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    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   */
231  @GwtIncompatible // java.lang.Math.getExponent, com.google.common.math.DoubleUtils
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   */
278  @GwtIncompatible // java.lang.Math.getExponent, com.google.common.math.DoubleUtils
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 integers, {@code 1} if
287   * {@code n == 0}, or {@code n!}, or {@link Double#POSITIVE_INFINITY} if
288   * {@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 static final int MAX_FACTORIAL = 170;
310
311  @VisibleForTesting
312  static final double[] everySixteenthFactorial = {
313    0x1.0p0,
314    0x1.30777758p44,
315    0x1.956ad0aae33a4p117,
316    0x1.ee69a78d72cb6p202,
317    0x1.fe478ee34844ap295,
318    0x1.c619094edabffp394,
319    0x1.3638dd7bd6347p498,
320    0x1.7cac197cfe503p605,
321    0x1.1e5dfc140e1e5p716,
322    0x1.8ce85fadb707ep829,
323    0x1.95d5f3d928edep945
324  };
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 {@code a}
338   *     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   *
343   * <p>This is reflexive and symmetric, but <em>not</em> transitive, so it is <em>not</em> an
344   * equivalence relation and <em>not</em> suitable for use in {@link Object#equals}
345   * implementations.
346   *
347   * @throws IllegalArgumentException if {@code tolerance} is {@code < 0} or NaN
348   * @since 13.0
349   */
350  public static boolean fuzzyEquals(double a, double b, double tolerance) {
351    MathPreconditions.checkNonNegative("tolerance", tolerance);
352    return Math.copySign(a - b, 1.0) <= tolerance
353        // copySign(x, 1.0) is a branch-free version of abs(x), but with different NaN semantics
354        || (a == b) // needed to ensure that infinities equal themselves
355        || (Double.isNaN(a) && Double.isNaN(b));
356  }
357
358  /**
359   * Compares {@code a} and {@code b} "fuzzily," with a tolerance for nearly-equal values.
360   *
361   * <p>This method is equivalent to
362   * {@code fuzzyEquals(a, b, tolerance) ? 0 : Double.compare(a, b)}. In particular, like
363   * {@link Double#compare(double, double)}, it treats all NaN values as equal and greater than all
364   * other values (including {@link Double#POSITIVE_INFINITY}).
365   *
366   * <p>This is <em>not</em> a total ordering and is <em>not</em> suitable for use in
367   * {@link Comparable#compareTo} implementations. In particular, it is not transitive.
368   *
369   * @throws IllegalArgumentException if {@code tolerance} is {@code < 0} or NaN
370   * @since 13.0
371   */
372  public static int fuzzyCompare(double a, double b, double tolerance) {
373    if (fuzzyEquals(a, b, tolerance)) {
374      return 0;
375    } else if (a < b) {
376      return -1;
377    } else if (a > b) {
378      return 1;
379    } else {
380      return Booleans.compare(Double.isNaN(a), Double.isNaN(b));
381    }
382  }
383
384  /**
385   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
386   * {@code values}.
387   *
388   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
389   * the arithmetic mean of the population.
390   *
391   * @param values a nonempty series of values
392   * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value
393   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
394   *     values. This method will be removed in February 2018.
395   */
396  @Deprecated
397  // com.google.common.math.DoubleUtils
398  @GwtIncompatible
399  public static double mean(double... values) {
400    checkArgument(values.length > 0, "Cannot take mean of 0 values");
401    long count = 1;
402    double mean = checkFinite(values[0]);
403    for (int index = 1; index < values.length; ++index) {
404      checkFinite(values[index]);
405      count++;
406      // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
407      mean += (values[index] - mean) / count;
408    }
409    return mean;
410  }
411
412  /**
413   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
414   * {@code values}.
415   *
416   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
417   * the arithmetic mean of the population.
418   *
419   * @param values a nonempty series of values
420   * @throws IllegalArgumentException if {@code values} is empty
421   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
422   *     values. This method will be removed in February 2018.
423   */
424  @Deprecated
425  public static double mean(int... values) {
426    checkArgument(values.length > 0, "Cannot take mean of 0 values");
427    // The upper bound on the the length of an array and the bounds on the int values mean that, in
428    // this case only, we can compute the sum as a long without risking overflow or loss of
429    // precision. So we do that, as it's slightly quicker than the Knuth algorithm.
430    long sum = 0;
431    for (int index = 0; index < values.length; ++index) {
432      sum += values[index];
433    }
434    return (double) sum / values.length;
435  }
436
437  /**
438   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
439   * {@code values}.
440   *
441   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
442   * the arithmetic mean of the population.
443   *
444   * @param values a nonempty series of values, which will be converted to {@code double} values
445   *     (this may cause loss of precision for longs of magnitude over 2^53 (slightly over 9e15))
446   * @throws IllegalArgumentException if {@code values} is empty
447   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
448   *     values. This method will be removed in February 2018.
449   */
450  @Deprecated
451  public static double mean(long... values) {
452    checkArgument(values.length > 0, "Cannot take mean of 0 values");
453    long count = 1;
454    double mean = values[0];
455    for (int index = 1; index < values.length; ++index) {
456      count++;
457      // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
458      mean += (values[index] - mean) / count;
459    }
460    return mean;
461  }
462
463  /**
464   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
465   * {@code values}.
466   *
467   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
468   * the arithmetic mean of the population.
469   *
470   * @param values a nonempty series of values, which will be converted to {@code double} values
471   *     (this may cause loss of precision)
472   * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value
473   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
474   *     values. This method will be removed in February 2018.
475   */
476  @Deprecated
477  // com.google.common.math.DoubleUtils
478  @GwtIncompatible
479  public static double mean(Iterable<? extends Number> values) {
480    return mean(values.iterator());
481  }
482
483  /**
484   * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of
485   * {@code values}.
486   *
487   * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of
488   * the arithmetic mean of the population.
489   *
490   * @param values a nonempty series of values, which will be converted to {@code double} values
491   *     (this may cause loss of precision)
492   * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value
493   * @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
494   *     values. This method will be removed in February 2018.
495   */
496  @Deprecated
497  // com.google.common.math.DoubleUtils
498  @GwtIncompatible
499  public static double mean(Iterator<? extends Number> values) {
500    checkArgument(values.hasNext(), "Cannot take mean of 0 values");
501    long count = 1;
502    double mean = checkFinite(values.next().doubleValue());
503    while (values.hasNext()) {
504      double value = checkFinite(values.next().doubleValue());
505      count++;
506      // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
507      mean += (value - mean) / count;
508    }
509    return mean;
510  }
511
512  @GwtIncompatible // com.google.common.math.DoubleUtils
513  @CanIgnoreReturnValue
514  private static double checkFinite(double argument) {
515    checkArgument(isFinite(argument));
516    return argument;
517  }
518
519  private DoubleMath() {}
520}