001/*
002 * Copyright (C) 2011 The Guava Authors
003 *
004 * Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except
005 * in compliance with the License. You may obtain a copy of the License at
006 *
007 * http://www.apache.org/licenses/LICENSE-2.0
008 *
009 * Unless required by applicable law or agreed to in writing, software distributed under the License
010 * is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express
011 * or implied. See the License for the specific language governing permissions and limitations under
012 * the License.
013 */
014
015package com.google.common.math;
016
017import static com.google.common.base.Preconditions.checkArgument;
018import static com.google.common.math.DoubleUtils.IMPLICIT_BIT;
019import static com.google.common.math.DoubleUtils.SIGNIFICAND_BITS;
020import static com.google.common.math.DoubleUtils.getSignificand;
021import static com.google.common.math.DoubleUtils.isFinite;
022import static com.google.common.math.DoubleUtils.isNormal;
023import static com.google.common.math.DoubleUtils.scaleNormalize;
024import static com.google.common.math.MathPreconditions.checkInRange;
025import static com.google.common.math.MathPreconditions.checkNonNegative;
026import static com.google.common.math.MathPreconditions.checkRoundingUnnecessary;
027import static java.lang.Math.abs;
028import static java.lang.Math.copySign;
029import static java.lang.Math.getExponent;
030import static java.lang.Math.log;
031import static java.lang.Math.rint;
032
033import com.google.common.annotations.GwtCompatible;
034import com.google.common.annotations.GwtIncompatible;
035import com.google.common.annotations.VisibleForTesting;
036import com.google.common.primitives.Booleans;
037import com.google.errorprone.annotations.CanIgnoreReturnValue;
038import java.math.BigInteger;
039import java.math.RoundingMode;
040import java.util.Iterator;
041
042/**
043 * A class for arithmetic on doubles that is not covered by {@link java.lang.Math}.
044 *
045 * @author Louis Wasserman
046 * @since 11.0
047 */
048@GwtCompatible(emulated = true)
049public final class DoubleMath {
050  /*
051   * This method returns a value y such that rounding y DOWN (towards zero) gives the same result as
052   * rounding x according to the specified mode.
053   */
054  @GwtIncompatible // #isMathematicalInteger, com.google.common.math.DoubleUtils
055  static double roundIntermediate(double x, RoundingMode mode) {
056    if (!isFinite(x)) {
057      throw new ArithmeticException("input is infinite or NaN");
058    }
059    switch (mode) {
060      case UNNECESSARY:
061        checkRoundingUnnecessary(isMathematicalInteger(x));
062        return x;
063
064      case FLOOR:
065        if (x >= 0.0 || isMathematicalInteger(x)) {
066          return x;
067        } else {
068          return (long) x - 1;
069        }
070
071      case CEILING:
072        if (x <= 0.0 || isMathematicalInteger(x)) {
073          return x;
074        } else {
075          return (long) x + 1;
076        }
077
078      case DOWN:
079        return x;
080
081      case UP:
082        if (isMathematicalInteger(x)) {
083          return x;
084        } else {
085          return (long) x + (x > 0 ? 1 : -1);
086        }
087
088      case HALF_EVEN:
089        return rint(x);
090
091      case HALF_UP:
092        {
093          double z = rint(x);
094          if (abs(x - z) == 0.5) {
095            return x + copySign(0.5, x);
096          } else {
097            return z;
098          }
099        }
100
101      case HALF_DOWN:
102        {
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 rounding
124   *         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
127   *         {@link RoundingMode#UNNECESSARY}
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 rounding
148   *         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
151   *         {@link RoundingMode#UNNECESSARY}
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
176   *         {@link RoundingMode#UNNECESSARY}
177   *     </ul>
178   */
179  // #roundIntermediate, java.lang.Math.getExponent, com.google.common.math.DoubleUtils
180  @GwtIncompatible
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   */
196  @GwtIncompatible // com.google.common.math.DoubleUtils
197  public static boolean isPowerOfTwo(double x) {
198    if (x > 0.0 && isFinite(x)) {
199      long significand = getSignificand(x);
200      return (significand & (significand - 1)) == 0;
201    }
202    return false;
203  }
204
205  /**
206   * Returns the base 2 logarithm of a double value.
207   *
208   * <p>Special cases:
209   * <ul>
210   * <li>If {@code x} is NaN or less than zero, the result is NaN.
211   * <li>If {@code x} is positive infinity, the result is positive infinity.
212   * <li>If {@code x} is positive or negative zero, the result is negative infinity.
213   * </ul>
214   *
215   * <p>The computed result is within 1 ulp of the exact result.
216   *
217   * <p>If the result of this method will be immediately rounded to an {@code int},
218   * {@link #log2(double, RoundingMode)} is faster.
219   */
220  public static double log2(double x) {
221    return log(x) / LN_2; // surprisingly within 1 ulp according to tests
222  }
223
224  private static final double LN_2 = log(2);
225
226  /**
227   * Returns the base 2 logarithm of a double value, rounded with the specified rounding mode to an
228   * {@code int}.
229   *
230   * <p>Regardless of the rounding mode, this is faster than {@code (int) log2(x)}.
231   *
232   * @throws IllegalArgumentException if {@code x <= 0.0}, {@code x} is NaN, or {@code x} is
233   *     infinite
234   */
235  @GwtIncompatible // java.lang.Math.getExponent, com.google.common.math.DoubleUtils
236  @SuppressWarnings("fallthrough")
237  public static int log2(double x, RoundingMode mode) {
238    checkArgument(x > 0.0 && isFinite(x), "x must be positive and finite");
239    int exponent = getExponent(x);
240    if (!isNormal(x)) {
241      return log2(x * IMPLICIT_BIT, mode) - SIGNIFICAND_BITS;
242      // Do the calculation on a normal value.
243    }
244    // x is positive, finite, and normal
245    boolean increment;
246    switch (mode) {
247      case UNNECESSARY:
248        checkRoundingUnnecessary(isPowerOfTwo(x));
249        // fall through
250      case FLOOR:
251        increment = false;
252        break;
253      case CEILING:
254        increment = !isPowerOfTwo(x);
255        break;
256      case DOWN:
257        increment = exponent < 0 & !isPowerOfTwo(x);
258        break;
259      case UP:
260        increment = exponent >= 0 & !isPowerOfTwo(x);
261        break;
262      case HALF_DOWN:
263      case HALF_EVEN:
264      case HALF_UP:
265        double xScaled = scaleNormalize(x);
266        // sqrt(2) is irrational, and the spec is relative to the "exact numerical result,"
267        // so log2(x) is never exactly exponent + 0.5.
268        increment = (xScaled * xScaled) > 2.0;
269        break;
270      default:
271        throw new AssertionError();
272    }
273    return increment ? exponent + 1 : exponent;
274  }
275
276  /**
277   * Returns {@code true} if {@code x} represents a mathematical integer.
278   *
279   * <p>This is equivalent to, but not necessarily implemented as, the expression {@code
280   * !Double.isNaN(x) && !Double.isInfinite(x) && x == Math.rint(x)}.
281   */
282  @GwtIncompatible // java.lang.Math.getExponent, com.google.common.math.DoubleUtils
283  public static boolean isMathematicalInteger(double x) {
284    return isFinite(x)
285        && (x == 0.0
286            || SIGNIFICAND_BITS - Long.numberOfTrailingZeros(getSignificand(x)) <= getExponent(x));
287  }
288
289  /**
290   * Returns {@code n!}, that is, the product of the first {@code n} positive integers, {@code 1} if
291   * {@code n == 0}, or {@code n!}, or {@link Double#POSITIVE_INFINITY} if
292   * {@code n! > Double.MAX_VALUE}.
293   *
294   * <p>The result is within 1 ulp of the true value.
295   *
296   * @throws IllegalArgumentException if {@code n < 0}
297   */
298  public static double factorial(int n) {
299    checkNonNegative("n", n);
300    if (n > MAX_FACTORIAL) {
301      return Double.POSITIVE_INFINITY;
302    } else {
303      // Multiplying the last (n & 0xf) values into their own accumulator gives a more accurate
304      // result than multiplying by everySixteenthFactorial[n >> 4] directly.
305      double accum = 1.0;
306      for (int i = 1 + (n & ~0xf); i <= n; i++) {
307        accum *= i;
308      }
309      return accum * everySixteenthFactorial[n >> 4];
310    }
311  }
312
313  @VisibleForTesting static final int MAX_FACTORIAL = 170;
314
315  @VisibleForTesting
316  static final double[] everySixteenthFactorial = {
317    0x1.0p0,
318    0x1.30777758p44,
319    0x1.956ad0aae33a4p117,
320    0x1.ee69a78d72cb6p202,
321    0x1.fe478ee34844ap295,
322    0x1.c619094edabffp394,
323    0x1.3638dd7bd6347p498,
324    0x1.7cac197cfe503p605,
325    0x1.1e5dfc140e1e5p716,
326    0x1.8ce85fadb707ep829,
327    0x1.95d5f3d928edep945
328  };
329
330  /**
331   * Returns {@code true} if {@code a} and {@code b} are within {@code tolerance} of each other.
332   *
333   * <p>Technically speaking, this is equivalent to {@code Math.abs(a - b) <= tolerance ||
334   * Double.valueOf(a).equals(Double.valueOf(b))}.
335   *
336   * <p>Notable special cases include:
337   *
338   * <ul>
339   *   <li>All NaNs are fuzzily equal.
340   *   <li>If {@code a == b}, then {@code a} and {@code b} are always fuzzily equal.
341   *   <li>Positive and negative zero are always fuzzily equal.
342   *   <li>If {@code tolerance} is zero, and neither {@code a} nor {@code b} is NaN, then {@code a}
343   *       and {@code b} are fuzzily equal if and only if {@code a == b}.
344   *   <li>With {@link Double#POSITIVE_INFINITY} tolerance, all non-NaN values are fuzzily equal.
345   *   <li>With finite tolerance, {@code Double.POSITIVE_INFINITY} and {@code
346   *       Double.NEGATIVE_INFINITY} are fuzzily equal only to themselves.
347   * </ul>
348   *
349   * <p>This is reflexive and symmetric, but <em>not</em> transitive, so it is <em>not</em> an
350   * equivalence relation and <em>not</em> suitable for use in {@link Object#equals}
351   * implementations.
352   *
353   * @throws IllegalArgumentException if {@code tolerance} is {@code < 0} or NaN
354   * @since 13.0
355   */
356  public static boolean fuzzyEquals(double a, double b, double tolerance) {
357    MathPreconditions.checkNonNegative("tolerance", tolerance);
358    return Math.copySign(a - b, 1.0) <= tolerance
359        // copySign(x, 1.0) is a branch-free version of abs(x), but with different NaN semantics
360        || (a == b) // needed to ensure that infinities equal themselves
361        || (Double.isNaN(a) && Double.isNaN(b));
362  }
363
364  /**
365   * Compares {@code a} and {@code b} "fuzzily," with a tolerance for nearly-equal values.
366   *
367   * <p>This method is equivalent to
368   * {@code fuzzyEquals(a, b, tolerance) ? 0 : Double.compare(a, b)}. In particular, like
369   * {@link Double#compare(double, double)}, it treats all NaN values as equal and greater than all
370   * other values (including {@link Double#POSITIVE_INFINITY}).
371   *
372   * <p>This is <em>not</em> a total ordering and is <em>not</em> suitable for use in
373   * {@link Comparable#compareTo} implementations. In particular, it is not transitive.
374   *
375   * @throws IllegalArgumentException if {@code tolerance} is {@code < 0} or NaN
376   * @since 13.0
377   */
378  public static int fuzzyCompare(double a, double b, double tolerance) {
379    if (fuzzyEquals(a, b, tolerance)) {
380      return 0;
381    } else if (a < b) {
382      return -1;
383    } else if (a > b) {
384      return 1;
385    } else {
386      return Booleans.compare(Double.isNaN(a), Double.isNaN(b));
387    }
388  }
389
390  /**
391   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
392   * {@code values}.
393   *
394   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
395   * the arithmetic mean of the population.
396   *
397   * @param values a nonempty series of values
398   * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value
399   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
400   *     values.
401   */
402  @Deprecated
403  // com.google.common.math.DoubleUtils
404  @GwtIncompatible
405  public static double mean(double... values) {
406    checkArgument(values.length > 0, "Cannot take mean of 0 values");
407    long count = 1;
408    double mean = checkFinite(values[0]);
409    for (int index = 1; index < values.length; ++index) {
410      checkFinite(values[index]);
411      count++;
412      // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
413      mean += (values[index] - mean) / count;
414    }
415    return mean;
416  }
417
418  /**
419   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
420   * {@code values}.
421   *
422   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
423   * the arithmetic mean of the population.
424   *
425   * @param values a nonempty series of values
426   * @throws IllegalArgumentException if {@code values} is empty
427   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
428   *     values.
429   */
430  @Deprecated
431  public static double mean(int... values) {
432    checkArgument(values.length > 0, "Cannot take mean of 0 values");
433    // The upper bound on the the length of an array and the bounds on the int values mean that, in
434    // this case only, we can compute the sum as a long without risking overflow or loss of
435    // precision. So we do that, as it's slightly quicker than the Knuth algorithm.
436    long sum = 0;
437    for (int index = 0; index < values.length; ++index) {
438      sum += values[index];
439    }
440    return (double) sum / values.length;
441  }
442
443  /**
444   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
445   * {@code values}.
446   *
447   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
448   * the arithmetic mean of the population.
449   *
450   * @param values a nonempty series of values, which will be converted to {@code double} values
451   *     (this may cause loss of precision for longs of magnitude over 2^53 (slightly over 9e15))
452   * @throws IllegalArgumentException if {@code values} is empty
453   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
454   *     values.
455   */
456  @Deprecated
457  public static double mean(long... values) {
458    checkArgument(values.length > 0, "Cannot take mean of 0 values");
459    long count = 1;
460    double mean = values[0];
461    for (int index = 1; index < values.length; ++index) {
462      count++;
463      // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
464      mean += (values[index] - mean) / count;
465    }
466    return mean;
467  }
468
469  /**
470   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
471   * {@code values}.
472   *
473   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
474   * the arithmetic mean of the population.
475   *
476   * @param values a nonempty series of values, which will be converted to {@code double} values
477   *     (this may cause loss of precision)
478   * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value
479   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
480   *     values.
481   */
482  @Deprecated
483  // com.google.common.math.DoubleUtils
484  @GwtIncompatible
485  public static double mean(Iterable<? extends Number> values) {
486    return mean(values.iterator());
487  }
488
489  /**
490   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
491   * {@code values}.
492   *
493   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
494   * the arithmetic mean of the population.
495   *
496   * @param values a nonempty series of values, which will be converted to {@code double} values
497   *     (this may cause loss of precision)
498   * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value
499   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
500   *     values.
501   */
502  @Deprecated
503  // com.google.common.math.DoubleUtils
504  @GwtIncompatible
505  public static double mean(Iterator<? extends Number> values) {
506    checkArgument(values.hasNext(), "Cannot take mean of 0 values");
507    long count = 1;
508    double mean = checkFinite(values.next().doubleValue());
509    while (values.hasNext()) {
510      double value = checkFinite(values.next().doubleValue());
511      count++;
512      // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
513      mean += (value - mean) / count;
514    }
515    return mean;
516  }
517
518  @GwtIncompatible // com.google.common.math.DoubleUtils
519  @CanIgnoreReturnValue
520  private static double checkFinite(double argument) {
521    checkArgument(isFinite(argument));
522    return argument;
523  }
524
525  private DoubleMath() {}
526}