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## src/java.base/share/classes/java/math/BigDecimal.java

```@@ -126,10 +126,11 @@
* <tr><th>Operation</th><th>Preferred Scale of Result</th></tr>
* <tr><td>Subtract</td><td>max(minuend.scale(), subtrahend.scale())</td>
* <tr><td>Multiply</td><td>multiplier.scale() + multiplicand.scale()</td>
* <tr><td>Divide</td><td>dividend.scale() - divisor.scale()</td>
* </table>
*
* These scales are the ones used by the methods which return exact
* arithmetic results; except that an exact divide may have to use a
* larger scale since the exact result may have more digits.  For
```

```@@ -344,10 +345,20 @@
* @since  1.5
*/
public static final BigDecimal TEN =
ZERO_THROUGH_TEN[10];

+    /**
+     * The value 0.1, with a scale of 1.
+     */
+    private static final BigDecimal ONE_TENTH = valueOf(1L, 1);
+
+    /**
+     * The value 0.5, with a scale of 1.
+     */
+    private static final BigDecimal ONE_HALF = valueOf(5L, 1);
+
// Constructors

/**
* Trusted package private constructor.
* Trusted simply means if val is INFLATED, intVal could not be null and
```

```@@ -1994,10 +2005,299 @@
result[1] = lhs.subtract(result[0].multiply(divisor));
return result;
}

/**
+     * Returns an approximation to the square root of {@code this}
+     * with rounding according to the context settings.
+     *
+     * <p>The preferred scale of the returned result is equal to
+     * {@code this.scale()/2}. The value of the returned result is
+     * always within one ulp of the exact decimal value for the
+     * precision in question.  If the rounding mode is {@link
+     * RoundingMode#HALF_UP HALF_UP}, {@link RoundingMode#HALF_DOWN
+     * HALF_DOWN}, or {@link RoundingMode#HALF_EVEN HALF_EVEN}, the
+     * result is within one half an ulp of the exact decimal value.
+     *
+     * <p>Special case:
+     * <ul>
+     * <li> The square root of a number numerically equal to {@code
+     * ZERO} is numerically equal to {@code ZERO} with a preferred
+     * scale according to the general rule above. In particular, for
+     * {@code ZERO}}, {@code ZERO.sqrt(mc).equals(ZERO)} is true with
+     * any {@code MathContext} as an argument.
+     * </ul>
+     *
+     * @param mc the context to use.
+     * @return the square root of {@code this}.
+     * @throws ArithmeticException if {@code this} is less than zero.
+     * @throws ArithmeticException if an exact result is requested
+     * ({@code mc.getPrecision()==0}) and there is no finite decimal
+     * expansion of the exact result
+     * @throws ArithmeticException if
+     * {@code (mc.getRoundingMode()==RoundingMode.UNNECESSARY}) and
+     * the exact result cannot fit in {@code mc.getPrecision()}
+     * digits.
+     * @since  9
+     */
+    public BigDecimal sqrt(MathContext mc) {
+        int signum = signum();
+        if (signum == 1) {
+            /*
+             * The following code draws on the algorithm presented in
+             * "Properly Rounded Variable Precision Square Root," Hull and
+             * Abrham, ACM Transactions on Mathematical Software, Vol 11,
+             * No. 3, September 1985, Pages 229-237.
+             *
+             * The BigDecimal computational model differs from the one
+             * presented in the paper in several ways: first BigDecimal
+             * numbers aren't necessarily normalized, second many more
+             * rounding modes are supported, including UNNECESSARY, and
+             * exact results can be requested.
+             *
+             * The main steps of the algorithm below are as follow, first
+             * argument reduce the value to the numerical range [1, 10)
+             * using the following relations:
+             *
+             * x = y * 10 ^ exp
+             * sqrt(x) = sqrt(y) * 10^(exp / 2) if exp is even
+             * sqrt(x) = sqrt(y/10) * 10 ^((exp+1)/2) is exp is odd
+             *
+             * Then use Newton's iteration on the reduced value to compute
+             * the numerical digits of the desired result.
+             *
+             * Finally, scale back to the desired exponent range and
+             * perform any adjustment to get the preferred scale in the
+             * representation.
+             */
+
+            // The code below favors relative simplicity over checking
+            // for special cases that could run faster.
+
+            int preferredScale = this.scale()/2;
+            BigDecimal zeroWithFinalPreferredScale = valueOf(0L, preferredScale);
+
+            // First phase of numerical normalization, strip trailing
+            // zeros and check for even powers of 10.
+            BigDecimal stripped = this.stripTrailingZeros();
+            int strippedScale = stripped.scale();
+
+            // Numerically sqrt(10^2N) = 10^N
+            if (stripped.isPowerOfTen() &&
+                strippedScale % 2 == 0) {
+                BigDecimal result = valueOf(1L, strippedScale/2);
+                if (result.scale() != preferredScale) {
+                    // Adjust to requested precision and preferred
+                    // scale as appropriate.
+                }
+                return result;
+            }
+
+            // After stripTrailingZeros, the representation is normalized as
+            //
+            // unscaledValue * 10^(-scale)
+            //
+            // where unscaledValue is an integer with the mimimum
+            // precision for the cohort of the numerical value. To
+            // allow binary floating-point hardware to be used to get
+            // approximately a 15 digit approximation to the square
+            // root, it is helpful to instead normalize this as so
+            // that the significand portion is to right of the decimal
+            // point by roughly (scale() - precision() +1).
+
+            // Now the precision / scale adjustment
+            int scale = stripped.scale() - stripped.precision() + 1;
+            if (scale % 2 == 0) {
+            } else {
+                scaleAdjust = scale - 1;
+            }
+
+
+            assert  // Verify 0.1 <= working < 10
+                ONE_TENTH.compareTo(working) <= 0 && working.compareTo(TEN) < 0;
+
+            // Use good ole' Math.sqrt to get the initial guess for
+            // the Newton iteration, good to at least 15 decimal
+            // digits. This approach does incur the cost of a
+            //
+            // BigDecimal -> double -> BigDecimal
+            //
+            // conversion cycle, but it avoids the need for several
+            // Newton iterations in BigDecimal arithmetic to get the
+            // working answer to 15 digits of precision. If many fewer
+            // than 15 digits were needed, it might be faster to do
+            // the loop entirely in BigDecimal arithmetic.
+            //
+            // (A double value might have as much many as 17 decimal
+            // digits of precision; it depends on the relative density
+            // of binary and decimal numbers at different regions of
+            // the number line.)
+            //
+            // (It would be possible to check for certain special
+            // cases to avoid doing any Newton iterations. For
+            // example, if the BigDecimal -> double conversion was
+            // known to be exact and the rounding mode had a
+            // low-enough precision, the post-Newton rounding logic
+            // could be applied directly.)
+
+            BigDecimal guess = new BigDecimal(Math.sqrt(working.doubleValue()));
+            int guessPrecision = 15;
+            int originalPrecision = mc.getPrecision();
+            int targetPrecision;
+
+            // If an exact value is requested, it must only need about
+            // half of the input digits to represent since multiplying
+            // an N digit number by itself yield a 2N-1 digit or 2N
+            // digit result.
+            if (originalPrecision == 0) {
+                targetPrecision = stripped.precision()/2 + 1;
+            } else {
+                targetPrecision = originalPrecision;
+            }
+
+            // When setting the precision to use inside the Newton
+            // iteration loop, take care to avoid the case where the
+            // precision of the input exceeds the requested precision
+            // and rounding the input value too soon.
+            BigDecimal approx = guess;
+            int workingPrecision = working.precision();
+            do {
+                int tmpPrecision = Math.max(Math.max(guessPrecision, targetPrecision + 2),
+                                           workingPrecision);
+                MathContext mcTmp = new MathContext(tmpPrecision, RoundingMode.HALF_EVEN);
+                // approx = 0.5 * (approx + fraction / approx)
+                approx = ONE_HALF.multiply(approx.add(working.divide(approx, mcTmp), mcTmp));
+                guessPrecision *= 2;
+            } while (guessPrecision < targetPrecision + 2);
+
+            BigDecimal result;
+            RoundingMode targetRm = mc.getRoundingMode();
+            if (targetRm == RoundingMode.UNNECESSARY || originalPrecision == 0) {
+                RoundingMode tmpRm =
+                    (targetRm == RoundingMode.UNNECESSARY) ? RoundingMode.DOWN : targetRm;
+                MathContext mcTmp = new MathContext(targetPrecision, tmpRm);
+
+                // If result*result != this numerically, the square
+                // root isn't exact
+                if (this.subtract(result.multiply(result)).compareTo(ZERO) != 0) {
+                    throw new ArithmeticException("Computed square root not exact.");
+                }
+            } else {
+            }
+
+            if (result.scale() != preferredScale) {
+                // The preferred scale of an add is
+                // max(addend.scale(), augend.scale()). Therefore, if
+                // the scale of the result is first minimized using
+                // stripTrailingZeros(), adding a zero of the
+                // preferred scale rounding the correct precision will
+                // perform the proper scale vs precision tradeoffs.
+                result = result.stripTrailingZeros().
+                        new MathContext(originalPrecision, RoundingMode.UNNECESSARY));
+            }
+            assert squareRootResultAssertions(result, mc);
+            return result;
+        } else {
+            switch (signum) {
+            case -1:
+                throw new ArithmeticException("Attempted square root " +
+                                              "of negative BigDecimal");
+            case 0:
+                return valueOf(0L, scale()/2);
+
+            default:
+                throw new AssertionError("Bad value from signum");
+            }
+        }
+    }
+
+    private boolean isPowerOfTen() {
+        return BigInteger.ONE.equals(this.unscaledValue());
+    }
+
+    /**
+     * For nonzero values, check numerical correctness properties of
+     * the computed result for the chosen rounding mode .
+     *
+     * For the directed roundings, for DOWN and FLOOR, result^2 must
+     * be {@code <=} the input and (result+ulp)^2 must be {@code >} the
+     * input. Conversely, for UP and CEIL, result^2 must be {@code >=} the
+     * input and (result-ulp)^2 must be {@code <} the input.
+     */
+    private boolean squareRootResultAssertions(BigDecimal result, MathContext mc) {
+        if (result.signum() == 0) {
+            return squareRootZeroResultAssertions(result, mc);
+        } else {
+            RoundingMode rm = mc.getRoundingMode();
+            BigDecimal ulp = result.ulp();
+            // Make neighbor down accurate even for powers of ten
+            if (this.isPowerOfTen()) {
+                ulp = ulp.divide(TEN);
+            }
+            BigDecimal neighborDown = result.subtract(ulp);
+
+            // Both the starting value and result should be nonzero and positive.
+            if (result.signum() != 1 ||
+                this.signum() != 1) {
+                return false;
+            }
+
+            switch (rm) {
+            case DOWN:
+            case FLOOR:
+                return
+                    result.multiply(result).compareTo(this)         <= 0 &&
+                    neighborUp.multiply(neighborUp).compareTo(this) > 0;
+
+            case UP:
+            case CEILING:
+                return
+                    result.multiply(result).compareTo(this)             >= 0 &&
+                    neighborDown.multiply(neighborDown).compareTo(this) < 0;
+
+            case HALF_DOWN:
+            case HALF_EVEN:
+            case HALF_UP:
+                BigDecimal err = result.multiply(result).subtract(this).abs();
+                BigDecimal errUp = neighborUp.multiply(neighborUp).subtract(this);
+                BigDecimal errDown =  this.subtract(neighborDown.multiply(neighborDown));
+                // All error values should be positive so don't need to
+                // compare absolute values.
+
+                int err_comp_errUp = err.compareTo(errUp);
+                int err_comp_errDown = err.compareTo(errDown);
+
+                return
+                    errUp.signum()   == 1 &&
+                    errDown.signum() == 1 &&
+
+                    err_comp_errUp   <= 0 &&
+                    err_comp_errDown <= 0 &&
+
+                    ((err_comp_errUp   == 0 ) ? err_comp_errDown < 0 : true) &&
+                    ((err_comp_errDown == 0 ) ? err_comp_errUp   < 0 : true);
+                // && could check for digit conditions for ties too
+
+            default: // Definition of UNNECESSARY already verified.
+                return true;
+            }
+        }
+    }
+
+    private boolean squareRootZeroResultAssertions(BigDecimal result, MathContext mc) {
+        return this.compareTo(ZERO) == 0;
+    }
+
+    /**
* Returns a {@code BigDecimal} whose value is
* <code>(this<sup>n</sup>)</code>, The power is computed exactly, to
* unlimited precision.
*
* <p>The parameter {@code n} must be in the range 0 through
```
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