Created an UncertainDouble class for uncertainty operations.

2 files changed, 552 insertions, 66 deletions

diff --git a/src/org/unitConverter/math/DecimalComparison.java b/src/org/unitConverter/math/DecimalComparison.java
index 859e8da..0f5b91e 100644
--- a/src/org/unitConverter/math/DecimalComparison.java
+++ b/src/org/unitConverter/math/DecimalComparison.java
@@ -27,42 +27,45 @@ import java.math.BigDecimal;
  */
 public final class DecimalComparison {
 	/**
-	 * The value used for double comparison. If two double values are within this value multiplied by the larger value,
-	 * they are considered equal.
+	 * The value used for double comparison. If two double values are within this
+	 * value multiplied by the larger value, they are considered equal.
 	 * 
 	 * @since 2019-03-18
 	 * @since v0.2.0
 	 */
 	public static final double DOUBLE_EPSILON = 1.0e-15;
-
+	
 	/**
-	 * The value used for float comparison. If two float values are within this value multiplied by the larger value,
-	 * they are considered equal.
+	 * The value used for float comparison. If two float values are within this
+	 * value multiplied by the larger value, they are considered equal.
 	 * 
 	 * @since 2019-03-18
 	 * @since v0.2.0
 	 */
 	public static final float FLOAT_EPSILON = 1.0e-6f;
-
+	
 	/**
 	 * Tests for equality of double values using {@link #DOUBLE_EPSILON}.
 	 * <p>
-	 * <strong>WARNING: </strong>this method is not technically transitive. If a and b are off by slightly less than
-	 * {@code epsilon * max(abs(a), abs(b))}, and b and c are off by slightly less than
-	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c) will both return true, but equals(a, c)
-	 * will return false. However, this situation is very unlikely to ever happen in a real programming situation.
+	 * <strong>WARNING: </strong>this method is not technically transitive. If a
+	 * and b are off by slightly less than {@code epsilon * max(abs(a), abs(b))},
+	 * and b and c are off by slightly less than
+	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c)
+	 * will both return true, but equals(a, c) will return false. However, this
+	 * situation is very unlikely to ever happen in a real programming situation.
 	 * <p>
 	 * If this does become a concern, some ways to solve this problem:
 	 * <ol>
-	 * <li>Raise the value of epsilon using {@link #equals(double, double, double)} (this does not make a violation of
-	 * transitivity impossible, it just significantly reduces the chances of it happening)
-	 * <li>Use {@link BigDecimal} instead of {@code double} (this will make a violation of transitivity 100% impossible)
+	 * <li>Raise the value of epsilon using
+	 * {@link #equals(double, double, double)} (this does not make a violation of
+	 * transitivity impossible, it just significantly reduces the chances of it
+	 * happening)
+	 * <li>Use {@link BigDecimal} instead of {@code double} (this will make a
+	 * violation of transitivity 100% impossible)
 	 * </ol>
 	 * 
-	 * @param a
-	 *            first value to test
-	 * @param b
-	 *            second value to test
+	 * @param a first value to test
+	 * @param b second value to test
 	 * @return whether they are equal
 	 * @since 2019-03-18
 	 * @since v0.2.0
@@ -71,57 +74,61 @@ public final class DecimalComparison {
 	public static final boolean equals(final double a, final double b) {
 		return DecimalComparison.equals(a, b, DOUBLE_EPSILON);
 	}
-
+	
 	/**
 	 * Tests for double equality using a custom epsilon value.
 	 * 
 	 * <p>
-	 * <strong>WARNING: </strong>this method is not technically transitive. If a and b are off by slightly less than
-	 * {@code epsilon * max(abs(a), abs(b))}, and b and c are off by slightly less than
-	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c) will both return true, but equals(a, c)
-	 * will return false. However, this situation is very unlikely to ever happen in a real programming situation.
+	 * <strong>WARNING: </strong>this method is not technically transitive. If a
+	 * and b are off by slightly less than {@code epsilon * max(abs(a), abs(b))},
+	 * and b and c are off by slightly less than
+	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c)
+	 * will both return true, but equals(a, c) will return false. However, this
+	 * situation is very unlikely to ever happen in a real programming situation.
 	 * <p>
 	 * If this does become a concern, some ways to solve this problem:
 	 * <ol>
-	 * <li>Raise the value of epsilon (this does not make a violation of transitivity impossible, it just significantly
-	 * reduces the chances of it happening)
-	 * <li>Use {@link BigDecimal} instead of {@code double} (this will make a violation of transitivity 100% impossible)
+	 * <li>Raise the value of epsilon (this does not make a violation of
+	 * transitivity impossible, it just significantly reduces the chances of it
+	 * happening)
+	 * <li>Use {@link BigDecimal} instead of {@code double} (this will make a
+	 * violation of transitivity 100% impossible)
 	 * </ol>
 	 * 
-	 * @param a
-	 *            first value to test
-	 * @param b
-	 *            second value to test
-	 * @param epsilon
-	 *            allowed difference
+	 * @param a       first value to test
+	 * @param b       second value to test
+	 * @param epsilon allowed difference
 	 * @return whether they are equal
 	 * @since 2019-03-18
 	 * @since v0.2.0
 	 */
-	public static final boolean equals(final double a, final double b, final double epsilon) {
+	public static final boolean equals(final double a, final double b,
+			final double epsilon) {
 		return Math.abs(a - b) <= epsilon * Math.max(Math.abs(a), Math.abs(b));
 	}
-
+	
 	/**
 	 * Tests for equality of float values using {@link #FLOAT_EPSILON}.
 	 * 
 	 * <p>
-	 * <strong>WARNING: </strong>this method is not technically transitive. If a and b are off by slightly less than
-	 * {@code epsilon * max(abs(a), abs(b))}, and b and c are off by slightly less than
-	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c) will both return true, but equals(a, c)
-	 * will return false. However, this situation is very unlikely to ever happen in a real programming situation.
+	 * <strong>WARNING: </strong>this method is not technically transitive. If a
+	 * and b are off by slightly less than {@code epsilon * max(abs(a), abs(b))},
+	 * and b and c are off by slightly less than
+	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c)
+	 * will both return true, but equals(a, c) will return false. However, this
+	 * situation is very unlikely to ever happen in a real programming situation.
 	 * <p>
 	 * If this does become a concern, some ways to solve this problem:
 	 * <ol>
-	 * <li>Raise the value of epsilon using {@link #equals(float, float, float)} (this does not make a violation of
-	 * transitivity impossible, it just significantly reduces the chances of it happening)
-	 * <li>Use {@link BigDecimal} instead of {@code float} (this will make a violation of transitivity 100% impossible)
+	 * <li>Raise the value of epsilon using {@link #equals(float, float, float)}
+	 * (this does not make a violation of transitivity impossible, it just
+	 * significantly reduces the chances of it happening)
+	 * <li>Use {@link BigDecimal} instead of {@code float} (this will make a
+	 * violation of transitivity 100% impossible)
 	 * </ol>
 	 * 
-	 * @param a
-	 *            first value to test
-	 * @param b
-	 *            second value to test
+	 * @param a first value to test
+	 * @param b second value to test
 	 * @return whether they are equal
 	 * @since 2019-03-18
 	 * @since v0.2.0
@@ -129,53 +136,121 @@ public final class DecimalComparison {
 	public static final boolean equals(final float a, final float b) {
 		return DecimalComparison.equals(a, b, FLOAT_EPSILON);
 	}
-
+	
 	/**
 	 * Tests for float equality using a custom epsilon value.
 	 * 
 	 * <p>
-	 * <strong>WARNING: </strong>this method is not technically transitive. If a and b are off by slightly less than
-	 * {@code epsilon * max(abs(a), abs(b))}, and b and c are off by slightly less than
-	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c) will both return true, but equals(a, c)
-	 * will return false. However, this situation is very unlikely to ever happen in a real programming situation.
+	 * <strong>WARNING: </strong>this method is not technically transitive. If a
+	 * and b are off by slightly less than {@code epsilon * max(abs(a), abs(b))},
+	 * and b and c are off by slightly less than
+	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c)
+	 * will both return true, but equals(a, c) will return false. However, this
+	 * situation is very unlikely to ever happen in a real programming situation.
 	 * <p>
 	 * If this does become a concern, some ways to solve this problem:
 	 * <ol>
-	 * <li>Raise the value of epsilon (this does not make a violation of transitivity impossible, it just significantly
-	 * reduces the chances of it happening)
-	 * <li>Use {@link BigDecimal} instead of {@code float} (this will make a violation of transitivity 100% impossible)
+	 * <li>Raise the value of epsilon (this does not make a violation of
+	 * transitivity impossible, it just significantly reduces the chances of it
+	 * happening)
+	 * <li>Use {@link BigDecimal} instead of {@code float} (this will make a
+	 * violation of transitivity 100% impossible)
 	 * </ol>
 	 * 
-	 * @param a
-	 *            first value to test
-	 * @param b
-	 *            second value to test
-	 * @param epsilon
-	 *            allowed difference
+	 * @param a       first value to test
+	 * @param b       second value to test
+	 * @param epsilon allowed difference
 	 * @return whether they are equal
 	 * @since 2019-03-18
 	 * @since v0.2.0
 	 */
-	public static final boolean equals(final float a, final float b, final float epsilon) {
+	public static final boolean equals(final float a, final float b,
+			final float epsilon) {
 		return Math.abs(a - b) <= epsilon * Math.max(Math.abs(a), Math.abs(b));
 	}
-
+	
+	/**
+	 * Tests for equality of {@code UncertainDouble} values using
+	 * {@link #DOUBLE_EPSILON}.
+	 * <p>
+	 * <strong>WARNING: </strong>this method is not technically transitive. If a
+	 * and b are off by slightly less than {@code epsilon * max(abs(a), abs(b))},
+	 * and b and c are off by slightly less than
+	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c)
+	 * will both return true, but equals(a, c) will return false. However, this
+	 * situation is very unlikely to ever happen in a real programming situation.
+	 * <p>
+	 * If this does become a concern, some ways to solve this problem:
+	 * <ol>
+	 * <li>Raise the value of epsilon using
+	 * {@link #equals(UncertainDouble, UncertainDouble, double)} (this does not
+	 * make a violation of transitivity impossible, it just significantly reduces
+	 * the chances of it happening)
+	 * <li>Use {@link BigDecimal} instead of {@code double} (this will make a
+	 * violation of transitivity 100% impossible)
+	 * </ol>
+	 * 
+	 * @param a first value to test
+	 * @param b second value to test
+	 * @return whether they are equal
+	 * @since 2020-09-07
+	 * @see #hashCode(double)
+	 */
+	public static final boolean equals(final UncertainDouble a,
+			final UncertainDouble b) {
+		return DecimalComparison.equals(a.value(), b.value())
+				&& DecimalComparison.equals(a.uncertainty(), b.uncertainty());
+	}
+	
+	/**
+	 * Tests for {@code UncertainDouble} equality using a custom epsilon value.
+	 * 
+	 * <p>
+	 * <strong>WARNING: </strong>this method is not technically transitive. If a
+	 * and b are off by slightly less than {@code epsilon * max(abs(a), abs(b))},
+	 * and b and c are off by slightly less than
+	 * {@code epsilon * max(abs(b), abs(c))}, then equals(a, b) and equals(b, c)
+	 * will both return true, but equals(a, c) will return false. However, this
+	 * situation is very unlikely to ever happen in a real programming situation.
+	 * <p>
+	 * If this does become a concern, some ways to solve this problem:
+	 * <ol>
+	 * <li>Raise the value of epsilon (this does not make a violation of
+	 * transitivity impossible, it just significantly reduces the chances of it
+	 * happening)
+	 * <li>Use {@link BigDecimal} instead of {@code double} (this will make a
+	 * violation of transitivity 100% impossible)
+	 * </ol>
+	 * 
+	 * @param a       first value to test
+	 * @param b       second value to test
+	 * @param epsilon allowed difference
+	 * @return whether they are equal
+	 * @since 2019-03-18
+	 * @since v0.2.0
+	 */
+	public static final boolean equals(final UncertainDouble a,
+			final UncertainDouble b, final double epsilon) {
+		return DecimalComparison.equals(a.value(), b.value(), epsilon)
+				&& DecimalComparison.equals(a.uncertainty(), b.uncertainty(),
+						epsilon);
+	}
+	
 	/**
-	 * Takes the hash code of doubles. Values that are equal according to {@link #equals(double, double)} will have the
-	 * same hash code.
+	 * Takes the hash code of doubles. Values that are equal according to
+	 * {@link #equals(double, double)} will have the same hash code.
 	 * 
-	 * @param d
-	 *            double to hash
+	 * @param d double to hash
 	 * @return hash code of double
 	 * @since 2019-10-16
 	 */
 	public static final int hash(final double d) {
 		return Float.hashCode((float) d);
 	}
-
+	
 	// You may NOT get any DecimalComparison instances
 	private DecimalComparison() {
 		throw new AssertionError();
 	}
-
+	
 }
diff --git a/src/org/unitConverter/math/UncertainDouble.java b/src/org/unitConverter/math/UncertainDouble.java
new file mode 100644
index 0000000..e948df9
--- /dev/null
+++ b/src/org/unitConverter/math/UncertainDouble.java
@@ -0,0 +1,411 @@
+/**
+ * @since 2020-09-07
+ */
+package org.unitConverter.math;
+
+import java.math.BigDecimal;
+import java.math.RoundingMode;
+import java.util.Objects;
+import java.util.regex.Matcher;
+import java.util.regex.Pattern;
+
+/**
+ * A double with an associated uncertainty value. For example, 3.2 ± 0.2.
+ * <p>
+ * All methods in this class throw a NullPointerException if any of their
+ * arguments is null.
+ *
+ * @since 2020-09-07
+ */
+public final class UncertainDouble implements Comparable<UncertainDouble> {
+	/**
+	 * The exact value 0
+	 */
+	public static final UncertainDouble ZERO = UncertainDouble.of(0, 0);
+	
+	/**
+	 * A regular expression that can recognize toString forms
+	 */
+	private static final Pattern TO_STRING = Pattern
+			.compile("([a-zA-Z_0-9\\.\\,]+)" // a number
+					// optional "± [number]"
+					+ "(?:\\s*(?:±|\\+-)\\s*([a-zA-Z_0-9\\.\\,]+))?");
+	
+	/**
+	 * Parses a string in the form of {@link UncertainDouble#toString(boolean)}
+	 * and returns the corresponding {@code UncertainDouble} instance.
+	 * <p>
+	 * This method allows some alternative forms of the string representation,
+	 * such as using "+-" instead of "±".
+	 * 
+	 * @param s string to parse
+	 * @return {@code UncertainDouble} instance
+	 * @throws IllegalArgumentException if the string is invalid
+	 * @since 2020-09-07
+	 */
+	public static final UncertainDouble fromString(String s) {
+		Objects.requireNonNull(s, "s may not be null");
+		final Matcher matcher = TO_STRING.matcher(s);
+		
+		double value, uncertainty;
+		try {
+			value = Double.parseDouble(matcher.group(1));
+		} catch (IllegalStateException | NumberFormatException e) {
+			throw new IllegalArgumentException(
+					"String " + s + " not in correct format.");
+		}
+		
+		final String uncertaintyString = matcher.group(2);
+		if (uncertaintyString == null) {
+			uncertainty = 0;
+		} else {
+			try {
+				uncertainty = Double.parseDouble(uncertaintyString);
+			} catch (final NumberFormatException e) {
+				throw new IllegalArgumentException(
+						"String " + s + " not in correct format.");
+			}
+		}
+		
+		return UncertainDouble.of(value, uncertainty);
+	}
+	
+	/**
+	 * Gets an {@code UncertainDouble} from its value and <b>absolute</b>
+	 * uncertainty.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public static final UncertainDouble of(double value, double uncertainty) {
+		return new UncertainDouble(value, uncertainty);
+	}
+	
+	/**
+	 * Gets an {@code UncertainDouble} from its value and <b>relative</b>
+	 * uncertainty.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public static final UncertainDouble ofRelative(double value,
+			double relativeUncertainty) {
+		return new UncertainDouble(value, value * relativeUncertainty);
+	}
+	
+	private final double value;
+	
+	private final double uncertainty;
+	
+	/**
+	 * @param value
+	 * @param uncertainty
+	 * @since 2020-09-07
+	 */
+	private UncertainDouble(double value, double uncertainty) {
+		this.value = value;
+		// uncertainty should only ever be positive
+		this.uncertainty = Math.abs(uncertainty);
+	}
+	
+	/**
+	 * Compares this {@code UncertainDouble} with another
+	 * {@code UncertainDouble}.
+	 * <p>
+	 * This method only compares the values, not the uncertainties. So 3.1 ± 0.5
+	 * is considered less than 3.2 ± 0.5, even though they are equivalent.
+	 * <p>
+	 * <b>Note:</b> The natural ordering of this class is inconsistent with
+	 * equals. Specifically, if two {@code UncertainDouble} instances {@code a}
+	 * and {@code b} have the same value but different uncertainties,
+	 * {@code a.compareTo(b)} will return 0 but {@code a.equals(b)} will return
+	 * {@code false}.
+	 */
+	@Override
+	public final int compareTo(UncertainDouble o) {
+		return Double.compare(this.value, o.value);
+	}
+	
+	/**
+	 * Returns the quotient of {@code this} and {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble dividedBy(UncertainDouble other) {
+		Objects.requireNonNull(other, "other may not be null");
+		return UncertainDouble.ofRelative(this.value / other.value, Math
+				.hypot(this.relativeUncertainty(), other.relativeUncertainty()));
+	}
+	
+	/**
+	 * Returns the quotient of {@code this} and the exact value {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble dividedByExact(double other) {
+		return UncertainDouble.of(this.value / other, this.uncertainty / other);
+	}
+	
+	@Override
+	public final boolean equals(Object obj) {
+		if (this == obj)
+			return true;
+		if (!(obj instanceof UncertainDouble))
+			return false;
+		final UncertainDouble other = (UncertainDouble) obj;
+		if (Double.doubleToLongBits(this.uncertainty) != Double
+				.doubleToLongBits(other.uncertainty))
+			return false;
+		if (Double.doubleToLongBits(this.value) != Double
+				.doubleToLongBits(other.value))
+			return false;
+		return true;
+	}
+	
+	/**
+	 * @param other another {@code UncertainDouble}
+	 * @return true iff this and {@code other} are within each other's
+	 *         uncertainty range.
+	 * @since 2020-09-07
+	 */
+	public final boolean equivalent(UncertainDouble other) {
+		Objects.requireNonNull(other, "other may not be null");
+		return Math.abs(this.value - other.value) <= Math.min(this.uncertainty,
+				other.uncertainty);
+	}
+	
+	/**
+	 * Gets the preferred scale for rounding a value for toString.
+	 * 
+	 * @since 2020-09-07
+	 */
+	private final int getDisplayScale() {
+		// round based on uncertainty
+		// if uncertainty starts with 1 (ignoring zeroes and the decimal
+		// point), rounds
+		// so that uncertainty has 2 significant digits.
+		// otherwise, rounds so that uncertainty has 1 significant digits.
+		// the value is rounded to the same number of decimal places as the
+		// uncertainty.
+		final BigDecimal bigUncertainty = BigDecimal.valueOf(this.uncertainty);
+		
+		// the scale that will give the uncertainty two decimal places
+		final int twoDecimalPlacesScale = bigUncertainty.scale()
+				- bigUncertainty.precision() + 2;
+		final BigDecimal roundedUncertainty = bigUncertainty
+				.setScale(twoDecimalPlacesScale, RoundingMode.HALF_EVEN);
+		
+		if (roundedUncertainty.unscaledValue().intValue() >= 20)
+			return twoDecimalPlacesScale - 1; // one decimal place
+		else
+			return twoDecimalPlacesScale;
+	}
+	
+	@Override
+	public final int hashCode() {
+		final int prime = 31;
+		int result = 1;
+		long temp;
+		temp = Double.doubleToLongBits(this.uncertainty);
+		result = prime * result + (int) (temp ^ temp >>> 32);
+		temp = Double.doubleToLongBits(this.value);
+		result = prime * result + (int) (temp ^ temp >>> 32);
+		return result;
+	}
+	
+	/**
+	 * @return true iff the value has no uncertainty
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final boolean isExact() {
+		return this.uncertainty == 0;
+	}
+	
+	/**
+	 * Returns the difference of {@code this} and {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble minus(UncertainDouble other) {
+		Objects.requireNonNull(other, "other may not be null");
+		return UncertainDouble.of(this.value - other.value,
+				Math.hypot(this.uncertainty, other.uncertainty));
+	}
+	
+	/**
+	 * Returns the difference of {@code this} and the exact value {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble minusExact(double other) {
+		return UncertainDouble.of(this.value - other, this.uncertainty);
+	}
+	
+	/**
+	 * Returns the sum of {@code this} and {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble plus(UncertainDouble other) {
+		Objects.requireNonNull(other, "other may not be null");
+		return UncertainDouble.of(this.value + other.value,
+				Math.hypot(this.uncertainty, other.uncertainty));
+	}
+	
+	/**
+	 * Returns the sum of {@code this} and the exact value {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble plusExact(double other) {
+		return UncertainDouble.of(this.value + other, this.uncertainty);
+	}
+	
+	/**
+	 * @return relative uncertainty
+	 * @since 2020-09-07
+	 */
+	public final double relativeUncertainty() {
+		return this.uncertainty / this.value;
+	}
+	
+	/**
+	 * Returns the product of {@code this} and {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble times(UncertainDouble other) {
+		Objects.requireNonNull(other, "other may not be null");
+		return UncertainDouble.ofRelative(this.value * other.value, Math
+				.hypot(this.relativeUncertainty(), other.relativeUncertainty()));
+	}
+	
+	/**
+	 * Returns the product of {@code this} and the exact value {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble timesExact(double other) {
+		return UncertainDouble.of(this.value * other, this.uncertainty * other);
+	}
+	
+	/**
+	 * Returns the result of {@code this} raised to the exponent {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble toExponent(UncertainDouble other) {
+		Objects.requireNonNull(other, "other may not be null");
+		
+		final double result = Math.pow(this.value, other.value);
+		final double relativeUncertainty = Math.hypot(
+				other.value * this.relativeUncertainty(),
+				Math.log(this.value) * other.uncertainty);
+		
+		return UncertainDouble.ofRelative(result, relativeUncertainty);
+	}
+	
+	/**
+	 * Returns the result of {@code this} raised the exact exponent
+	 * {@code other}.
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final UncertainDouble toExponentExact(double other) {
+		return UncertainDouble.ofRelative(Math.pow(this.value, other),
+				this.relativeUncertainty() * other);
+	}
+	
+	/**
+	 * Returns a string representation of this {@code UncertainDouble}.
+	 * <p>
+	 * This method returns the same value as {@link #toString(boolean)}, but
+	 * {@code showUncertainty} is true if and only if the uncertainty is
+	 * non-zero.
+	 * 
+	 * <p>
+	 * Examples:
+	 * 
+	 * <pre>
+	 * UncertainDouble.of(3.27, 0.22).toString() = "3.3 ± 0.2"
+	 * UncertainDouble.of(3.27, 0.13).toString() = "3.27 ± 0.13"
+	 * UncertainDouble.of(-5.01, 0).toString() = "-5.01"
+	 * </pre>
+	 * 
+	 * @since 2020-09-07
+	 */
+	@Override
+	public final String toString() {
+		return this.toString(!this.isExact());
+	}
+	
+	/**
+	 * Returns a string representation of this {@code UncertainDouble}.
+	 * <p>
+	 * If {@code showUncertainty} is true, the string will be of the form "VALUE
+	 * ± UNCERTAINTY", and if it is false the string will be of the form "VALUE"
+	 * <p>
+	 * VALUE represents a string representation of this {@code UncertainDouble}'s
+	 * value. If the uncertainty is non-zero, the string will be rounded to the
+	 * same precision as the uncertainty, otherwise it will not be rounded. The
+	 * string is still rounded if {@code showUncertainty} is false.<br>
+	 * UNCERTAINTY represents a string representation of this
+	 * {@code UncertainDouble}'s uncertainty. If the uncertainty ends in 1X
+	 * (where X represents any digit) it will be rounded to two significant
+	 * digits otherwise it will be rounded to one significant digit.
+	 * <p>
+	 * Examples:
+	 * 
+	 * <pre>
+	 * UncertainDouble.of(3.27, 0.22).toString(false) = "3.3"
+	 * UncertainDouble.of(3.27, 0.22).toString(true) = "3.3 ± 0.2"
+	 * UncertainDouble.of(3.27, 0.13).toString(false) = "3.27"
+	 * UncertainDouble.of(3.27, 0.13).toString(true) = "3.27 ± 0.13"
+	 * UncertainDouble.of(-5.01, 0).toString(false) = "-5.01"
+	 * UncertainDouble.of(-5.01, 0).toString(true) = "-5.01 ± 0.0"
+	 * </pre>
+	 * 
+	 * @since 2020-09-07
+	 */
+	public final String toString(boolean showUncertainty) {
+		String valueString, uncertaintyString;
+		
+		// generate the string representation of value and uncertainty
+		if (this.isExact()) {
+			uncertaintyString = "0.0";
+			valueString = Double.toString(this.value);
+			
+		} else {
+			// round the value and uncertainty according to getDisplayScale()
+			final BigDecimal bigValue = BigDecimal.valueOf(this.value);
+			final BigDecimal bigUncertainty = BigDecimal.valueOf(this.uncertainty);
+			
+			final int displayScale = this.getDisplayScale();
+			final BigDecimal roundedUncertainty = bigUncertainty
+					.setScale(displayScale, RoundingMode.HALF_EVEN);
+			final BigDecimal roundedValue = bigValue.setScale(displayScale,
+					RoundingMode.HALF_EVEN);
+			
+			valueString = roundedValue.toString();
+			uncertaintyString = roundedUncertainty.toString();
+		}
+		
+		// return "value" or "value ± uncertainty" depending on showUncertainty
+		return valueString + (showUncertainty ? " ± " + uncertaintyString : "");
+	}
+	
+	/**
+	 * @return absolute uncertainty
+	 * @since 2020-09-07
+	 */
+	public final double uncertainty() {
+		return this.uncertainty;
+	}
+	
+	/**
+	 * @return value without uncertainty
+	 * @since 2020-09-07
+	 */
+	public final double value() {
+		return this.value;
+	}
+}