Java example source code file (Math.java)
The Math.java Java example source code/*
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package java.lang;
import java.util.Random;
import sun.misc.FloatConsts;
import sun.misc.DoubleConsts;
/**
* The class {@code Math} contains methods for performing basic
* numeric operations such as the elementary exponential, logarithm,
* square root, and trigonometric functions.
*
* <p>Unlike some of the numeric methods of class
* {@code StrictMath}, all implementations of the equivalent
* functions of class {@code Math} are not defined to return the
* bit-for-bit same results. This relaxation permits
* better-performing implementations where strict reproducibility is
* not required.
*
* <p>By default many of the {@code Math} methods simply call
* the equivalent method in {@code StrictMath} for their
* implementation. Code generators are encouraged to use
* platform-specific native libraries or microprocessor instructions,
* where available, to provide higher-performance implementations of
* {@code Math} methods. Such higher-performance
* implementations still must conform to the specification for
* {@code Math}.
*
* <p>The quality of implementation specifications concern two
* properties, accuracy of the returned result and monotonicity of the
* method. Accuracy of the floating-point {@code Math} methods is
* measured in terms of <i>ulps, units in the last place. For a
* given floating-point format, an {@linkplain #ulp(double) ulp} of a
* specific real number value is the distance between the two
* floating-point values bracketing that numerical value. When
* discussing the accuracy of a method as a whole rather than at a
* specific argument, the number of ulps cited is for the worst-case
* error at any argument. If a method always has an error less than
* 0.5 ulps, the method always returns the floating-point number
* nearest the exact result; such a method is <i>correctly
* rounded</i>. A correctly rounded method is generally the best a
* floating-point approximation can be; however, it is impractical for
* many floating-point methods to be correctly rounded. Instead, for
* the {@code Math} class, a larger error bound of 1 or 2 ulps is
* allowed for certain methods. Informally, with a 1 ulp error bound,
* when the exact result is a representable number, the exact result
* should be returned as the computed result; otherwise, either of the
* two floating-point values which bracket the exact result may be
* returned. For exact results large in magnitude, one of the
* endpoints of the bracket may be infinite. Besides accuracy at
* individual arguments, maintaining proper relations between the
* method at different arguments is also important. Therefore, most
* methods with more than 0.5 ulp errors are required to be
* <i>semi-monotonic: whenever the mathematical function is
* non-decreasing, so is the floating-point approximation, likewise,
* whenever the mathematical function is non-increasing, so is the
* floating-point approximation. Not all approximations that have 1
* ulp accuracy will automatically meet the monotonicity requirements.
*
* <p>
* The platform uses signed two's complement integer arithmetic with
* int and long primitive types. The developer should choose
* the primitive type to ensure that arithmetic operations consistently
* produce correct results, which in some cases means the operations
* will not overflow the range of values of the computation.
* The best practice is to choose the primitive type and algorithm to avoid
* overflow. In cases where the size is {@code int} or {@code long} and
* overflow errors need to be detected, the methods {@code addExact},
* {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact}
* throw an {@code ArithmeticException} when the results overflow.
* For other arithmetic operations such as divide, absolute value,
* increment, decrement, and negation overflow occurs only with
* a specific minimum or maximum value and should be checked against
* the minimum or maximum as appropriate.
*
* @author unascribed
* @author Joseph D. Darcy
* @since JDK1.0
*/
public final class Math {
/**
* Don't let anyone instantiate this class.
*/
private Math() {}
/**
* The {@code double} value that is closer than any other to
* <i>e, the base of the natural logarithms.
*/
public static final double E = 2.7182818284590452354;
/**
* The {@code double} value that is closer than any other to
* <i>pi, the ratio of the circumference of a circle to its
* diameter.
*/
public static final double PI = 3.14159265358979323846;
/**
* Returns the trigonometric sine of an angle. Special cases:
* <ul>* <ul> * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } * </ul> * </li> * </ul> * <p> * If the signs of arguments are unknown and a positive modulus * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}. * * @param x the dividend * @param y the divisor * @return the floor modulus {@code x - (floorDiv(x, y) * y)} * @throws ArithmeticException if the divisor {@code y} is zero * @see #floorDiv(int, int) * @since 1.8 */ public static int floorMod(int x, int y) { int r = x - floorDiv(x, y) * y; return r; } /** * Returns the floor modulus of the {@code long} arguments. * <p> * The floor modulus is {@code x - (floorDiv(x, y) * y)}, * has the same sign as the divisor {@code y}, and * is in the range of {@code -abs(y) < r < +abs(y)}. * * <p> * The relationship between {@code floorDiv} and {@code floorMod} is such that: * <ul> * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} * </ul> * <p> * For examples, see {@link #floorMod(int, int)}. * * @param x the dividend * @param y the divisor * @return the floor modulus {@code x - (floorDiv(x, y) * y)} * @throws ArithmeticException if the divisor {@code y} is zero * @see #floorDiv(long, long) * @since 1.8 */ public static long floorMod(long x, long y) { return x - floorDiv(x, y) * y; } /** * Returns the absolute value of an {@code int} value. * If the argument is not negative, the argument is returned. * If the argument is negative, the negation of the argument is returned. * * <p>Note that if the argument is equal to the value of * {@link Integer#MIN_VALUE}, the most negative representable * {@code int} value, the result is that same value, which is * negative. * * @param a the argument whose absolute value is to be determined * @return the absolute value of the argument. */ public static int abs(int a) { return (a < 0) ? -a : a; } /** * Returns the absolute value of a {@code long} value. * If the argument is not negative, the argument is returned. * If the argument is negative, the negation of the argument is returned. * * <p>Note that if the argument is equal to the value of * {@link Long#MIN_VALUE}, the most negative representable * {@code long} value, the result is that same value, which * is negative. * * @param a the argument whose absolute value is to be determined * @return the absolute value of the argument. */ public static long abs(long a) { return (a < 0) ? -a : a; } /** * Returns the absolute value of a {@code float} value. * If the argument is not negative, the argument is returned. * If the argument is negative, the negation of the argument is returned. * Special cases: * <ul> ulp(-x) == ulp(x).
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive or negative infinity, then the
* result is positive infinity.
* <li> If the argument is positive or negative zero, then the result is
* {@code Double.MIN_VALUE}.
* <li> If the argument is ±{@code Double.MAX_VALUE}, then
* the result is equal to 2<sup>971.
* </ul>
*
* @param d the floating-point value whose ulp is to be returned
* @return the size of an ulp of the argument
* @author Joseph D. Darcy
* @since 1.5
*/
public static double ulp(double d) {
int exp = getExponent(d);
switch(exp) {
case DoubleConsts.MAX_EXPONENT+1: // NaN or infinity
return Math.abs(d);
case DoubleConsts.MIN_EXPONENT-1: // zero or subnormal
return Double.MIN_VALUE;
default:
assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;
// ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
if (exp >= DoubleConsts.MIN_EXPONENT) {
return powerOfTwoD(exp);
}
else {
// return a subnormal result; left shift integer
// representation of Double.MIN_VALUE appropriate
// number of positions
return Double.longBitsToDouble(1L <<
(exp - (DoubleConsts.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
}
}
}
/**
* Returns the size of an ulp of the argument. An ulp, unit in
* the last place, of a {@code float} value is the positive
* distance between this floating-point value and the {@code
* float} value next larger in magnitude. Note that for non-NaN
* <i>x, ulp(-x) == ulp(x).
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive or negative infinity, then the
* result is positive infinity.
* <li> If the argument is positive or negative zero, then the result is
* {@code Float.MIN_VALUE}.
* <li> If the argument is ±{@code Float.MAX_VALUE}, then
* the result is equal to 2<sup>104.
* </ul>
*
* @param f the floating-point value whose ulp is to be returned
* @return the size of an ulp of the argument
* @author Joseph D. Darcy
* @since 1.5
*/
public static float ulp(float f) {
int exp = getExponent(f);
switch(exp) {
case FloatConsts.MAX_EXPONENT+1: // NaN or infinity
return Math.abs(f);
case FloatConsts.MIN_EXPONENT-1: // zero or subnormal
return FloatConsts.MIN_VALUE;
default:
assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;
// ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
if (exp >= FloatConsts.MIN_EXPONENT) {
return powerOfTwoF(exp);
}
else {
// return a subnormal result; left shift integer
// representation of FloatConsts.MIN_VALUE appropriate
// number of positions
return Float.intBitsToFloat(1 <<
(exp - (FloatConsts.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
}
}
}
/**
* Returns the signum function of the argument; zero if the argument
* is zero, 1.0 if the argument is greater than zero, -1.0 if the
* argument is less than zero.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive zero or negative zero, then the
* result is the same as the argument.
* </ul>
*
* @param d the floating-point value whose signum is to be returned
* @return the signum function of the argument
* @author Joseph D. Darcy
* @since 1.5
*/
public static double signum(double d) {
return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
}
/**
* Returns the signum function of the argument; zero if the argument
* is zero, 1.0f if the argument is greater than zero, -1.0f if the
* argument is less than zero.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive zero or negative zero, then the
* result is the same as the argument.
* </ul>
*
* @param f the floating-point value whose signum is to be returned
* @return the signum function of the argument
* @author Joseph D. Darcy
* @since 1.5
*/
public static float signum(float f) {
return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
}
/**
* Returns the hyperbolic sine of a {@code double} value.
* The hyperbolic sine of <i>x is defined to be
* (<i>ex - e-x)/2
* where <i>e is {@linkplain Math#E Euler's number}.
*
* <p>Special cases:
* <ul>
*
* <li>If the argument is NaN, then the result is NaN.
*
* <li>If the argument is infinite, then the result is an infinity
* with the same sign as the argument.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* </ul>
*
* <p>The computed result must be within 2.5 ulps of the exact result.
*
* @param x The number whose hyperbolic sine is to be returned.
* @return The hyperbolic sine of {@code x}.
* @since 1.5
*/
public static double sinh(double x) {
return StrictMath.sinh(x);
}
/**
* Returns the hyperbolic cosine of a {@code double} value.
* The hyperbolic cosine of <i>x is defined to be
* (<i>ex + e-x)/2
* where <i>e is {@linkplain Math#E Euler's number}.
*
* <p>Special cases:
* <ul>
*
* <li>If the argument is NaN, then the result is NaN.
*
* <li>If the argument is infinite, then the result is positive
* infinity.
*
* <li>If the argument is zero, then the result is {@code 1.0}.
*
* </ul>
*
* <p>The computed result must be within 2.5 ulps of the exact result.
*
* @param x The number whose hyperbolic cosine is to be returned.
* @return The hyperbolic cosine of {@code x}.
* @since 1.5
*/
public static double cosh(double x) {
return StrictMath.cosh(x);
}
/**
* Returns the hyperbolic tangent of a {@code double} value.
* The hyperbolic tangent of <i>x is defined to be
* (<i>ex - e-x)/(ex + e-x),
* in other words, {@linkplain Math#sinh
* sinh(<i>x)}/{@linkplain Math#cosh cosh(x)}. Note
* that the absolute value of the exact tanh is always less than
* 1.
*
* <p>Special cases:
* <ul>
*
* <li>If the argument is NaN, then the result is NaN.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* <li>If the argument is positive infinity, then the result is
* {@code +1.0}.
*
* <li>If the argument is negative infinity, then the result is
* {@code -1.0}.
*
* </ul>
*
* <p>The computed result must be within 2.5 ulps of the exact result.
* The result of {@code tanh} for any finite input must have
* an absolute value less than or equal to 1. Note that once the
* exact result of tanh is within 1/2 of an ulp of the limit value
* of ±1, correctly signed ±{@code 1.0} should
* be returned.
*
* @param x The number whose hyperbolic tangent is to be returned.
* @return The hyperbolic tangent of {@code x}.
* @since 1.5
*/
public static double tanh(double x) {
return StrictMath.tanh(x);
}
/**
* Returns sqrt(<i>x2 +y2)
* without intermediate overflow or underflow.
*
* <p>Special cases:
* <ul>
*
* <li> If either argument is infinite, then the result
* is positive infinity.
*
* <li> If either argument is NaN and neither argument is infinite,
* then the result is NaN.
*
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact
* result. If one parameter is held constant, the results must be
* semi-monotonic in the other parameter.
*
* @param x a value
* @param y a value
* @return sqrt(<i>x2 +y2)
* without intermediate overflow or underflow
* @since 1.5
*/
public static double hypot(double x, double y) {
return StrictMath.hypot(x, y);
}
/**
* Returns <i>ex -1. Note that for values of
* <i>x near 0, the exact sum of
* {@code expm1(x)} + 1 is much closer to the true
* result of <i>ex than {@code exp(x)}.
*
* <p>Special cases:
* <ul>
* <li>If the argument is NaN, the result is NaN.
*
* <li>If the argument is positive infinity, then the result is
* positive infinity.
*
* <li>If the argument is negative infinity, then the result is
* -1.0.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic. The result of
* {@code expm1} for any finite input must be greater than or
* equal to {@code -1.0}. Note that once the exact result of
* <i>e{@code x} - 1 is within 1/2
* ulp of the limit value -1, {@code -1.0} should be
* returned.
*
* @param x the exponent to raise <i>e to in the computation of
* <i>e{@code x} -1.
* @return the value <i>e{@code x} - 1.
* @since 1.5
*/
public static double expm1(double x) {
return StrictMath.expm1(x);
}
/**
* Returns the natural logarithm of the sum of the argument and 1.
* Note that for small values {@code x}, the result of
* {@code log1p(x)} is much closer to the true result of ln(1
* + {@code x}) than the floating-point evaluation of
* {@code log(1.0+x)}.
*
* <p>Special cases:
*
* <ul>
*
* <li>If the argument is NaN or less than -1, then the result is
* NaN.
*
* <li>If the argument is positive infinity, then the result is
* positive infinity.
*
* <li>If the argument is negative one, then the result is
* negative infinity.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param x a value
* @return the value ln({@code x} + 1), the natural
* log of {@code x} + 1
* @since 1.5
*/
public static double log1p(double x) {
return StrictMath.log1p(x);
}
/**
* Returns the first floating-point argument with the sign of the
* second floating-point argument. Note that unlike the {@link
* StrictMath#copySign(double, double) StrictMath.copySign}
* method, this method does not require NaN {@code sign}
* arguments to be treated as positive values; implementations are
* permitted to treat some NaN arguments as positive and other NaN
* arguments as negative to allow greater performance.
*
* @param magnitude the parameter providing the magnitude of the result
* @param sign the parameter providing the sign of the result
* @return a value with the magnitude of {@code magnitude}
* and the sign of {@code sign}.
* @since 1.6
*/
public static double copySign(double magnitude, double sign) {
return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
(DoubleConsts.SIGN_BIT_MASK)) |
(Double.doubleToRawLongBits(magnitude) &
(DoubleConsts.EXP_BIT_MASK |
DoubleConsts.SIGNIF_BIT_MASK)));
}
/**
* Returns the first floating-point argument with the sign of the
* second floating-point argument. Note that unlike the {@link
* StrictMath#copySign(float, float) StrictMath.copySign}
* method, this method does not require NaN {@code sign}
* arguments to be treated as positive values; implementations are
* permitted to treat some NaN arguments as positive and other NaN
* arguments as negative to allow greater performance.
*
* @param magnitude the parameter providing the magnitude of the result
* @param sign the parameter providing the sign of the result
* @return a value with the magnitude of {@code magnitude}
* and the sign of {@code sign}.
* @since 1.6
*/
public static float copySign(float magnitude, float sign) {
return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
(FloatConsts.SIGN_BIT_MASK)) |
(Float.floatToRawIntBits(magnitude) &
(FloatConsts.EXP_BIT_MASK |
FloatConsts.SIGNIF_BIT_MASK)));
}
/**
* Returns the unbiased exponent used in the representation of a
* {@code float}. Special cases:
*
* <ul>
* <li>If the argument is NaN or infinite, then the result is
* {@link Float#MAX_EXPONENT} + 1.
* <li>If the argument is zero or subnormal, then the result is
* {@link Float#MIN_EXPONENT} -1.
* </ul>
* @param f a {@code float} value
* @return the unbiased exponent of the argument
* @since 1.6
*/
public static int getExponent(float f) {
/*
* Bitwise convert f to integer, mask out exponent bits, shift
* to the right and then subtract out float's bias adjust to
* get true exponent value
*/
return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
(FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
}
/**
* Returns the unbiased exponent used in the representation of a
* {@code double}. Special cases:
*
* <ul>
* <li>If the argument is NaN or infinite, then the result is
* {@link Double#MAX_EXPONENT} + 1.
* <li>If the argument is zero or subnormal, then the result is
* {@link Double#MIN_EXPONENT} -1.
* </ul>
* @param d a {@code double} value
* @return the unbiased exponent of the argument
* @since 1.6
*/
public static int getExponent(double d) {
/*
* Bitwise convert d to long, mask out exponent bits, shift
* to the right and then subtract out double's bias adjust to
* get true exponent value.
*/
return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
(DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
}
/**
* Returns the floating-point number adjacent to the first
* argument in the direction of the second argument. If both
* arguments compare as equal the second argument is returned.
*
* <p>
* Special cases:
* <ul>
* <li> If either argument is a NaN, then NaN is returned.
*
* <li> If both arguments are signed zeros, {@code direction}
* is returned unchanged (as implied by the requirement of
* returning the second argument if the arguments compare as
* equal).
*
* <li> If {@code start} is
* ±{@link Double#MIN_VALUE} and {@code direction}
* has a value such that the result should have a smaller
* magnitude, then a zero with the same sign as {@code start}
* is returned.
*
* <li> If {@code start} is infinite and
* {@code direction} has a value such that the result should
* have a smaller magnitude, {@link Double#MAX_VALUE} with the
* same sign as {@code start} is returned.
*
* <li> If {@code start} is equal to ±
* {@link Double#MAX_VALUE} and {@code direction} has a
* value such that the result should have a larger magnitude, an
* infinity with same sign as {@code start} is returned.
* </ul>
*
* @param start starting floating-point value
* @param direction value indicating which of
* {@code start}'s neighbors or {@code start} should
* be returned
* @return The floating-point number adjacent to {@code start} in the
* direction of {@code direction}.
* @since 1.6
*/
public static double nextAfter(double start, double direction) {
/*
* The cases:
*
* nextAfter(+infinity, 0) == MAX_VALUE
* nextAfter(+infinity, +infinity) == +infinity
* nextAfter(-infinity, 0) == -MAX_VALUE
* nextAfter(-infinity, -infinity) == -infinity
*
* are naturally handled without any additional testing
*/
// First check for NaN values
if (Double.isNaN(start) || Double.isNaN(direction)) {
// return a NaN derived from the input NaN(s)
return start + direction;
} else if (start == direction) {
return direction;
} else { // start > direction or start < direction
// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
// then bitwise convert start to integer.
long transducer = Double.doubleToRawLongBits(start + 0.0d);
/*
* IEEE 754 floating-point numbers are lexicographically
* ordered if treated as signed- magnitude integers .
* Since Java's integers are two's complement,
* incrementing" the two's complement representation of a
* logically negative floating-point value *decrements*
* the signed-magnitude representation. Therefore, when
* the integer representation of a floating-point values
* is less than zero, the adjustment to the representation
* is in the opposite direction than would be expected at
* first .
*/
if (direction > start) { // Calculate next greater value
transducer = transducer + (transducer >= 0L ? 1L:-1L);
} else { // Calculate next lesser value
assert direction < start;
if (transducer > 0L)
--transducer;
else
if (transducer < 0L )
++transducer;
/*
* transducer==0, the result is -MIN_VALUE
*
* The transition from zero (implicitly
* positive) to the smallest negative
* signed magnitude value must be done
* explicitly.
*/
else
transducer = DoubleConsts.SIGN_BIT_MASK | 1L;
}
return Double.longBitsToDouble(transducer);
}
}
/**
* Returns the floating-point number adjacent to the first
* argument in the direction of the second argument. If both
* arguments compare as equal a value equivalent to the second argument
* is returned.
*
* <p>
* Special cases:
* <ul>
* <li> If either argument is a NaN, then NaN is returned.
*
* <li> If both arguments are signed zeros, a value equivalent
* to {@code direction} is returned.
*
* <li> If {@code start} is
* ±{@link Float#MIN_VALUE} and {@code direction}
* has a value such that the result should have a smaller
* magnitude, then a zero with the same sign as {@code start}
* is returned.
*
* <li> If {@code start} is infinite and
* {@code direction} has a value such that the result should
* have a smaller magnitude, {@link Float#MAX_VALUE} with the
* same sign as {@code start} is returned.
*
* <li> If {@code start} is equal to ±
* {@link Float#MAX_VALUE} and {@code direction} has a
* value such that the result should have a larger magnitude, an
* infinity with same sign as {@code start} is returned.
* </ul>
*
* @param start starting floating-point value
* @param direction value indicating which of
* {@code start}'s neighbors or {@code start} should
* be returned
* @return The floating-point number adjacent to {@code start} in the
* direction of {@code direction}.
* @since 1.6
*/
public static float nextAfter(float start, double direction) {
/*
* The cases:
*
* nextAfter(+infinity, 0) == MAX_VALUE
* nextAfter(+infinity, +infinity) == +infinity
* nextAfter(-infinity, 0) == -MAX_VALUE
* nextAfter(-infinity, -infinity) == -infinity
*
* are naturally handled without any additional testing
*/
// First check for NaN values
if (Float.isNaN(start) || Double.isNaN(direction)) {
// return a NaN derived from the input NaN(s)
return start + (float)direction;
} else if (start == direction) {
return (float)direction;
} else { // start > direction or start < direction
// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
// then bitwise convert start to integer.
int transducer = Float.floatToRawIntBits(start + 0.0f);
/*
* IEEE 754 floating-point numbers are lexicographically
* ordered if treated as signed- magnitude integers .
* Since Java's integers are two's complement,
* incrementing" the two's complement representation of a
* logically negative floating-point value *decrements*
* the signed-magnitude representation. Therefore, when
* the integer representation of a floating-point values
* is less than zero, the adjustment to the representation
* is in the opposite direction than would be expected at
* first.
*/
if (direction > start) {// Calculate next greater value
transducer = transducer + (transducer >= 0 ? 1:-1);
} else { // Calculate next lesser value
assert direction < start;
if (transducer > 0)
--transducer;
else
if (transducer < 0 )
++transducer;
/*
* transducer==0, the result is -MIN_VALUE
*
* The transition from zero (implicitly
* positive) to the smallest negative
* signed magnitude value must be done
* explicitly.
*/
else
transducer = FloatConsts.SIGN_BIT_MASK | 1;
}
return Float.intBitsToFloat(transducer);
}
}
/**
* Returns the floating-point value adjacent to {@code d} in
* the direction of positive infinity. This method is
* semantically equivalent to {@code nextAfter(d,
* Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
* implementation may run faster than its equivalent
* {@code nextAfter} call.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
*
* <li> If the argument is positive infinity, the result is
* positive infinity.
*
* <li> If the argument is zero, the result is
* {@link Double#MIN_VALUE}
*
* </ul>
*
* @param d starting floating-point value
* @return The adjacent floating-point value closer to positive
* infinity.
* @since 1.6
*/
public static double nextUp(double d) {
if( Double.isNaN(d) || d == Double.POSITIVE_INFINITY)
return d;
else {
d += 0.0d;
return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
((d >= 0.0d)?+1L:-1L));
}
}
/**
* Returns the floating-point value adjacent to {@code f} in
* the direction of positive infinity. This method is
* semantically equivalent to {@code nextAfter(f,
* Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
* implementation may run faster than its equivalent
* {@code nextAfter} call.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
*
* <li> If the argument is positive infinity, the result is
* positive infinity.
*
* <li> If the argument is zero, the result is
* {@link Float#MIN_VALUE}
*
* </ul>
*
* @param f starting floating-point value
* @return The adjacent floating-point value closer to positive
* infinity.
* @since 1.6
*/
public static float nextUp(float f) {
if( Float.isNaN(f) || f == FloatConsts.POSITIVE_INFINITY)
return f;
else {
f += 0.0f;
return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
((f >= 0.0f)?+1:-1));
}
}
/**
* Returns the floating-point value adjacent to {@code d} in
* the direction of negative infinity. This method is
* semantically equivalent to {@code nextAfter(d,
* Double.NEGATIVE_INFINITY)}; however, a
* {@code nextDown} implementation may run faster than its
* equivalent {@code nextAfter} call.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
*
* <li> If the argument is negative infinity, the result is
* negative infinity.
*
* <li> If the argument is zero, the result is
* {@code -Double.MIN_VALUE}
*
* </ul>
*
* @param d starting floating-point value
* @return The adjacent floating-point value closer to negative
* infinity.
* @since 1.8
*/
public static double nextDown(double d) {
if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
return d;
else {
if (d == 0.0)
return -Double.MIN_VALUE;
else
return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
((d > 0.0d)?-1L:+1L));
}
}
/**
* Returns the floating-point value adjacent to {@code f} in
* the direction of negative infinity. This method is
* semantically equivalent to {@code nextAfter(f,
* Float.NEGATIVE_INFINITY)}; however, a
* {@code nextDown} implementation may run faster than its
* equivalent {@code nextAfter} call.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
*
* <li> If the argument is negative infinity, the result is
* negative infinity.
*
* <li> If the argument is zero, the result is
* {@code -Float.MIN_VALUE}
*
* </ul>
*
* @param f starting floating-point value
* @return The adjacent floating-point value closer to negative
* infinity.
* @since 1.8
*/
public static float nextDown(float f) {
if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
return f;
else {
if (f == 0.0f)
return -Float.MIN_VALUE;
else
return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
((f > 0.0f)?-1:+1));
}
}
/**
* Returns {@code d} ×
* 2<sup>{@code scaleFactor} rounded as if performed
* by a single correctly rounded floating-point multiply to a
* member of the double value set. See the Java
* Language Specification for a discussion of floating-point
* value sets. If the exponent of the result is between {@link
* Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
* answer is calculated exactly. If the exponent of the result
* would be larger than {@code Double.MAX_EXPONENT}, an
* infinity is returned. Note that if the result is subnormal,
* precision may be lost; that is, when {@code scalb(x, n)}
* is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
* <i>x. When the result is non-NaN, the result has the same
* sign as {@code d}.
*
* <p>Special cases:
* <ul>
* <li> If the first argument is NaN, NaN is returned.
* <li> If the first argument is infinite, then an infinity of the
* same sign is returned.
* <li> If the first argument is zero, then a zero of the same
* sign is returned.
* </ul>
*
* @param d number to be scaled by a power of two.
* @param scaleFactor power of 2 used to scale {@code d}
* @return {@code d} × 2<sup>{@code scaleFactor}
* @since 1.6
*/
public static double scalb(double d, int scaleFactor) {
/*
* This method does not need to be declared strictfp to
* compute the same correct result on all platforms. When
* scaling up, it does not matter what order the
* multiply-store operations are done; the result will be
* finite or overflow regardless of the operation ordering.
* However, to get the correct result when scaling down, a
* particular ordering must be used.
*
* When scaling down, the multiply-store operations are
* sequenced so that it is not possible for two consecutive
* multiply-stores to return subnormal results. If one
* multiply-store result is subnormal, the next multiply will
* round it away to zero. This is done by first multiplying
* by 2 ^ (scaleFactor % n) and then multiplying several
* times by by 2^n as needed where n is the exponent of number
* that is a covenient power of two. In this way, at most one
* real rounding error occurs. If the double value set is
* being used exclusively, the rounding will occur on a
* multiply. If the double-extended-exponent value set is
* being used, the products will (perhaps) be exact but the
* stores to d are guaranteed to round to the double value
* set.
*
* It is _not_ a valid implementation to first multiply d by
* 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
* MIN_EXPONENT) since even in a strictfp program double
* rounding on underflow could occur; e.g. if the scaleFactor
* argument was (MIN_EXPONENT - n) and the exponent of d was a
* little less than -(MIN_EXPONENT - n), meaning the final
* result would be subnormal.
*
* Since exact reproducibility of this method can be achieved
* without any undue performance burden, there is no
* compelling reason to allow double rounding on underflow in
* scalb.
*/
// magnitude of a power of two so large that scaling a finite
// nonzero value by it would be guaranteed to over or
// underflow; due to rounding, scaling down takes takes an
// additional power of two which is reflected here
final int MAX_SCALE = DoubleConsts.MAX_EXPONENT + -DoubleConsts.MIN_EXPONENT +
DoubleConsts.SIGNIFICAND_WIDTH + 1;
int exp_adjust = 0;
int scale_increment = 0;
double exp_delta = Double.NaN;
// Make sure scaling factor is in a reasonable range
if(scaleFactor < 0) {
scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
scale_increment = -512;
exp_delta = twoToTheDoubleScaleDown;
}
else {
scaleFactor = Math.min(scaleFactor, MAX_SCALE);
scale_increment = 512;
exp_delta = twoToTheDoubleScaleUp;
}
// Calculate (scaleFactor % +/-512), 512 = 2^9, using
// technique from "Hacker's Delight" section 10-2.
int t = (scaleFactor >> 9-1) >>> 32 - 9;
exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
d *= powerOfTwoD(exp_adjust);
scaleFactor -= exp_adjust;
while(scaleFactor != 0) {
d *= exp_delta;
scaleFactor -= scale_increment;
}
return d;
}
/**
* Returns {@code f} ×
* 2<sup>{@code scaleFactor} rounded as if performed
* by a single correctly rounded floating-point multiply to a
* member of the float value set. See the Java
* Language Specification for a discussion of floating-point
* value sets. If the exponent of the result is between {@link
* Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
* answer is calculated exactly. If the exponent of the result
* would be larger than {@code Float.MAX_EXPONENT}, an
* infinity is returned. Note that if the result is subnormal,
* precision may be lost; that is, when {@code scalb(x, n)}
* is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
* <i>x. When the result is non-NaN, the result has the same
* sign as {@code f}.
*
* <p>Special cases:
* <ul>
* <li> If the first argument is NaN, NaN is returned.
* <li> If the first argument is infinite, then an infinity of the
* same sign is returned.
* <li> If the first argument is zero, then a zero of the same
* sign is returned.
* </ul>
*
* @param f number to be scaled by a power of two.
* @param scaleFactor power of 2 used to scale {@code f}
* @return {@code f} × 2<sup>{@code scaleFactor}
* @since 1.6
*/
public static float scalb(float f, int scaleFactor) {
// magnitude of a power of two so large that scaling a finite
// nonzero value by it would be guaranteed to over or
// underflow; due to rounding, scaling down takes takes an
// additional power of two which is reflected here
final int MAX_SCALE = FloatConsts.MAX_EXPONENT + -FloatConsts.MIN_EXPONENT +
FloatConsts.SIGNIFICAND_WIDTH + 1;
// Make sure scaling factor is in a reasonable range
scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
/*
* Since + MAX_SCALE for float fits well within the double
* exponent range and + float -> double conversion is exact
* the multiplication below will be exact. Therefore, the
* rounding that occurs when the double product is cast to
* float will be the correctly rounded float result. Since
* all operations other than the final multiply will be exact,
* it is not necessary to declare this method strictfp.
*/
return (float)((double)f*powerOfTwoD(scaleFactor));
}
// Constants used in scalb
static double twoToTheDoubleScaleUp = powerOfTwoD(512);
static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
/**
* Returns a floating-point power of two in the normal range.
*/
static double powerOfTwoD(int n) {
assert(n >= DoubleConsts.MIN_EXPONENT && n <= DoubleConsts.MAX_EXPONENT);
return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
(DoubleConsts.SIGNIFICAND_WIDTH-1))
& DoubleConsts.EXP_BIT_MASK);
}
/**
* Returns a floating-point power of two in the normal range.
*/
static float powerOfTwoF(int n) {
assert(n >= FloatConsts.MIN_EXPONENT && n <= FloatConsts.MAX_EXPONENT);
return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
(FloatConsts.SIGNIFICAND_WIDTH-1))
& FloatConsts.EXP_BIT_MASK);
}
}
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