20, 0x2b, 0x20, 0x62, 0x20, 0x2a, 0x20, 0x62,
+  0x20, 0x2d, 0x20, 0x31, 0x36, 0x29, 0x20, 0x2f, 0x20, 0x28, 0x32, 0x35,
+  0x36, 0x20, 0x2d, 0x20, 0x31, 0x36, 0x29, 0x3b, 0x0a, 0x20, 0x20, 0x20,
+  0x20, 0x2f, 0x2f, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20, 0x20,
+  0x20, 0x20, 0x41, 0x5b, 0x67, 0x65, 0x74, 0x5f, 0x67, 0x6c, 0x6f, 0x62,
+  0x61, 0x6c, 0x5f, 0x69, 0x64, 0x28, 0x30, 0x29, 0x5d, 0x20, 0x3d, 0x20,
+  0x35, 0x3b, 0x0a, 0x7d, 0x0a, 0x0a
+};
+unsigned int hexdouble_cl_len = 6966;

libmandel/tests/accurate.cpp

@@ -1,10 +1,11 @@
 #include "../include/TripleDouble.h"
 #include "../include/Types.h"
+#include "../include/HexDouble.h"
 
 #include <vector>
 #include <boost/random.hpp>
 
-const int nTests = 1000;
+const int nTests = 10000;
 
 using namespace boost::multiprecision;
 using namespace boost::random;
@@ -12,7 +13,7 @@ using namespace boost::random;
 std::vector<mnd::Real> generateRandom(int len, int seed)
 {
     uniform_real_distribution<mnd::Real> dist(0, 1);
-    uniform_int_distribution<int> expDist(-30, 30);
+    uniform_int_distribution<int> expDist(-120, 120);
     independent_bits_engine<mt19937, std::numeric_limits<cpp_bin_float_50>::digits, cpp_int> gen(seed);
 
     std::vector<mnd::Real> numbers;
@@ -59,10 +60,19 @@ int main()
 
     mnd::Real tripleDoubleAdd = maxErr<mnd::TripleDouble>([] (const auto& a, const auto& b) { return a + b; });
     mnd::Real tripleDoubleMul = maxErr<mnd::TripleDouble>([] (const auto& a, const auto& b) { return a * b; });
+    mnd::Real tripleDoubleAABBA = maxErr<mnd::TripleDouble>([] (const auto& a, const auto& b) { return a * a - b * b + a; });
     std::cout << std::setprecision(10) << std::scientific;
     std::cout << "max triple double add error: " << tripleDoubleAdd << std::endl;
     std::cout << "max triple double mul error: " << tripleDoubleMul << std::endl;
+    std::cout << "max triple double aa - bb + a error: " << tripleDoubleAABBA << std::endl;
 
+    mnd::Real hexDoubleAdd = maxErr<mnd::HexDouble>([] (const auto& a, const auto& b) { return a + b; });
+    mnd::Real hexDoubleMul = maxErr<mnd::HexDouble>([] (const auto& a, const auto& b) { return a * b; });
+    mnd::Real hexDoubleAABBA = maxErr<mnd::HexDouble>([] (const auto& a, const auto& b) { return a * a - b * b + a; });
+    std::cout << std::setprecision(10) << std::scientific;
+    std::cout << "max hex double add error: " << hexDoubleAdd << std::endl;
+    std::cout << "max hex double mul error: " << hexDoubleMul << std::endl;
+    std::cout << "max hex double aa - bb + a error: " << hexDoubleAABBA << std::endl;
 
 }