```001/*
002 * Copyright (C) 2011 The Guava Authors
003 *
005 * you may not use this file except in compliance with the License.
006 * You may obtain a copy of the License at
007 *
009 *
010 * Unless required by applicable law or agreed to in writing, software
012 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
013 * See the License for the specific language governing permissions and
014 * limitations under the License.
015 */
016
018
029import static java.lang.Math.abs;
030import static java.lang.Math.copySign;
031import static java.lang.Math.getExponent;
032import static java.lang.Math.log;
033import static java.lang.Math.rint;
034
039
040import java.math.BigInteger;
041import java.math.RoundingMode;
042import java.util.Iterator;
043
044/**
045 * A class for arithmetic on doubles that is not covered by {@link java.lang.Math}.
046 *
047 * @author Louis Wasserman
048 * @since 11.0
049 */
050@GwtCompatible(emulated = true)
051public final class DoubleMath {
052  /*
053   * This method returns a value y such that rounding y DOWN (towards zero) gives the same result
054   * as rounding x according to the specified mode.
055   */
057  static double roundIntermediate(double x, RoundingMode mode) {
058    if (!isFinite(x)) {
059      throw new ArithmeticException("input is infinite or NaN");
060    }
061    switch (mode) {
062      case UNNECESSARY:
063        checkRoundingUnnecessary(isMathematicalInteger(x));
064        return x;
065
066      case FLOOR:
067        if (x >= 0.0 || isMathematicalInteger(x)) {
068          return x;
069        } else {
070          return x - 1.0;
071        }
072
073      case CEILING:
074        if (x <= 0.0 || isMathematicalInteger(x)) {
075          return x;
076        } else {
077          return x + 1.0;
078        }
079
080      case DOWN:
081        return x;
082
083      case UP:
084        if (isMathematicalInteger(x)) {
085          return x;
086        } else {
087          return x + Math.copySign(1.0, x);
088        }
089
090      case HALF_EVEN:
091        return rint(x);
092
093      case HALF_UP: {
094        double z = rint(x);
095        if (abs(x - z) == 0.5) {
096          return x + copySign(0.5, x);
097        } else {
098          return z;
099        }
100      }
101
102      case HALF_DOWN: {
103        double z = rint(x);
104        if (abs(x - z) == 0.5) {
105          return x;
106        } else {
107          return z;
108        }
109      }
110
111      default:
112        throw new AssertionError();
113    }
114  }
115
116  /**
117   * Returns the {@code int} value that is equal to {@code x} rounded with the specified rounding
118   * mode, if possible.
119   *
120   * @throws ArithmeticException if
121   *         <ul>
122   *         <li>{@code x} is infinite or NaN
123   *         <li>{@code x}, after being rounded to a mathematical integer using the specified
124   *         rounding mode, is either less than {@code Integer.MIN_VALUE} or greater than {@code
125   *         Integer.MAX_VALUE}
126   *         <li>{@code x} is not a mathematical integer and {@code mode} is
128   *         </ul>
129   */
130  @GwtIncompatible("#roundIntermediate")
131  public static int roundToInt(double x, RoundingMode mode) {
132    double z = roundIntermediate(x, mode);
133    checkInRange(z > MIN_INT_AS_DOUBLE - 1.0 & z < MAX_INT_AS_DOUBLE + 1.0);
134    return (int) z;
135  }
136
137  private static final double MIN_INT_AS_DOUBLE = -0x1p31;
138  private static final double MAX_INT_AS_DOUBLE = 0x1p31 - 1.0;
139
140  /**
141   * Returns the {@code long} value that is equal to {@code x} rounded with the specified rounding
142   * mode, if possible.
143   *
144   * @throws ArithmeticException if
145   *         <ul>
146   *         <li>{@code x} is infinite or NaN
147   *         <li>{@code x}, after being rounded to a mathematical integer using the specified
148   *         rounding mode, is either less than {@code Long.MIN_VALUE} or greater than {@code
149   *         Long.MAX_VALUE}
150   *         <li>{@code x} is not a mathematical integer and {@code mode} is
152   *         </ul>
153   */
154  @GwtIncompatible("#roundIntermediate")
155  public static long roundToLong(double x, RoundingMode mode) {
156    double z = roundIntermediate(x, mode);
157    checkInRange(MIN_LONG_AS_DOUBLE - z < 1.0 & z < MAX_LONG_AS_DOUBLE_PLUS_ONE);
158    return (long) z;
159  }
160
161  private static final double MIN_LONG_AS_DOUBLE = -0x1p63;
162  /*
163   * We cannot store Long.MAX_VALUE as a double without losing precision.  Instead, we store
164   * Long.MAX_VALUE + 1 == -Long.MIN_VALUE, and then offset all comparisons by 1.
165   */
166  private static final double MAX_LONG_AS_DOUBLE_PLUS_ONE = 0x1p63;
167
168  /**
169   * Returns the {@code BigInteger} value that is equal to {@code x} rounded with the specified
170   * rounding mode, if possible.
171   *
172   * @throws ArithmeticException if
173   *         <ul>
174   *         <li>{@code x} is infinite or NaN
175   *         <li>{@code x} is not a mathematical integer and {@code mode} is
177   *         </ul>
178   */
179  @GwtIncompatible("#roundIntermediate, java.lang.Math.getExponent, "
181  public static BigInteger roundToBigInteger(double x, RoundingMode mode) {
182    x = roundIntermediate(x, mode);
183    if (MIN_LONG_AS_DOUBLE - x < 1.0 & x < MAX_LONG_AS_DOUBLE_PLUS_ONE) {
184      return BigInteger.valueOf((long) x);
185    }
186    int exponent = getExponent(x);
187    long significand = getSignificand(x);
188    BigInteger result = BigInteger.valueOf(significand).shiftLeft(exponent - SIGNIFICAND_BITS);
189    return (x < 0) ? result.negate() : result;
190  }
191
192  /**
193   * Returns {@code true} if {@code x} is exactly equal to {@code 2^k} for some finite integer
194   * {@code k}.
195   */
197  public static boolean isPowerOfTwo(double x) {
198    return x > 0.0 && isFinite(x) && LongMath.isPowerOfTwo(getSignificand(x));
199  }
200
201  /**
202   * Returns the base 2 logarithm of a double value.
203   *
204   * <p>Special cases:
205   * <ul>
206   * <li>If {@code x} is NaN or less than zero, the result is NaN.
207   * <li>If {@code x} is positive infinity, the result is positive infinity.
208   * <li>If {@code x} is positive or negative zero, the result is negative infinity.
209   * </ul>
210   *
211   * <p>The computed result is within 1 ulp of the exact result.
212   *
213   * <p>If the result of this method will be immediately rounded to an {@code int},
214   * {@link #log2(double, RoundingMode)} is faster.
215   */
216  public static double log2(double x) {
217    return log(x) / LN_2; // surprisingly within 1 ulp according to tests
218  }
219
220  private static final double LN_2 = log(2);
221
222  /**
223   * Returns the base 2 logarithm of a double value, rounded with the specified rounding mode to an
224   * {@code int}.
225   *
226   * <p>Regardless of the rounding mode, this is faster than {@code (int) log2(x)}.
227   *
228   * @throws IllegalArgumentException if {@code x <= 0.0}, {@code x} is NaN, or {@code x} is
229   *         infinite
230   */
232  @SuppressWarnings("fallthrough")
233  public static int log2(double x, RoundingMode mode) {
234    checkArgument(x > 0.0 && isFinite(x), "x must be positive and finite");
235    int exponent = getExponent(x);
236    if (!isNormal(x)) {
237      return log2(x * IMPLICIT_BIT, mode) - SIGNIFICAND_BITS;
238      // Do the calculation on a normal value.
239    }
240    // x is positive, finite, and normal
241    boolean increment;
242    switch (mode) {
243      case UNNECESSARY:
244        checkRoundingUnnecessary(isPowerOfTwo(x));
245        // fall through
246      case FLOOR:
247        increment = false;
248        break;
249      case CEILING:
250        increment = !isPowerOfTwo(x);
251        break;
252      case DOWN:
253        increment = exponent < 0 & !isPowerOfTwo(x);
254        break;
255      case UP:
256        increment = exponent >= 0 & !isPowerOfTwo(x);
257        break;
258      case HALF_DOWN:
259      case HALF_EVEN:
260      case HALF_UP:
261        double xScaled = scaleNormalize(x);
262        // sqrt(2) is irrational, and the spec is relative to the "exact numerical result,"
263        // so log2(x) is never exactly exponent + 0.5.
264        increment = (xScaled * xScaled) > 2.0;
265        break;
266      default:
267        throw new AssertionError();
268    }
269    return increment ? exponent + 1 : exponent;
270  }
271
272  /**
273   * Returns {@code true} if {@code x} represents a mathematical integer.
274   *
275   * <p>This is equivalent to, but not necessarily implemented as, the expression {@code
276   * !Double.isNaN(x) && !Double.isInfinite(x) && x == Math.rint(x)}.
277   */
279  public static boolean isMathematicalInteger(double x) {
280    return isFinite(x)
281        && (x == 0.0 ||
282            SIGNIFICAND_BITS - Long.numberOfTrailingZeros(getSignificand(x)) <= getExponent(x));
283  }
284
285  /**
286   * Returns {@code n!}, that is, the product of the first {@code n} positive
287   * integers, {@code 1} if {@code n == 0}, or {@code n!}, or
288   * {@link Double#POSITIVE_INFINITY} if {@code n! > Double.MAX_VALUE}.
289   *
290   * <p>The result is within 1 ulp of the true value.
291   *
292   * @throws IllegalArgumentException if {@code n < 0}
293   */
294  public static double factorial(int n) {
295    checkNonNegative("n", n);
296    if (n > MAX_FACTORIAL) {
297      return Double.POSITIVE_INFINITY;
298    } else {
299      // Multiplying the last (n & 0xf) values into their own accumulator gives a more accurate
300      // result than multiplying by everySixteenthFactorial[n >> 4] directly.
301      double accum = 1.0;
302      for (int i = 1 + (n & ~0xf); i <= n; i++) {
303        accum *= i;
304      }
305      return accum * everySixteenthFactorial[n >> 4];
306    }
307  }
308
309  @VisibleForTesting
310  static final int MAX_FACTORIAL = 170;
311
312  @VisibleForTesting
313  static final double[] everySixteenthFactorial = {
314      0x1.0p0,
315      0x1.30777758p44,
317      0x1.ee69a78d72cb6p202,
318      0x1.fe478ee34844ap295,
319      0x1.c619094edabffp394,
320      0x1.3638dd7bd6347p498,
321      0x1.7cac197cfe503p605,
322      0x1.1e5dfc140e1e5p716,
324      0x1.95d5f3d928edep945};
325
326  /**
327   * Returns {@code true} if {@code a} and {@code b} are within {@code tolerance} of each other.
328   *
329   * <p>Technically speaking, this is equivalent to
330   * {@code Math.abs(a - b) <= tolerance || Double.valueOf(a).equals(Double.valueOf(b))}.
331   *
332   * <p>Notable special cases include:
333   * <ul>
334   * <li>All NaNs are fuzzily equal.
335   * <li>If {@code a == b}, then {@code a} and {@code b} are always fuzzily equal.
336   * <li>Positive and negative zero are always fuzzily equal.
337   * <li>If {@code tolerance} is zero, and neither {@code a} nor {@code b} is NaN, then
338   * {@code a} and {@code b} are fuzzily equal if and only if {@code a == b}.
339   * <li>With {@link Double#POSITIVE_INFINITY} tolerance, all non-NaN values are fuzzily equal.
340   * <li>With finite tolerance, {@code Double.POSITIVE_INFINITY} and {@code
341   * Double.NEGATIVE_INFINITY} are fuzzily equal only to themselves.
342   * </li>
343   *
344   * <p>This is reflexive and symmetric, but <em>not</em> transitive, so it is <em>not</em> an
345   * equivalence relation and <em>not</em> suitable for use in {@link Object#equals}
346   * implementations.
347   *
348   * @throws IllegalArgumentException if {@code tolerance} is {@code < 0} or NaN
349   * @since 13.0
350   */
351  public static boolean fuzzyEquals(double a, double b, double tolerance) {
352    MathPreconditions.checkNonNegative("tolerance", tolerance);
353    return
354          Math.copySign(a - b, 1.0) <= tolerance
355           // copySign(x, 1.0) is a branch-free version of abs(x), but with different NaN semantics
356          || (a == b) // needed to ensure that infinities equal themselves
357          || (Double.isNaN(a) && Double.isNaN(b));
358  }
359
360  /**
361   * Compares {@code a} and {@code b} "fuzzily," with a tolerance for nearly-equal values.
362   *
363   * <p>This method is equivalent to
364   * {@code fuzzyEquals(a, b, tolerance) ? 0 : Double.compare(a, b)}. In particular, like
365   * {@link Double#compare(double, double)}, it treats all NaN values as equal and greater than all
366   * other values (including {@link Double#POSITIVE_INFINITY}).
367   *
368   * <p>This is <em>not</em> a total ordering and is <em>not</em> suitable for use in
369   * {@link Comparable#compareTo} implementations.  In particular, it is not transitive.
370   *
371   * @throws IllegalArgumentException if {@code tolerance} is {@code < 0} or NaN
372   * @since 13.0
373   */
374  public static int fuzzyCompare(double a, double b, double tolerance) {
375    if (fuzzyEquals(a, b, tolerance)) {
376      return 0;
377    } else if (a < b) {
378      return -1;
379    } else if (a > b) {
380      return 1;
381    } else {
382      return Booleans.compare(Double.isNaN(a), Double.isNaN(b));
383    }
384  }
385
386  /**
387   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
388   * {@code values}.
389   *
390   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
391   * the arithmetic mean of the population.
392   *
393   * @param values a nonempty series of values
394   * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value
395   */
397  public static double mean(double... values) {
398    checkArgument(values.length > 0, "Cannot take mean of 0 values");
399    long count = 1;
400    double mean = checkFinite(values[0]);
401    for (int index = 1; index < values.length; ++index) {
402      checkFinite(values[index]);
403      count++;
404      // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
405      mean += (values[index] - mean) / count;
406    }
407    return mean;
408  }
409
410  /**
411   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
412   * {@code values}.
413   *
414   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
415   * the arithmetic mean of the population.
416   *
417   * @param values a nonempty series of values
418   * @throws IllegalArgumentException if {@code values} is empty
419   */
420  public static double mean(int... values) {
421    checkArgument(values.length > 0, "Cannot take mean of 0 values");
422    // The upper bound on the the length of an array and the bounds on the int values mean that, in
423    // this case only, we can compute the sum as a long without risking overflow or loss of
424    // precision. So we do that, as it's slightly quicker than the Knuth algorithm.
425    long sum = 0;
426    for (int index = 0; index < values.length; ++index) {
427      sum += values[index];
428    }
429    return (double) sum / values.length;
430  }
431
432  /**
433   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
434   * {@code values}.
435   *
436   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
437   * the arithmetic mean of the population.
438   *
439   * @param values a nonempty series of values, which will be converted to {@code double} values
440   *     (this may cause loss of precision for longs of magnitude over 2^53 (slightly over 9e15))
441   * @throws IllegalArgumentException if {@code values} is empty
442   */
443  public static double mean(long... values) {
444    checkArgument(values.length > 0, "Cannot take mean of 0 values");
445    long count = 1;
446    double mean = values[0];
447    for (int index = 1; index < values.length; ++index) {
448      count++;
449      // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
450      mean += (values[index] - mean) / count;
451    }
452    return mean;
453  }
454
455  /**
456   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
457   * {@code values}.
458   *
459   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
460   * the arithmetic mean of the population.
461   *
462   * @param values a nonempty series of values, which will be converted to {@code double} values
463   *     (this may cause loss of precision)
464   * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value
465   */
467  public static double mean(Iterable<? extends Number> values) {
468    return mean(values.iterator());
469  }
470
471  /**
472   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
473   * {@code values}.
474   *
475   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
476   * the arithmetic mean of the population.
477   *
478   * @param values a nonempty series of values, which will be converted to {@code double} values
479   *     (this may cause loss of precision)
480   * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value
481   */
483  public static double mean(Iterator<? extends Number> values) {
484    checkArgument(values.hasNext(), "Cannot take mean of 0 values");
485    long count = 1;
486    double mean = checkFinite(values.next().doubleValue());
487    while (values.hasNext()) {
488      double value = checkFinite(values.next().doubleValue());
489      count++;
490      // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
491      mean += (value - mean) / count;
492    }
493    return mean;
494  }
495
497  private static double checkFinite(double argument) {
498    checkArgument(isFinite(argument));
499    return argument;
500  }
501
502  private DoubleMath() {}
503}

```