/*
* Javolution - Java(TM) Solution for Real-Time and Embedded Systems
* Copyright (C) 2012 - Javolution (http://javolution.org/)
* All rights reserved.
*
* Permission to use, copy, modify, and distribute this software is
* freely granted, provided that this notice is preserved.
*/
package javolution.lang;
/**
* <p> An utility class providing a {@link Realtime} implementation of
* the math library.</p>
*
* @author <a href="mailto:jean-marie@dautelle.com">Jean-Marie Dautelle</a>
* @version 4.2, January 6, 2007
*/
@Realtime
public final class MathLib {
/**
* Default constructor.
*/
private MathLib() {}
/**
* Returns the number of bits in the minimal two's-complement representation
* of the specified <code>int</code>, excluding a sign bit.
* For positive <code>int</code>, this is equivalent to the number of bits
* in the ordinary binary representation. For negative <code>int</code>,
* it is equivalent to the number of bits of the positive value
* <code>-(i + 1)</code>.
*
* @param i the <code>int</code> value for which the bit length is returned.
* @return the bit length of <code>i</code>.
*/
public static int bitLength(int i) {
if (i < 0)
i = -++i;
return (i < 1 << 16) ? (i < 1 << 8) ? BIT_LENGTH[i]
: BIT_LENGTH[i >>> 8] + 8
: (i < 1 << 24) ? BIT_LENGTH[i >>> 16] + 16
: BIT_LENGTH[i >>> 24] + 24;
}
private static final byte[] BIT_LENGTH = { 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4,
4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6,
6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
8, 8, 8 };
/**
* Returns the number of bits in the minimal two's-complement representation
* of the specified <code>long</code>, excluding a sign bit.
* For positive <code>long</code>, this is equivalent to the number of bits
* in the ordinary binary representation. For negative <code>long</code>,
* it is equivalent to the number of bits of the positive value
* <code>-(l + 1)</code>.
*
* @param l the <code>long</code> value for which the bit length is returned.
* @return the bit length of <code>l</code>.
*/
public static int bitLength(long l) {
int i = (int) (l >> 32);
if (i > 0)
return (i < 1 << 16) ? (i < 1 << 8) ? BIT_LENGTH[i] + 32
: BIT_LENGTH[i >>> 8] + 40
: (i < 1 << 24) ? BIT_LENGTH[i >>> 16] + 48
: BIT_LENGTH[i >>> 24] + 56;
if (i < 0)
return bitLength(-++l);
i = (int) l;
return (i < 0) ? 32 : (i < 1 << 16) ? (i < 1 << 8) ? BIT_LENGTH[i]
: BIT_LENGTH[i >>> 8] + 8
: (i < 1 << 24) ? BIT_LENGTH[i >>> 16] + 16
: BIT_LENGTH[i >>> 24] + 24;
}
/**
* Returns the number of one-bits in the two's complement binary
* representation of the specified <code>long</code> value.
* This function is sometimes referred to as the <i>population count</i>.
*
* @param longValue the <code>long</code> value.
* @return the number of one-bits in the two's complement binary
* representation of the specified <code>longValue</code>.
*/
public static int bitCount(long longValue) {
longValue = longValue - ((longValue >>> 1) & 0x5555555555555555L);
longValue = (longValue & 0x3333333333333333L)
+ ((longValue >>> 2) & 0x3333333333333333L);
longValue = (longValue + (longValue >>> 4)) & 0x0f0f0f0f0f0f0f0fL;
longValue = longValue + (longValue >>> 8);
longValue = longValue + (longValue >>> 16);
longValue = longValue + (longValue >>> 32);
return (int) longValue & 0x7f;
}
/**
* Returns the number of zero bits preceding the highest-order
* ("leftmost") one-bit in the two's complement binary representation
* of the specified <code>long</code> value. Returns 64 if the specifed
* value is zero.
*
* @param longValue the <code>long</code> value.
* @return the number of leading zero bits.
*/
public static int numberOfLeadingZeros(long longValue) {
// From Hacker's Delight
if (longValue == 0)
return 64;
int n = 1;
int x = (int)(longValue >>> 32);
if (x == 0) { n += 32; x = (int)longValue; }
if (x >>> 16 == 0) { n += 16; x <<= 16; }
if (x >>> 24 == 0) { n += 8; x <<= 8; }
if (x >>> 28 == 0) { n += 4; x <<= 4; }
if (x >>> 30 == 0) { n += 2; x <<= 2; }
n -= x >>> 31;
return n;
}
/**
* Returns the number of zero bits following the lowest-order ("rightmost")
* one-bit in the two's complement binary representation of the specified
* <code>long</code> value. Returns 64 if the specifed value is zero.
*
* @param longValue the <code>long</code> value.
* @return the number of trailing zero bits.
*/
public static int numberOfTrailingZeros(long longValue) {
// From Hacker's Delight
int x, y;
if (longValue == 0) return 64;
int n = 63;
y = (int)longValue; if (y != 0) { n = n -32; x = y; } else x = (int)(longValue>>>32);
y = x <<16; if (y != 0) { n = n -16; x = y; }
y = x << 8; if (y != 0) { n = n - 8; x = y; }
y = x << 4; if (y != 0) { n = n - 4; x = y; }
y = x << 2; if (y != 0) { n = n - 2; x = y; }
return n - ((x << 1) >>> 31);
}
/**
* Returns the number of digits of the decimal representation of the
* specified <code>int</code> value, excluding the sign character if any.
*
* @param i the <code>int</code> value for which the digit length is returned.
* @return <code>String.valueOf(i).length()</code> for zero or positive values;
* <code>String.valueOf(i).length() - 1</code> for negative values.
*/
public static int digitLength(int i) {
if (i >= 0)
return (i >= 100000) ? (i >= 10000000) ? (i >= 1000000000) ? 10
: (i >= 100000000) ? 9 : 8 : (i >= 1000000) ? 7 : 6
: (i >= 100) ? (i >= 10000) ? 5 : (i >= 1000) ? 4 : 3
: (i >= 10) ? 2 : 1;
if (i == Integer.MIN_VALUE)
return 10; // "2147483648".length()
return digitLength(-i); // No overflow possible.
}
/**
* Returns the number of digits of the decimal representation of the
* the specified <code>long</code>, excluding the sign character if any.
*
* @param l the <code>long</code> value for which the digit length is returned.
* @return <code>String.valueOf(l).length()</code> for zero or positive values;
* <code>String.valueOf(l).length() - 1</code> for negative values.
*/
public static int digitLength(long l) {
if (l >= 0)
return (l <= Integer.MAX_VALUE) ? digitLength((int) l)
: // At least 10 digits or more.
(l >= 100000000000000L) ? (l >= 10000000000000000L) ? (l >= 1000000000000000000L) ? 19
: (l >= 100000000000000000L) ? 18 : 17
: (l >= 1000000000000000L) ? 16 : 15
: (l >= 100000000000L) ? (l >= 10000000000000L) ? 14
: (l >= 1000000000000L) ? 13 : 12
: (l >= 10000000000L) ? 11 : 10;
if (l == Long.MIN_VALUE)
return 19; // "9223372036854775808".length()
return digitLength(-l);
}
/**
* Returns the closest <code>double</code> representation of the
* specified <code>long</code> number multiplied by a power of two.
*
* @param m the <code>long</code> multiplier.
* @param n the power of two exponent.
* @return <code>m * 2<sup>n</sup></code>.
*/
public static double toDoublePow2(long m, int n) {
if (m == 0)
return 0.0;
if (m == Long.MIN_VALUE)
return toDoublePow2(Long.MIN_VALUE >> 1, n + 1);
if (m < 0)
return -toDoublePow2(-m, n);
int bitLength = MathLib.bitLength(m);
int shift = bitLength - 53;
long exp = 1023L + 52 + n + shift; // Use long to avoid overflow.
if (exp >= 0x7FF)
return Double.POSITIVE_INFINITY;
if (exp <= 0) { // Degenerated number (subnormal, assume 0 for bit 52)
if (exp <= -54)
return 0.0;
return toDoublePow2(m, n + 54) / 18014398509481984L; // 2^54 Exact.
}
// Normal number.
long bits = (shift > 0) ? (m >> shift) + ((m >> (shift - 1)) & 1) : // Rounding.
m << -shift;
if (((bits >> 52) != 1) && (++exp >= 0x7FF))
return Double.POSITIVE_INFINITY;
bits &= 0x000fffffffffffffL; // Clears MSB (bit 52)
bits |= exp << 52;
return Double.longBitsToDouble(bits);
}
/**/
/**
* Returns the closest <code>double</code> representation of the
* specified <code>long</code> number multiplied by a power of ten.
*
* @param m the <code>long</code> multiplier.
* @param n the power of ten exponent.
* @return <code>multiplier * 10<sup>n</sup></code>.
**/
public static double toDoublePow10(long m, int n) {
if (m == 0)
return 0.0;
if (m == Long.MIN_VALUE)
return toDoublePow10(Long.MIN_VALUE / 10, n + 1);
if (m < 0)
return -toDoublePow10(-m, n);
if (n >= 0) { // Positive power.
if (n > 308)
return Double.POSITIVE_INFINITY;
// Works with 4 x 32 bits registers (x3:x2:x1:x0)
long x0 = 0; // 32 bits.
long x1 = 0; // 32 bits.
long x2 = m & MASK_32; // 32 bits.
long x3 = m >>> 32; // 32 bits.
int pow2 = 0;
while (n != 0) {
int i = (n >= POW5_INT.length) ? POW5_INT.length - 1 : n;
int coef = POW5_INT[i]; // 31 bits max.
if (((int) x0) != 0)
x0 *= coef; // 63 bits max.
if (((int) x1) != 0)
x1 *= coef; // 63 bits max.
x2 *= coef; // 63 bits max.
x3 *= coef; // 63 bits max.
x1 += x0 >>> 32;
x0 &= MASK_32;
x2 += x1 >>> 32;
x1 &= MASK_32;
x3 += x2 >>> 32;
x2 &= MASK_32;
// Adjusts powers.
pow2 += i;
n -= i;
// Normalizes (x3 should be 32 bits max).
long carry = x3 >>> 32;
if (carry != 0) { // Shift.
x0 = x1;
x1 = x2;
x2 = x3 & MASK_32;
x3 = carry;
pow2 += 32;
}
}
// Merges registers to a 63 bits mantissa.
int shift = 31 - MathLib.bitLength(x3); // -1..30
pow2 -= shift;
long mantissa = (shift < 0) ? (x3 << 31) | (x2 >>> 1) : // x3 is 32 bits.
(((x3 << 32) | x2) << shift) | (x1 >>> (32 - shift));
return toDoublePow2(mantissa, pow2);
} else { // n < 0
if (n < -324 - 20)
return 0.0;
// Works with x1:x0 126 bits register.
long x1 = m; // 63 bits.
long x0 = 0; // 63 bits.
int pow2 = 0;
while (true) {
// Normalizes x1:x0
int shift = 63 - MathLib.bitLength(x1);
x1 <<= shift;
x1 |= x0 >>> (63 - shift);
x0 = (x0 << shift) & MASK_63;
pow2 -= shift;
// Checks if division has to be performed.
if (n == 0)
break; // Done.
// Retrieves power of 5 divisor.
int i = (-n >= POW5_INT.length) ? POW5_INT.length - 1 : -n;
int divisor = POW5_INT[i];
// Performs the division (126 bits by 31 bits).
long wh = (x1 >>> 32);
long qh = wh / divisor;
long r = wh - qh * divisor;
long wl = (r << 32) | (x1 & MASK_32);
long ql = wl / divisor;
r = wl - ql * divisor;
x1 = (qh << 32) | ql;
wh = (r << 31) | (x0 >>> 32);
qh = wh / divisor;
r = wh - qh * divisor;
wl = (r << 32) | (x0 & MASK_32);
ql = wl / divisor;
x0 = (qh << 32) | ql;
// Adjusts powers.
n += i;
pow2 -= i;
}
return toDoublePow2(x1, pow2);
}
}
private static final long MASK_63 = 0x7FFFFFFFFFFFFFFFL;
private static final long MASK_32 = 0xFFFFFFFFL;
private static final int[] POW5_INT = { 1, 5, 25, 125, 625, 3125, 15625,
78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125 };
/**/
/**
* Returns the closest <code>long</code> representation of the
* specified <code>double</code> number multiplied by a power of two.
*
* @param d the <code>double</code> multiplier.
* @param n the power of two exponent.
* @return <code>d * 2<sup>n</sup></code>
* @throws ArithmeticException if the conversion cannot be performed
* (NaN, Infinity or overflow).
**/
public static long toLongPow2(double d, int n) {
long bits = Double.doubleToLongBits(d);
boolean isNegative = (bits >> 63) != 0;
int exp = ((int) (bits >> 52)) & 0x7FF;
long m = bits & 0x000fffffffffffffL;
if (exp == 0x7FF)
throw new ArithmeticException(
"Cannot convert to long (Infinity or NaN)");
if (exp == 0) {
if (m == 0)
return 0L;
return toLongPow2(d * 18014398509481984L, n - 54); // 2^54 Exact.
}
m |= 0x0010000000000000L; // Sets MSB (bit 52)
long shift = exp - 1023L - 52 + n; // Use long to avoid overflow.
if (shift <= -64)
return 0L;
if (shift >= 11)
throw new ArithmeticException("Cannot convert to long (overflow)");
m = (shift >= 0) ? m << shift : (m >> -shift)
+ ((m >> -(shift + 1)) & 1); // Rounding.
return isNegative ? -m : m;
}
/**/
/**
* Returns the closest <code>long</code> representation of the
* specified <code>double</code> number multiplied by a power of ten.
*
* @param d the <code>double</code> multiplier.
* @param n the power of two exponent.
* @return <code>d * 10<sup>n</sup></code>.
*/
public static long toLongPow10(double d, int n) {
long bits = Double.doubleToLongBits(d);
boolean isNegative = (bits >> 63) != 0;
int exp = ((int) (bits >> 52)) & 0x7FF;
long m = bits & 0x000fffffffffffffL;
if (exp == 0x7FF)
throw new ArithmeticException(
"Cannot convert to long (Infinity or NaN)");
if (exp == 0) {
if (m == 0)
return 0L;
return toLongPow10(d * 1E16, n - 16);
}
m |= 0x0010000000000000L; // Sets MSB (bit 52)
int pow2 = exp - 1023 - 52;
// Retrieves 63 bits m with n == 0.
if (n >= 0) {
// Works with 4 x 32 bits registers (x3:x2:x1:x0)
long x0 = 0; // 32 bits.
long x1 = 0; // 32 bits.
long x2 = m & MASK_32; // 32 bits.
long x3 = m >>> 32; // 32 bits.
while (n != 0) {
int i = (n >= POW5_INT.length) ? POW5_INT.length - 1 : n;
int coef = POW5_INT[i]; // 31 bits max.
if (((int) x0) != 0)
x0 *= coef; // 63 bits max.
if (((int) x1) != 0)
x1 *= coef; // 63 bits max.
x2 *= coef; // 63 bits max.
x3 *= coef; // 63 bits max.
x1 += x0 >>> 32;
x0 &= MASK_32;
x2 += x1 >>> 32;
x1 &= MASK_32;
x3 += x2 >>> 32;
x2 &= MASK_32;
// Adjusts powers.
pow2 += i;
n -= i;
// Normalizes (x3 should be 32 bits max).
long carry = x3 >>> 32;
if (carry != 0) { // Shift.
x0 = x1;
x1 = x2;
x2 = x3 & MASK_32;
x3 = carry;
pow2 += 32;
}
}
// Merges registers to a 63 bits mantissa.
int shift = 31 - MathLib.bitLength(x3); // -1..30
pow2 -= shift;
m = (shift < 0) ? (x3 << 31) | (x2 >>> 1) : // x3 is 32 bits.
(((x3 << 32) | x2) << shift) | (x1 >>> (32 - shift));
} else { // n < 0
// Works with x1:x0 126 bits register.
long x1 = m; // 63 bits.
long x0 = 0; // 63 bits.
while (true) {
// Normalizes x1:x0
int shift = 63 - MathLib.bitLength(x1);
x1 <<= shift;
x1 |= x0 >>> (63 - shift);
x0 = (x0 << shift) & MASK_63;
pow2 -= shift;
// Checks if division has to be performed.
if (n == 0)
break; // Done.
// Retrieves power of 5 divisor.
int i = (-n >= POW5_INT.length) ? POW5_INT.length - 1 : -n;
int divisor = POW5_INT[i];
// Performs the division (126 bits by 31 bits).
long wh = (x1 >>> 32);
long qh = wh / divisor;
long r = wh - qh * divisor;
long wl = (r << 32) | (x1 & MASK_32);
long ql = wl / divisor;
r = wl - ql * divisor;
x1 = (qh << 32) | ql;
wh = (r << 31) | (x0 >>> 32);
qh = wh / divisor;
r = wh - qh * divisor;
wl = (r << 32) | (x0 & MASK_32);
ql = wl / divisor;
x0 = (qh << 32) | ql;
// Adjusts powers.
n += i;
pow2 -= i;
}
m = x1;
}
if (pow2 > 0)
throw new ArithmeticException("Overflow");
if (pow2 < -63)
return 0;
m = (m >> -pow2) + ((m >> -(pow2 + 1)) & 1); // Rounding.
return isNegative ? -m : m;
}
/**/
/**
* Returns the largest power of 2 that is less than or equal to the
* the specified positive value.
*
* @param d the <code>double</code> number.
* @return <code>floor(Log2(abs(d)))</code>
* @throws ArithmeticException if <code>d <= 0<code> or <code>d</code>
* is <code>NaN</code> or <code>Infinity</code>.
**/
public static int floorLog2(double d) {
if (d <= 0)
throw new ArithmeticException("Negative number or zero");
long bits = Double.doubleToLongBits(d);
int exp = ((int) (bits >> 52)) & 0x7FF;
if (exp == 0x7FF)
throw new ArithmeticException("Infinity or NaN");
if (exp == 0)
return floorLog2(d * 18014398509481984L) - 54; // 2^54 Exact.
return exp - 1023;
}
/**/
/**
* Returns the largest power of 10 that is less than or equal to the
* the specified positive value.
*
* @param d the <code>double</code> number.
* @return <code>floor(Log10(abs(d)))</code>
* @throws ArithmeticException if <code>d <= 0<code> or <code>d</code>
* is <code>NaN</code> or <code>Infinity</code>.
**/
public static int floorLog10(double d) {
int guess = (int) (LOG2_DIV_LOG10 * MathLib.floorLog2(d));
double pow10 = MathLib.toDoublePow10(1, guess);
if ((pow10 <= d) && (pow10 * 10 > d))
return guess;
if (pow10 > d)
return guess - 1;
return guess + 1;
}
private static final double LOG2_DIV_LOG10 = 0.3010299956639811952137388947;
/**
* The natural logarithm.
**/
public static final double E = 2.71828182845904523536028747135266;
/**
* The ratio of the circumference of a circle to its diameter.
**/
public static final double PI = 3.1415926535897932384626433832795;
/**
* Half the ratio of the circumference of a circle to its diameter.
**/
public static final double HALF_PI = 1.5707963267948966192313216916398;
/**
* Twice the ratio of the circumference of a circle to its diameter.
**/
public static final double TWO_PI = 6.283185307179586476925286766559;
/**
* Four time the ratio of the circumference of a circle to its diameter.
**/
public static final double FOUR_PI = 12.566370614359172953850573533118;
/**
* Holds {@link #PI} * {@link #PI}.
**/
public static final double PI_SQUARE = 9.8696044010893586188344909998762;
/**
* The natural logarithm of two.
**/
public static final double LOG2 = 0.69314718055994530941723212145818;
/**
* The natural logarithm of ten.
**/
public static final double LOG10 = 2.3025850929940456840179914546844;
/**
* The square root of two.
**/
public static final double SQRT2 = 1.4142135623730950488016887242097;
/**
* Not-A-Number.
**/
public static final double NaN = 0.0 / 0.0;
/**
* Infinity.
**/
public static final double Infinity = 1.0 / 0.0;
/**/
/**
* Converts an angle in degrees to radians.
*
* @param degrees the angle in degrees.
* @return the specified angle in radians.
**/
public static double toRadians(double degrees) {
return degrees * (PI / 180.0);
}
/**/
/**
* Converts an angle in radians to degrees.
*
* @param radians the angle in radians.
* @return the specified angle in degrees.
**/
public static double toDegrees(double radians) {
return radians * (180.0 / PI);
}
/**/
/**
* Returns the positive square root of the specified value.
*
* @param x the value.
* @return <code>java.lang.Math.sqrt(x)</code>
**/
public static double sqrt(double x) {
return Math.sqrt(x); // CLDC 1.1
}
/**/
/**
* Returns the remainder of the division of the specified two arguments.
*
* @param x the dividend.
* @param y the divisor.
* @return <code>x - round(x / y) * y</code>
**/
public static double rem(double x, double y) {
double tmp = x / y;
if (MathLib.abs(tmp) <= Long.MAX_VALUE)
return x - MathLib.round(tmp) * y;
else
return NaN;
}
/**/
/**
* Returns the smallest (closest to negative infinity)
* <code>double</code> value that is not less than the argument and is
* equal to a mathematical integer.
*
* @param x the value.
* @return <code>java.lang.Math.ceil(x)</code>
**/
public static double ceil(double x) {
return Math.ceil(x); // CLDC 1.1
}
/**/
/**
* Returns the largest (closest to positive infinity)
* <code>double</code> value that is not greater than the argument and
* is equal to a mathematical integer.
*
* @param x the value.
* @return <code>java.lang.Math.ceil(x)</code>
**/
public static double floor(double x) {
return Math.floor(x); // CLDC 1.1
}
/**/
/**
* Returns the trigonometric sine of the specified angle in radians.
*
* @param radians the angle in radians.
* @return <code>java.lang.Math.sin(radians)</code>
**/
public static double sin(double radians) {
return Math.sin(radians); // CLDC 1.1
}
/**/
/**
* Returns the trigonometric cosine of the specified angle in radians.
*
* @param radians the angle in radians.
* @return <code>java.lang.Math.cos(radians)</code>
**/
public static double cos(double radians) {
return Math.cos(radians); // CLDC 1.1
}
/**/
/**
* Returns the trigonometric tangent of the specified angle in radians.
*
* @param radians the angle in radians.
* @return <code>java.lang.Math.tan(radians)</code>
**/
public static double tan(double radians) {
return Math.tan(radians); // CLDC 1.1
}
/**/
/**
* Returns the arc sine of the specified value,
* in the range of -<i>pi</i>/2 through <i>pi</i>/2.
*
* @param x the value whose arc sine is to be returned.
* @return the arc sine in radians for the specified value.
**/
public static double asin(double x) {
if (x < -1.0 || x > 1.0)
return MathLib.NaN;
if (x == -1.0)
return -HALF_PI;
if (x == 1.0)
return HALF_PI;
return MathLib.atan(x / MathLib.sqrt(1.0 - x * x));
}
/**/
/**
* Returns the arc cosine of the specified value,
* in the range of 0.0 through <i>pi</i>.
*
* @param x the value whose arc cosine is to be returned.
* @return the arc cosine in radians for the specified value.
**/
public static double acos(double x) {
return HALF_PI - MathLib.asin(x);
}
/**/
/**
* Returns the arc tangent of the specified value,
* in the range of -<i>pi</i>/2 through <i>pi</i>/2.
*
* @param x the value whose arc tangent is to be returned.
* @return the arc tangent in radians for the specified value.
* @see <a href="http://mathworld.wolfram.com/InverseTangent.html">
* Inverse Tangent -- from MathWorld</a>
**/
public static double atan(double x) {
return MathLib._atan(x);
}
/**/
/**
* Returns the angle theta such that
* <code>(x == cos(theta)) && (y == sin(theta))</code>.
*
* @param y the y value.
* @param x the x value.
* @return the angle theta in radians.
* @see <a href="http://en.wikipedia.org/wiki/Atan2">Wikipedia: Atan2</a>
**/
public static double atan2(double y, double x) {
// From Wikipedia.
if (x > 0) return MathLib.atan(y / x);
if ((y >= 0) && (x < 0)) return MathLib.atan(y / x) + PI;
if ((y < 0) && (x < 0)) return MathLib.atan(y / x) - PI;
if ((y > 0) && (x == 0)) return PI / 2;
if ((y < 0) && (x == 0)) return -PI / 2;
return Double.NaN; // ((y == 0) && (x == 0))
}
/**/
/**
* Returns the hyperbolic sine of x.
*
* @param x the value for which the hyperbolic sine is calculated.
* @return <code>(exp(x) - exp(-x)) / 2</code>
**/
public static double sinh(double x) {
return (MathLib.exp(x) - MathLib.exp(-x)) * 0.5;
}
/**/
/**
* Returns the hyperbolic cosine of x.
*
* @param x the value for which the hyperbolic cosine is calculated.
* @return <code>(exp(x) + exp(-x)) / 2</code>
**/
public static double cosh(double x) {
return (MathLib.exp(x) + MathLib.exp(-x)) * 0.5;
}
/**/
/**
* Returns the hyperbolic tangent of x.
*
* @param x the value for which the hyperbolic tangent is calculated.
* @return <code>(exp(2 * x) - 1) / (exp(2 * x) + 1)</code>
**/
public static double tanh(double x) {
return (MathLib.exp(2 * x) - 1) / (MathLib.exp(2 * x) + 1);
}
/**/
/**
* Returns <i>{@link #E e}</i> raised to the specified power.
*
* @param x the exponent.
* @return <code><i>e</i><sup>x</sup></code>
* @see <a href="http://mathworld.wolfram.com/ExponentialFunction.html">
* Exponential Function -- from MathWorld</a>
**/
public static double exp(double x) {
return MathLib._ieee754_exp(x);
}
/**/
/**
* Returns the natural logarithm (base <i>{@link #E e}</i>) of the specified
* value.
*
* @param x the value greater than <code>0.0</code>.
* @return the value y such as <code><i>e</i><sup>y</sup> == x</code>
**/
public static double log(double x) {
return MathLib._ieee754_log(x);
}
/**/
/**
* Returns the decimal logarithm of the specified value.
*
* @param x the value greater than <code>0.0</code>.
* @return the value y such as <code>10<sup>y</sup> == x</code>
**/
public static double log10(double x) {
return log(x) * INV_LOG10;
}
private static double INV_LOG10 = 0.43429448190325182765112891891661;
/**
* Returns the value of the first argument raised to the power of the
* second argument.
*
* @param x the base.
* @param y the exponent.
* @return <code>x<sup>y</sup></code>
**/
public static double pow(double x, double y) {
// Use close approximation (+/- LSB)
if ((x < 0) && (y == (int) y))
return (((int) y) & 1) == 0 ? pow(-x, y) : -pow(-x, y);
return MathLib.exp(y * MathLib.log(x));
}
/**
* Returns the closest <code>int</code> to the specified argument.
*
* @param f the <code>float</code> value to be rounded to a <code>int</code>
* @return the nearest <code>int</code> value.
**/
public static int round(float f) {
return (int) floor(f + 0.5f);
}
/**/
/**
* Returns the closest <code>long</code> to the specified argument.
*
* @param d the <code>double</code> value to be rounded to a
* <code>long</code>
* @return the nearest <code>long</code> value.
**/
public static long round(double d) {
return (long) floor(d + 0.5d);
}
/**
* Returns the absolute value of the specified <code>int</code> argument.
*
* @param i the <code>int</code> value.
* @return <code>i</code> or <code>-i</code>
*/
public static int abs(int i) {
return (i < 0) ? -i : i;
}
/**
* Returns the absolute value of the specified <code>long</code> argument.
*
* @param l the <code>long</code> value.
* @return <code>l</code> or <code>-l</code>
*/
public static long abs(long l) {
return (l < 0) ? -l : l;
}
/**
* Returns the absolute value of the specified <code>float</code> argument.
*
* @param f the <code>float</code> value.
* @return <code>f</code> or <code>-f</code>
**/
public static float abs(float f) {
return (f < 0) ? -f : f;
}
/**
* Returns the absolute value of the specified <code>double</code> argument.
*
* @param d the <code>double</code> value.
* @return <code>d</code> or <code>-d</code>
**/
public static double abs(double d) {
return (d < 0) ? -d : d;
}
/**/
/**
* Returns the greater of two <code>int</code> values.
*
* @param x the first value.
* @param y the second value.
* @return the larger of <code>x</code> and <code>y</code>.
*/
public static int max(int x, int y) {
return (x >= y) ? x : y;
}
/**
* Returns the greater of two <code>long</code> values.
*
* @param x the first value.
* @param y the second value.
* @return the larger of <code>x</code> and <code>y</code>.
*/
public static long max(long x, long y) {
return (x >= y) ? x : y;
}
/**
* Returns the greater of two <code>float</code> values.
*
* @param x the first value.
* @param y the second value.
* @return the larger of <code>x</code> and <code>y</code>.
**/
public static float max(float x, float y) {
return (x >= y) ? x : y;
}
/**
* Returns the greater of two <code>double</code> values.
*
* @param x the first value.
* @param y the second value.
* @return the larger of <code>x</code> and <code>y</code>.
**/
public static double max(double x, double y) {
return (x >= y) ? x : y;
}
/**/
/**
* Returns the smaller of two <code>int</code> values.
*
* @param x the first value.
* @param y the second value.
* @return the smaller of <code>x</code> and <code>y</code>.
*/
public static int min(int x, int y) {
return (x < y) ? x : y;
}
/**
* Returns the smaller of two <code>long</code> values.
*
* @param x the first value.
* @param y the second value.
* @return the smaller of <code>x</code> and <code>y</code>.
*/
public static long min(long x, long y) {
return (x < y) ? x : y;
}
/**
* Returns the smaller of two <code>float</code> values.
*
* @param x the first value.
* @param y the second value.
* @return the smaller of <code>x</code> and <code>y</code>.
**/
public static float min(float x, float y) {
return (x < y) ? x : y;
}
/**/
/**
* Returns the smaller of two <code>double</code> values.
*
* @param x the first value.
* @param y the second value.
* @return the smaller of <code>x</code> and <code>y</code>.
**/
public static double min(double x, double y) {
return (x < y) ? x : y;
}
////////////////////////////////////////////////////////////////////////////
/* @(#)s_atan.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
*/
/* atan(x)
* Method
* 1. Reduce x to positive by atan(x) = -atan(-x).
* 2. According to the integer k=4t+0.25 chopped, t=x, the argument
* is further reduced to one of the following intervals and the
* arctangent of t is evaluated by the corresponding formula:
*
* [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
* [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
* [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
* [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
* [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
static final double atanhi[] = { 4.63647609000806093515e-01, // atan(0.5)hi 0x3FDDAC67, 0x0561BB4F
7.85398163397448278999e-01, // atan(1.0)hi 0x3FE921FB, 0x54442D18
9.82793723247329054082e-01, // atan(1.5)hi 0x3FEF730B, 0xD281F69B
1.57079632679489655800e+00, // atan(inf)hi 0x3FF921FB, 0x54442D18
};
static final double atanlo[] = { 2.26987774529616870924e-17, // atan(0.5)lo 0x3C7A2B7F, 0x222F65E2
3.06161699786838301793e-17, // atan(1.0)lo 0x3C81A626, 0x33145C07
1.39033110312309984516e-17, // atan(1.5)lo 0x3C700788, 0x7AF0CBBD
6.12323399573676603587e-17, // atan(inf)lo 0x3C91A626, 0x33145C07
};
static final double aT[] = { 3.33333333333329318027e-01, // 0x3FD55555, 0x5555550D
-1.99999999998764832476e-01, // 0xBFC99999, 0x9998EBC4
1.42857142725034663711e-01, // 0x3FC24924, 0x920083FF
-1.11111104054623557880e-01, // 0xBFBC71C6, 0xFE231671
9.09088713343650656196e-02, // 0x3FB745CD, 0xC54C206E
-7.69187620504482999495e-02, // 0xBFB3B0F2, 0xAF749A6D
6.66107313738753120669e-02, // 0x3FB10D66, 0xA0D03D51
-5.83357013379057348645e-02, // 0xBFADDE2D, 0x52DEFD9A
4.97687799461593236017e-02, // 0x3FA97B4B, 0x24760DEB
-3.65315727442169155270e-02, // 0xBFA2B444, 0x2C6A6C2F
1.62858201153657823623e-02, // 0x3F90AD3A, 0xE322DA11
};
static final double one = 1.0, huge = 1.0e300;
static double _atan(double x) {
double w, s1, s2, z;
int ix, hx, id;
long xBits = Double.doubleToLongBits(x);
int __HIx = (int) (xBits >> 32);
int __LOx = (int) xBits;
hx = __HIx;
ix = hx & 0x7fffffff;
if (ix >= 0x44100000) { // if |x| >= 2^66
if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (__LOx != 0)))
return x + x; // NaN
if (hx > 0)
return atanhi[3] + atanlo[3];
else
return -atanhi[3] - atanlo[3];
}
if (ix < 0x3fdc0000) { // |x| < 0.4375
if (ix < 0x3e200000) // |x| < 2^-29
if (huge + x > one)
return x;
id = -1;
} else {
x = MathLib.abs(x);
if (ix < 0x3ff30000) // |x| < 1.1875
if (ix < 0x3fe60000) { // 7/16 <=|x|<11/16
id = 0;
x = (2.0 * x - one) / (2.0 + x);
} else { // 11/16<=|x|< 19/16
id = 1;
x = (x - one) / (x + one);
}
else if (ix < 0x40038000) { // |x| < 2.4375
id = 2;
x = (x - 1.5) / (one + 1.5 * x);
} else { // 2.4375 <= |x| < 2^66
id = 3;
x = -1.0 / x;
}
}
// end of argument reduction
z = x * x;
w = z * z;
// break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly
s1 = z
* (aT[0] + w
* (aT[2] + w
* (aT[4] + w
* (aT[6] + w * (aT[8] + w * aT[10])))));
s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
if (id < 0)
return x - x * (s1 + s2);
else {
z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
return (hx < 0) ? -z : z;
}
}
/**/
////////////////////////////////////////////////////////////////////////////
/* @(#)e_log.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __ieee754_log(x)
* Return the logrithm of x
*
* Method :
* 1. Argument Reduction: find k and f such that
* x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* 2. Approximation of log(1+f).
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
* = 2s + s*R
* We use a special Reme algorithm on [0,0.1716] to generate
* a polynomial of degree 14 to approximate R The maximum error
* of this polynomial approximation is bounded by 2**-58.45. In
* other words,
* 2 4 6 8 10 12 14
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
* (the values of Lg1 to Lg7 are listed in the program)
* and
* | 2 14 | -58.45
* | Lg1*s +...+Lg7*s - R(z) | <= 2
* | |
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
* In order to guarantee error in log below 1ulp, we compute log
* by
* log(1+f) = f - s*(f - R) (if f is not too large)
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
*
* 3. Finally, log(x) = k*ln2 + log(1+f).
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
* Here ln2 is split into two floating point number:
* ln2_hi + ln2_lo,
* where n*ln2_hi is always exact for |n| < 2000.
*
* Special cases:
* log(x) is NaN with signal if x < 0 (including -INF) ;
* log(+INF) is +INF; log(0) is -INF with signal;
* log(NaN) is that NaN with no signal.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
static final double ln2_hi = 6.93147180369123816490e-01, // 3fe62e42 fee00000
ln2_lo = 1.90821492927058770002e-10, // 3dea39ef 35793c76
two54 = 1.80143985094819840000e+16, // 43500000 00000000
Lg1 = 6.666666666666735130e-01, // 3FE55555 55555593
Lg2 = 3.999999999940941908e-01, // 3FD99999 9997FA04
Lg3 = 2.857142874366239149e-01, // 3FD24924 94229359
Lg4 = 2.222219843214978396e-01, // 3FCC71C5 1D8E78AF
Lg5 = 1.818357216161805012e-01, // 3FC74664 96CB03DE
Lg6 = 1.531383769920937332e-01, // 3FC39A09 D078C69F
Lg7 = 1.479819860511658591e-01; // 3FC2F112 DF3E5244
static final double zero = 0.0;
static double _ieee754_log(double x) {
double hfsq, f, s, z, R, w, t1, t2, dk;
int k, hx, i, j;
int lx; // unsigned
long xBits = Double.doubleToLongBits(x);
hx = (int) (xBits >> 32);
lx = (int) xBits;
k = 0;
if (hx < 0x00100000) { // x < 2**-1022
if (((hx & 0x7fffffff) | lx) == 0)
return -two54 / zero; // log(+-0)=-inf
if (hx < 0)
return (x - x) / zero; // log(-#) = NaN
k -= 54;
x *= two54; // subnormal number, scale up x
xBits = Double.doubleToLongBits(x);
hx = (int) (xBits >> 32); // high word of x
}
if (hx >= 0x7ff00000)
return x + x;
k += (hx >> 20) - 1023;
hx &= 0x000fffff;
i = (hx + 0x95f64) & 0x100000;
xBits = Double.doubleToLongBits(x);
int HIx = hx | (i ^ 0x3ff00000); // normalize x or x/2
xBits = ((HIx & 0xFFFFFFFFL) << 32) | (xBits & 0xFFFFFFFFL);
x = Double.longBitsToDouble(xBits);
k += (i >> 20);
f = x - 1.0;
if ((0x000fffff & (2 + hx)) < 3) { // |f| < 2**-20
if (f == zero)
if (k == 0)
return zero;
else {
dk = (double) k;
return dk * ln2_hi + dk * ln2_lo;
}
R = f * f * (0.5 - 0.33333333333333333 * f);
if (k == 0)
return f - R;
else {
dk = (double) k;
return dk * ln2_hi - ((R - dk * ln2_lo) - f);
}
}
s = f / (2.0 + f);
dk = (double) k;
z = s * s;
i = hx - 0x6147a;
w = z * z;
j = 0x6b851 - hx;
t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
i |= j;
R = t2 + t1;
if (i > 0) {
hfsq = 0.5 * f * f;
if (k == 0)
return f - (hfsq - s * (hfsq + R));
else
return dk * ln2_hi
- ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
} else if (k == 0)
return f - s * (f - R);
else
return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
}
/**/
////////////////////////////////////////////////////////////////////////////
/* @(#)e_exp.c 1.6 04/04/22 */
/*
* ====================================================
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __ieee754_exp(x)
* Returns the exponential of x.
*
* Method
* 1. Argument reduction:
* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
* Given x, find r and integer k such that
*
* x = k*ln2 + r, |r| <= 0.5*ln2.
*
* Here r will be represented as r = hi-lo for better
* accuracy.
*
* 2. Approximation of exp(r) by a special rational function on
* the interval [0,0.34658]:
* Write
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
* We use a special Remes algorithm on [0,0.34658] to generate
* a polynomial of degree 5 to approximate R. The maximum error
* of this polynomial approximation is bounded by 2**-59. In
* other words,
* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
* (where z=r*r, and the values of P1 to P5 are listed below)
* and
* | 5 | -59
* | 2.0+P1*z+...+P5*z - R(z) | <= 2
* | |
* The computation of exp(r) thus becomes
* 2*r
* exp(r) = 1 + -------
* R - r
* r*R1(r)
* = 1 + r + ----------- (for better accuracy)
* 2 - R1(r)
* where
* 2 4 10
* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
*
* 3. Scale back to obtain exp(x):
* From step 1, we have
* exp(x) = 2^k * exp(r)
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF) is 0, and
* for finite argument, only exp(0)=1 is exact.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* Misc. info.
* For IEEE double
* if x > 7.09782712893383973096e+02 then exp(x) overflow
* if x < -7.45133219101941108420e+02 then exp(x) underflow
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
static final double halF[] = { 0.5, -0.5, },
twom1000 = 9.33263618503218878990e-302, // 2**-1000=0x01700000,0
o_threshold = 7.09782712893383973096e+02, // 0x40862E42, 0xFEFA39EF
u_threshold = -7.45133219101941108420e+02, // 0xc0874910, 0xD52D3051
ln2HI[] = { 6.93147180369123816490e-01, // 0x3fe62e42, 0xfee00000
-6.93147180369123816490e-01, },// 0xbfe62e42, 0xfee00000
ln2LO[] = { 1.90821492927058770002e-10, // 0x3dea39ef, 0x35793c76
-1.90821492927058770002e-10, },// 0xbdea39ef, 0x35793c76
invln2 = 1.44269504088896338700e+00, // 0x3ff71547, 0x652b82fe
P1 = 1.66666666666666019037e-01, // 0x3FC55555, 0x5555553E
P2 = -2.77777777770155933842e-03, // 0xBF66C16C, 0x16BEBD93
P3 = 6.61375632143793436117e-05, // 0x3F11566A, 0xAF25DE2C
P4 = -1.65339022054652515390e-06, // 0xBEBBBD41, 0xC5D26BF1
P5 = 4.13813679705723846039e-08; // 0x3E663769, 0x72BEA4D0
static double _ieee754_exp(double x) // default IEEE double exp
{
double y, hi = 0, lo = 0, c, t;
int k = 0, xsb;
int hx; // Unsigned.
long xBits = Double.doubleToLongBits(x);
int __HIx = (int) (xBits >> 32);
int __LOx = (int) xBits;
hx = __HIx; // high word of x
xsb = (hx >> 31) & 1; // sign bit of x
hx &= 0x7fffffff; // high word of |x|
// filter out non-finite argument
if (hx >= 0x40862E42) { // if |x|>=709.78...
if (hx >= 0x7ff00000)
if (((hx & 0xfffff) | __LOx) != 0)
return x + x; // NaN
else
return (xsb == 0) ? x : 0.0;
if (x > o_threshold)
return huge * huge; // overflow
if (x < u_threshold)
return twom1000 * twom1000; // underflow
}
// argument reduction
if (hx > 0x3fd62e42) { // if |x| > 0.5 ln2
if (hx < 0x3FF0A2B2) { // and |x| < 1.5 ln2
hi = x - ln2HI[xsb];
lo = ln2LO[xsb];
k = 1 - xsb - xsb;
} else {
k = (int) (invln2 * x + halF[xsb]);
t = k;
hi = x - t * ln2HI[0]; // t*ln2HI is exact here
lo = t * ln2LO[0];
}
x = hi - lo;
} else if (hx < 0x3e300000) { // when |x|<2**-28
if (huge + x > one)
return one + x;// trigger inexact
} else
k = 0;
// x is now in primary range
t = x * x;
c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
if (k == 0)
return one - ((x * c) / (c - 2.0) - x);
else
y = one - ((lo - (x * c) / (2.0 - c)) - hi);
long yBits = Double.doubleToLongBits(y);
int __HIy = (int) (yBits >> 32);
if (k >= -1021) {
__HIy += (k << 20); // add k to y's exponent
yBits = ((__HIy & 0xFFFFFFFFL) << 32) | (yBits & 0xFFFFFFFFL);
y = Double.longBitsToDouble(yBits);
return y;
} else {
__HIy += ((k + 1000) << 20);// add k to y's exponent
yBits = ((__HIy & 0xFFFFFFFFL) << 32) | (yBits & 0xFFFFFFFFL);
y = Double.longBitsToDouble(yBits);
return y * twom1000;
}
}
}