/**
 * The MIT License (MIT)
 *
 * Copyright (c) 2015-2016 decimal4j (tools4j), Marco Terzer
 *
 * Permission is hereby granted, free of charge, to any person obtaining a copy
 * of this software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the Software is
 * furnished to do so, subject to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included in all
 * copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 * SOFTWARE.
 */
package org.decimal4j.arithmetic;

import static org.decimal4j.arithmetic.Square.SQRT_MAX_VALUE;

import org.decimal4j.api.DecimalArithmetic;
import org.decimal4j.scale.Scale9f;
import org.decimal4j.scale.ScaleMetrics;
import org.decimal4j.scale.Scales;
import org.decimal4j.truncate.DecimalRounding;

/**
 * Provides methods to calculate multiplication results.
 */
final class Mul {

        private static final ScaleMetrics SCALE9F = Scale9f.INSTANCE;

        //sufficient (but not necessary) condition that product fits in long
        private static final boolean doesProductFitInLong(long uDecimal1, long uDecimal2) {
                if (-SQRT_MAX_VALUE <= uDecimal1 & uDecimal1 <= SQRT_MAX_VALUE & -SQRT_MAX_VALUE <= uDecimal2 & uDecimal2 <= SQRT_MAX_VALUE) {
                        return true;
                }
                return false;
                //NOTE: not worth checking (too much overhead for too few special cases):
//              final int leadingZeros = Long.numberOfLeadingZeros(uDecimal1) + Long.numberOfLeadingZeros(~uDecimal1) + Long.numberOfLeadingZeros(uDecimal2) + Long.numberOfLeadingZeros(~uDecimal2);
//              return leadingZeros > Long.SIZE + 1;
        }
        
        /**
         * Calculates the multiple {@code uDecimal1 * uDecimal2 / scaleFactor}
         * without rounding.
         * 
         * @param arith
         *            the arithmetic with access to scale metrics etc.
         * @param uDecimal1
         *            the first unscaled decimal factor
         * @param uDecimal2
         *            the second unscaled decimal factor
         * @return the multiplication result without rounding
         */
        public static final long multiply(DecimalArithmetic arith, long uDecimal1, long uDecimal2) {
                final SpecialMultiplicationResult special = SpecialMultiplicationResult.getFor(arith, uDecimal1, uDecimal2);
                if (special != null) {
                        return special.multiply(arith, uDecimal1, uDecimal2);
                }
                return multiply(uDecimal1, arith.getScaleMetrics(), uDecimal2);
        }
        
        /**
         * Calculates unchecked multiplication by an unscaled value with the given scale
         * without rounding.
         * 
         * @param uDecimal
         *            the unscaled decimal factor
         * @param unscaled
         *            the second unscaled factor
         * @param scale
         *            the scale of the second factor
         * @return the multiplication result without rounding and without overflow checks
         */
        public static final long multiplyByUnscaled(long uDecimal, long unscaled, int scale) {
                if (scale > Scales.MAX_SCALE) {
                        throw new IllegalArgumentException("Illegal scale, must be <=" + Scales.MAX_SCALE + " but was " + scale);
                }
                if (uDecimal == 0 | unscaled == 0) {
                        return 0;
                } else if (scale == 0) {
                        return uDecimal * unscaled;
                } else if (scale < 0) {
                        return Pow10.divideByPowerOf10(uDecimal * unscaled, scale);
                }
                final ScaleMetrics scaleMetrics = Scales.getScaleMetrics(scale);
                return multiply(uDecimal, scaleMetrics, unscaled);
        }
        
        /**
         * Calculates the multiple {@code uDecimal1 * uDecimal2 / scaleFactor}
         * without rounding.
         * 
         * @param uDecimal1
         *            the first unscaled decimal factor
         * @param scaleMetrics2
         *            the scale metrics associated with the second factor
         * @param uDecimal2
         *            the second unscaled decimal factor
         * @return the multiplication result without rounding
         */
        private static final long multiply(long uDecimal1, ScaleMetrics scaleMetrics2, long uDecimal2) {
                if (doesProductFitInLong(uDecimal1, uDecimal2)) {
                        //product fits in long, just do it
                        return scaleMetrics2.divideByScaleFactor(uDecimal1 * uDecimal2);
                }
                final int scale = scaleMetrics2.getScale();
                if (scale <= 9) {
                        //use scale to split into 2 parts: i (integral) and f (fractional)
                        //with this scale, the low order product f1*f2 fits in a long
                        final long i1 = scaleMetrics2.divideByScaleFactor(uDecimal1);
                        final long i2 = scaleMetrics2.divideByScaleFactor(uDecimal2);
                        final long f1 = uDecimal1 - scaleMetrics2.multiplyByScaleFactor(i1);
                        final long f2 = uDecimal2 - scaleMetrics2.multiplyByScaleFactor(i2);
                        return uDecimal1 * i2 + i1 * f2 + scaleMetrics2.divideByScaleFactor(f1 * f2);
                } else {
                        //use scale9 to split into 2 parts: h (high) and l (low)
                        final ScaleMetrics scaleDiff09 = Scales.getScaleMetrics(scale - 9);
                        final ScaleMetrics scaleDiff18 = Scales.getScaleMetrics(18 - scale);
                        final long h1 = SCALE9F.divideByScaleFactor(uDecimal1);
                        final long h2 = SCALE9F.divideByScaleFactor(uDecimal2);
                        final long l1 = uDecimal1 - SCALE9F.multiplyByScaleFactor(h1);
                        final long l2 = uDecimal2 - SCALE9F.multiplyByScaleFactor(h2);
                        final long h1xl2 = h1 * l2;
                        final long h2xl1 = h2 * l1;
                        final long l1xl2d = SCALE9F.divideByScaleFactor(l1 * l2);
                        final long h1xl2d = scaleDiff09.divideByScaleFactor(h1xl2);
                        final long h2xl1d = scaleDiff09.divideByScaleFactor(h2xl1);
                        final long h1xl2r = h1xl2 - scaleDiff09.multiplyByScaleFactor(h1xl2d);
                        final long h2xl1r = h2xl1 - scaleDiff09.multiplyByScaleFactor(h2xl1d);
                        return scaleDiff18.multiplyByScaleFactor(h1 * h2) + h1xl2d + h2xl1d + scaleDiff09.divideByScaleFactor(h1xl2r + h2xl1r + l1xl2d); 
                }
        }

        /**
         * Calculates the multiple {@code uDecimal1 * uDecimal2 / scaleFactor}
         * applying the specified rounding if necessary.
         * 
         * @param arith
         *            the arithmetic with access to scale metrics etc.
         * @param rounding
         *            the rounding to apply if necessary
         * @param uDecimal1
         *            the first unscaled decimal factor
         * @param uDecimal2
         *            the second unscaled decimal factor
         * @return the multiplication result with rounding
         */
        public static final long multiply(DecimalArithmetic arith, DecimalRounding rounding, long uDecimal1, long uDecimal2) {
                final SpecialMultiplicationResult special = SpecialMultiplicationResult.getFor(arith, uDecimal1, uDecimal2);
                if (special != null) {
                        return special.multiply(arith, uDecimal1, uDecimal2);
                }
                return multiply(rounding, uDecimal1, arith.getScaleMetrics(), uDecimal2);
        }

        /**
         * Calculates unchecked multiplication by an unscaled value with the given
         * scale with rounding.
         * 
         * @param rounding
         *            the rounding to apply
         * @param uDecimal
         *            the unscaled decimal factor
         * @param unscaled
         *            the second unscaled factor
         * @param scale
         *            the scale of the second factor
         * @return the multiplication result with rounding and without overflow checks
         */
        public static final long multiplyByUnscaled(DecimalRounding rounding, long uDecimal, long unscaled, int scale) {
                if (scale > Scales.MAX_SCALE) {
                        throw new IllegalArgumentException("Illegal scale, must be <=" + Scales.MAX_SCALE + " but was " + scale);
                }
                if (uDecimal == 0 | unscaled == 0) {
                        return 0;
                } else if (scale == 0) {
                        return uDecimal * unscaled;
                } else if (scale < 0) {
                        return Pow10.divideByPowerOf10(rounding, uDecimal * unscaled, scale);
                }
                final ScaleMetrics scaleMetrics = Scales.getScaleMetrics(scale);
                return multiply(rounding, uDecimal, scaleMetrics, unscaled);
        }

        /**
         * Calculates unchecked multiplication by an unscaled value with the given
         * scale with rounding.
         * 
         * @param rounding
         *            the rounding to apply
         * @param uDecimal1
         *            the first unscaled decimal factor
         * @param scaleMetrics2
         *            the scale metrics associated with the second factor
         * @param uDecimal2
         *            the second unscaled decimal factor
         * @return the multiplication result with rounding and without overflow checks
         */
        private static final long multiply(DecimalRounding rounding, long uDecimal1, ScaleMetrics scaleMetrics2, long uDecimal2) {
                if (doesProductFitInLong(uDecimal1, uDecimal2)) {
                        //product fits in long, just do it
                        return multiply32(rounding, uDecimal1, scaleMetrics2, uDecimal2);
                }
                
                final int scale = scaleMetrics2.getScale();
                if (scale <= 9) {
                        //use scale to split into 2 parts: i (integral) and f (fractional)
                        //with this scale, the low order product f1*f2 fits in a long
                        final long i1 = scaleMetrics2.divideByScaleFactor(uDecimal1);
                        final long i2 = scaleMetrics2.divideByScaleFactor(uDecimal2);
                        final long f1 = uDecimal1 - scaleMetrics2.multiplyByScaleFactor(i1);
                        final long f2 = uDecimal2 - scaleMetrics2.multiplyByScaleFactor(i2);
                        final long f1xf2 = f1 * f2;
                        final long f1xf2d = scaleMetrics2.divideByScaleFactor(f1xf2);
                        final long f1xf2r = f1xf2 - scaleMetrics2.multiplyByScaleFactor(f1xf2d);
                        final long unrounded = uDecimal1 * i2 + i1 * f2 + f1xf2d;
                        return unrounded + Rounding.calculateRoundingIncrement(rounding, unrounded, f1xf2r, scaleMetrics2.getScaleFactor());
                } else {
                        //use scale9 to split into 2 parts: h (high) and l (low)
                        final ScaleMetrics scaleDiff09 = Scales.getScaleMetrics(scale - 9);
                        final ScaleMetrics scaleDiff18 = Scales.getScaleMetrics(18 - scale);
                        final long h1 = SCALE9F.divideByScaleFactor(uDecimal1);
                        final long h2 = SCALE9F.divideByScaleFactor(uDecimal2);
                        final long l1 = uDecimal1 - SCALE9F.multiplyByScaleFactor(h1);
                        final long l2 = uDecimal2 - SCALE9F.multiplyByScaleFactor(h2);
                        final long h1xl2 = h1 * l2;
                        final long h2xl1 = h2 * l1;
                        final long l1xl2 = l1 * l2;
                        final long l1xl2d = SCALE9F.divideByScaleFactor(l1xl2);
                        final long h1xl2d = scaleDiff09.divideByScaleFactor(h1xl2);
                        final long h2xl1d = scaleDiff09.divideByScaleFactor(h2xl1);
                        final long h1xl2r = h1xl2 - scaleDiff09.multiplyByScaleFactor(h1xl2d);
                        final long h2xl1r = h2xl1 - scaleDiff09.multiplyByScaleFactor(h2xl1d);
                        final long l1xl2r = l1xl2 - SCALE9F.multiplyByScaleFactor(l1xl2d);
                        final long h1xl2_h2xl1_l1xl1 = h1xl2r + h2xl1r + l1xl2d; 
                        final long h1xl2_h2xl1_l1xl1d = scaleDiff09.divideByScaleFactor(h1xl2_h2xl1_l1xl1); 
                        final long h1xl2_h2xl1_l1xl1r = h1xl2_h2xl1_l1xl1 - scaleDiff09.multiplyByScaleFactor(h1xl2_h2xl1_l1xl1d); 
                        final long unrounded = scaleDiff18.multiplyByScaleFactor(h1 * h2) + h1xl2d + h2xl1d + h1xl2_h2xl1_l1xl1d;
                        final long remainder = SCALE9F.multiplyByScaleFactor(h1xl2_h2xl1_l1xl1r) + l1xl2r;
                        return unrounded + Rounding.calculateRoundingIncrement(rounding, unrounded, remainder, scaleMetrics2.getScaleFactor());
                }
        }
        
        /**
         * Calculates {@code round((uDecimal1 * uDecimal2) / scaleFactor2)} treating
         * the factors as 32 bit values whose product must fit in a long result.
         * 
         * @param rounding
         *            the rounding to use
         * @param uDecimal1
         *            the first factor
         * @param scaleMetrics2
         *            the scale metrics to apply to the product
         * @param uDecimal2
         *            the second factor
         * @return the product rounded if necessary
         */
        private static final long multiply32(DecimalRounding rounding, long uDecimal1, ScaleMetrics scaleMetrics2, long uDecimal2) {
                final long u1xu2 = uDecimal1 * uDecimal2;
                final long u1xu2d = scaleMetrics2.divideByScaleFactor(u1xu2);
                final long u1xu2r = u1xu2 - scaleMetrics2.multiplyByScaleFactor(u1xu2d);
                return u1xu2d + Rounding.calculateRoundingIncrement(rounding, u1xu2d, u1xu2r, scaleMetrics2.getScaleFactor());
        }

        /**
         * Calculates the multiple {@code uDecimal1 * uDecimal2 / scaleFactor}
         * without rounding checking for overflows.
         * 
         * @param arith
         *            the arithmetic with access to scale metrics etc.
         * @param uDecimal1
         *            the first unscaled decimal factor
         * @param uDecimal2
         *            the second unscaled decimal factor
         * @return the multiplication result without rounding and with overflow checks
         */
        public static final long multiplyChecked(final DecimalArithmetic arith, final long uDecimal1, final long uDecimal2) {
                final SpecialMultiplicationResult special = SpecialMultiplicationResult.getFor(arith, uDecimal1, uDecimal2);
                if (special != null) {
                        return special.multiply(arith, uDecimal1, uDecimal2);
                }
                final ScaleMetrics scaleMetrics = arith.getScaleMetrics();
                return multiplyChecked(scaleMetrics, uDecimal1, scaleMetrics, uDecimal2);
        }
        
        /**
         * Calculates checked multiplication by an unscaled value with the given scale
         * without rounding.
         * 
         * @param arith
         *            the decimal arithmetics associated with the first factor
         * @param uDecimal
         *            the unscaled decimal factor
         * @param unscaled
         *            the second unscaled factor
         * @param scale
         *            the scale of the second factor
         * @return the multiplication result without rounding and with overflow checks
         */
        public static final long multiplyByUnscaledChecked(DecimalArithmetic arith, long uDecimal, long unscaled, int scale) {
                if (scale > Scales.MAX_SCALE) {
                        throw new IllegalArgumentException("Illegal scale, must be <=" + Scales.MAX_SCALE + " but was " + scale);
                }
                if (uDecimal == 0 | unscaled == 0) {
                        return 0;
                } else if (scale == 0) {
                        return arith.multiplyByLong(uDecimal, unscaled);
                } else if (scale < 0) {
                        final long unscaledResult = Checked.multiplyLong(uDecimal, unscaled);
                        return Pow10.divideByPowerOf10Checked(arith, unscaledResult, scale);
                }
                final ScaleMetrics scaleMetrics = Scales.getScaleMetrics(scale);
                return multiplyChecked(arith.getScaleMetrics(), uDecimal, scaleMetrics, unscaled);
        }
        
        /**
         * Calculates checked multiplication by an unscaled value with the given scale
         * without rounding.
         * 
         * @param scaleMetrics1
         *            the scale matrics associated with the first factor
         * @param uDecimal1
         *            the first unscaled decimal factor
         * @param scaleMetrics2
         *            the scale matrics associated with the second factor
         * @param uDecimal2
         *            the second unscaled decimal factor
         * @return the multiplication result without rounding and with overflow checks
         */
        private static final long multiplyChecked(ScaleMetrics scaleMetrics1, long uDecimal1, ScaleMetrics scaleMetrics2, long uDecimal2) {
                try {
                        if (doesProductFitInLong(uDecimal1, uDecimal2)) {
                                return scaleMetrics2.divideByScaleFactor(uDecimal1 * uDecimal2);
                        }
                        
                        final int scale = scaleMetrics2.getScale();
                        if (scale <= 9) {
                                //use scale to split into 2 parts: i (integral) and f (fractional)
                                //with this scale, the low order product f1*f2 fits in a long
                                final long i1 = scaleMetrics2.divideByScaleFactor(uDecimal1);
                                final long i2 = scaleMetrics2.divideByScaleFactor(uDecimal2);
                                final long f1 = uDecimal1 - scaleMetrics2.multiplyByScaleFactor(i1);
                                final long f2 = uDecimal2 - scaleMetrics2.multiplyByScaleFactor(i2);
                                final long i1xf2 = i1 * f2;//cannot overflow
                                final long f1xf2 = scaleMetrics2.divideByScaleFactor(f1 * f2);//product fits for this scale, hence unchecked
                                //add it all up now, every operation checked
                                long result = Checked.multiplyLong(uDecimal1, i2);
                                result = Checked.addLong(result, i1xf2);
                                result = Checked.addLong(result, f1xf2);
                                return result;
                        } else {
                                //use scale9 to split into 2 parts: h (high) and l (low)
                                final ScaleMetrics scaleDiff09 = Scales.getScaleMetrics(scale - 9);
                                final ScaleMetrics scaleDiff18 = Scales.getScaleMetrics(18 - scale);
                                final long h1 = SCALE9F.divideByScaleFactor(uDecimal1);
                                final long h2 = SCALE9F.divideByScaleFactor(uDecimal2);
                                final long l1 = uDecimal1 - SCALE9F.multiplyByScaleFactor(h1);
                                final long l2 = uDecimal2 - SCALE9F.multiplyByScaleFactor(h2);
                                final long h1xh2 = Checked.multiplyLong(h1, h2);//checked
                                final long h1xl2 = h1 * l2;//cannot overflow
                                final long h2xl1 = h2 * l1;//cannot overflow
                                final long l1xl2d = SCALE9F.divideByScaleFactor(l1 * l2);//product fits for scale 9, hence unchecked
                                final long h1xl2d = scaleDiff09.divideByScaleFactor(h1xl2);
                                final long h2xl1d = scaleDiff09.divideByScaleFactor(h2xl1);
                                final long h1xl2r = h1xl2 - scaleDiff09.multiplyByScaleFactor(h1xl2d);
                                final long h2xl1r = h2xl1 - scaleDiff09.multiplyByScaleFactor(h2xl1d);
                                //add it all up now, every operation checked
                                long result = scaleDiff18.multiplyByScaleFactorExact(h1xh2);
                                result = Checked.addLong(result, h1xl2d);
                                result = Checked.addLong(result, h2xl1d);
                                result = Checked.addLong(result, scaleDiff09.divideByScaleFactor(h1xl2r + h2xl1r + l1xl2d));
                                return result;
                        }
                } catch (ArithmeticException e) {
                        throw Exceptions.newArithmeticExceptionWithCause("Overflow: " + scaleMetrics1.toString(uDecimal1) + " * " + scaleMetrics2.toString(uDecimal2), e);
                }
        }

        /**
         * Calculates the multiple {@code uDecimal1 * uDecimal2 / scaleFactor}
         * with rounding.
         * 
         * @param arith
         *            the arithmetic with access to scale metrics etc.
         * @param rounding
         *            the rounding to apply for truncated decimals
         * @param uDecimal1
         *            the first unscaled decimal factor
         * @param uDecimal2
         *            the second unscaled decimal factor
         *            
         * @return the multiplication result with rounding and overflow checking
         */
        public static final long multiplyChecked(DecimalArithmetic arith, DecimalRounding rounding, long uDecimal1, long uDecimal2) {
                final SpecialMultiplicationResult special = SpecialMultiplicationResult.getFor(arith, uDecimal1, uDecimal2);
                if (special != null) {
                        return special.multiply(arith, uDecimal1, uDecimal2);
                }
                final ScaleMetrics scaleMetrics = arith.getScaleMetrics();
                return multiplyChecked(rounding, scaleMetrics, uDecimal1, scaleMetrics, uDecimal2);
        }

        /**
         * Calculates checked multiplication by an unscaled value with the given
         * scale with rounding.
         * 
         * @param arith
         *            the arithmetics associated with {@code uDecimal}
         * @param rounding
         *            the rounding to apply
         * @param uDecimal
         *            the unscaled decimal factor
         * @param unscaled
         *            the second unscaled factor
         * @param scale
         *            the scale of the second factor
         * @return the multiplication result with rounding and overflow checks
         */
        public static final long multiplyByUnscaledChecked(DecimalArithmetic arith, DecimalRounding rounding, long uDecimal, long unscaled, int scale) {
                if (scale > Scales.MAX_SCALE) {
                        throw new IllegalArgumentException("Illegal scale, must be <=" + Scales.MAX_SCALE + " but was " + scale);
                }
                if (uDecimal == 0 | unscaled == 0) {
                        return 0;
                } else if (scale == 0) {
                        return arith.multiplyByLong(uDecimal, unscaled);
                } else if (scale < 0) {
                        final long unscaledResult = Checked.multiplyLong(uDecimal, unscaled);
                        return Pow10.divideByPowerOf10Checked(arith, rounding, unscaledResult, scale);
                }
                final ScaleMetrics scaleMetrics2 = Scales.getScaleMetrics(scale);
                return multiplyChecked(rounding, arith.getScaleMetrics(), uDecimal, scaleMetrics2, unscaled);
        }

        /**
         * Calculates the checked multiple
         * {@code uDecimal1 * uDecimal2 / scaleFactor2} with rounding.
         * 
         * @param rounding
         *            the rounding to apply for truncated decimals
         * @param scaleMetrics1
         *            the scale metrics of the first factor
         * @param uDecimal1
         *            the first unscaled decimal factor
         * @param scaleMetrics2
         *            the scale metrics of the second factor
         * @param uDecimal2
         *            the second unscaled decimal factor
         * @return the multiplication result with rounding and overflow checking
         */
        private static final long multiplyChecked(DecimalRounding rounding, ScaleMetrics scaleMetrics1, long uDecimal1, ScaleMetrics scaleMetrics2, long uDecimal2) {
                try {
                        if (doesProductFitInLong(uDecimal1, uDecimal2)) {
                                //product fits in long, just do it
                                return multiply32(rounding, uDecimal1, scaleMetrics2, uDecimal2);
                        }

                        final int scale = scaleMetrics2.getScale();
                        if (scale <= 9) {
                                //use scale to split into 2 parts: i (integral) and f (fractional)
                                //with this scale, the low order product f1*f2 fits in a long
                                final long i1 = scaleMetrics2.divideByScaleFactor(uDecimal1);
                                final long i2 = scaleMetrics2.divideByScaleFactor(uDecimal2);
                                final long f1 = uDecimal1 - scaleMetrics2.multiplyByScaleFactor(i1);
                                final long f2 = uDecimal2 - scaleMetrics2.multiplyByScaleFactor(i2);
                                final long i1xf2 = i1 * f2;//cannot overflow
                                final long f1xf2 = f1 * f2;//cannot overflow for this scale
                                final long f1xf2d = scaleMetrics2.divideByScaleFactor(f1xf2);
                                final long f1xf2r = f1xf2 - scaleMetrics2.multiplyByScaleFactor(f1xf2d);
                                //add it all up now, every operation checked
                                long result = Checked.multiplyLong(uDecimal1, i2);
                                result = Checked.addLong(result, i1xf2);
                                result = Checked.addLong(result, f1xf2d);
                                
                                return result + Rounding.calculateRoundingIncrement(rounding, result, f1xf2r, scaleMetrics2.getScaleFactor());
                        } else {
                                //use scale9 to split into 2 parts: h (high) and l (low)
                                final ScaleMetrics scaleDiff09 = Scales.getScaleMetrics(scale - 9);
                                final ScaleMetrics scaleDiff18 = Scales.getScaleMetrics(18 - scale);
                                final long h1 = SCALE9F.divideByScaleFactor(uDecimal1);
                                final long h2 = SCALE9F.divideByScaleFactor(uDecimal2);
                                final long l1 = uDecimal1 - SCALE9F.multiplyByScaleFactor(h1);
                                final long l2 = uDecimal2 - SCALE9F.multiplyByScaleFactor(h2);
                                final long h1xl2 = h1 * l2;
                                final long h2xl1 = h2 * l1;
                                final long l1xl2 = l1 * l2;
                                final long l1xl2d = SCALE9F.divideByScaleFactor(l1xl2);
                                final long h1xl2d = scaleDiff09.divideByScaleFactor(h1xl2);
                                final long h2xl1d = scaleDiff09.divideByScaleFactor(h2xl1);
                                final long h1xl2r = h1xl2 - scaleDiff09.multiplyByScaleFactor(h1xl2d);
                                final long h2xl1r = h2xl1 - scaleDiff09.multiplyByScaleFactor(h2xl1d);
                                final long l1xl2r = l1xl2 - SCALE9F.multiplyByScaleFactor(l1xl2d);
                                final long h1xl2_h2xl1_l1xl1 = h1xl2r + h2xl1r + l1xl2d; 
                                final long h1xl2_h2xl1_l1xl1d = scaleDiff09.divideByScaleFactor(h1xl2_h2xl1_l1xl1); 
                                final long h1xl2_h2xl1_l1xl1r = h1xl2_h2xl1_l1xl1 - scaleDiff09.multiplyByScaleFactorExact(h1xl2_h2xl1_l1xl1d); 
                                
                                final long h1xh2 = Checked.multiplyLong(h1, h2);//checked
                                //add it all up now, every operation checked
                                long result = scaleDiff18.multiplyByScaleFactorExact(h1xh2);
                                result = Checked.addLong(result, h1xl2d);
                                result = Checked.addLong(result, h2xl1d);
                                result = Checked.addLong(result, scaleDiff09.divideByScaleFactor(h1xl2r + h2xl1r + l1xl2d));//inner sum cannot overflow
                                
                                final long remainder = SCALE9F.multiplyByScaleFactor(h1xl2_h2xl1_l1xl1r) + l1xl2r;//cannot overflow
                                return Checked.addLong(result, Rounding.calculateRoundingIncrement(rounding, result, remainder, scaleMetrics2.getScaleFactor()));
                        }
                } catch (ArithmeticException e) {
                        Exceptions.rethrowIfRoundingNecessary(e);
                        throw Exceptions.newArithmeticExceptionWithCause("Overflow: " + scaleMetrics1.toString(uDecimal1) + " * " + scaleMetrics2.toString(uDecimal2), e);
                }
        }
        
        //no instances
        private Mul() {
        }
}