Mathematical functions

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The java.lang.Math class allows for the use of many common mathematical functions that can be used while creating programs.

Since it is in the java.lang package, the Math class need not be imported. However, in programs extensively utilizing these functions, a static import can be used.

Math constants

There are two constants in the Math class that are fairly accurate approximations of irrational mathematical numbers.

Math.E

The Math.E constant, or the base of natural logarithms, represents the value of e, the base of the natural logarithms, about 2.718282.

Code section 3.20: Math.E public static final double E = 2.718281828459045;

Math.PI

The Math.PI constant represents the value of pi, the ratio of a circle's circumference to its diameter, about 3.14159265.

Code section 3.21: Math.PI public static final double PI = 3.141592653589793;

Math methods

Exponential methods

There are several methods in the Math class that deal with exponential functions.

Exponentiation

The power method, double Math.pow(double, double), returns the first parameter to the power of the second parameter. For example, a call to Math.pow(2, 10) will return a value of 1024.

The double Math.exp(double) method, a special case of pow, returns e to the power of the parameter. In addition, double Math.expm1(double) returns (e^x - 1). Both of these methods are more accurate and convenient in these special cases.

Java also provides special cases of the pow function for square roots and cube roots of doubles, double Math.sqrt(double) and double Math.cbrt(double).

Logarithms

Java has no general logarithm function; when needed this can be simulated using the change-of-base theorem.

double Math.log(double) returns the natural logarithm of the parameter (not the common logarithm, as its name suggests!).

double Math.log10(double) returns the common (base-10) logarithm of the parameter.

double Math.log1p(double) returns ln(parameter+1). It is recommended for small values.

Trigonometric and hyperbolic methods

The trigonometric methods of the Math class allow users to easily deal with trigonometric functions in programs. All accept only doubles. Please note that all values using these methods are initially passed and returned in radians, not degrees. However, conversions are possible.

Trigonometric functions

The three main trigonometric methods are Math.sin(x), Math.cos(x), and Math.tan(x), which are used to find the sine, cosine, and tangent, respectively, of any given number. So, for example, a call to Math.sin(Math.PI/2) would return a value of about 1. Although methods for finding the cosecant, secant, and cotangent are not available, these values can be found by taking the reciprocal of the sine, cosine, and tangent, respectively. For example, the cosecant of pi/2 could be found using 1/Math.sin(Math.PI/2).

Inverse trigonometric functions

Java provides inverse counterparts to the trigonometric functions: Math.asin(x), and Math.acos(x), Math.atan(x).

Hyperbolic functions

In addition, hyperbolic functions are available: Math.sinh(x), Math.cosh(x), and Math.tanh(x).

Radian/degree conversion

To convert between degree and radian measures of angles, two methods are available, Math.toRadians(x) and Math.toDegrees(x). While using Math.toRadians(x), a degrees value must be passed in, and that value in radians (the degree value multiplied by pi/180) will be returned. The Math.toDegrees(x) method takes in a value in radians and the value in degrees (the radian value multiplied by 180/pi) is returned.

Absolute value: Math.abs

The absolute value method of the Math class is compatible with the int, long, float, and double types. The data returned is the absolute value of parameter (how far away it is from zero) in the same date type. For example:

Code section 3.22: Math.abs int result = Math.abs(-3);

In this example, result will contain a value of 3.

Maximum and minimum values

These methods are very simple comparing functions. Instead of using if...else statements, one can use the Math.max(x1, x2) and Math.min(x1, x2) methods. The Math.max(x1, x2) simply returns the greater of the two values, while the Math.min(x1, x2) returns the lesser of the two. Acceptable types for these methods include int, long, float, and double.

Functions dealing with floating-point representation

Java 1.5 and 1.6 introduced several non-mathematical functions specific to the computer floating-point representation of numbers.

Math.ulp(double) and Math.ulp(float) return an ulp, the smallest value which, when added to the argument, would be recognized as larger than the argument.

Math.copySign returns the value of the first argument with the sign of the second argument. It can be used to determine the sign of a zero value.

Math.getExponent returns (as an int) the exponent used to scale the floating-point argument in computer representation.

Rounding number example

Sometimes, we are not only interested in mathematically correct rounded numbers, but we want that a fixed number of significant digits are always displayed, regardless of the number used. Here is an example program that returns always the correct string. You are invited to modify it such that it does the same and is simpler!

The constant class contains repeating constants that should exist only once in the code so that to avoid inadvertent changes. (If the one constant is changed inadvertently, it is most likely to be seen, as it is used at several locations.)

Code listing 3.20: Common.java /** * Class that comprises of common constant values. */ public class Common { /** Dash or minus constant */ public static final char DASH = '-'; /** The exponent sign in a scientific number, or the capital letter E */ public static final char EXPONENT = 'E'; /** The full stop or period */ public static final char PERIOD = '.'; /** The zero string constant used at several places */ public static final String ZERO = "0"; }

The Maths class is like an addition to the java.lang.Math class and contains the rounding calculations.

Code listing 3.21: Maths.java package common; /** * Class for special mathematical calculations. * ATTENTION: Should not depend on any other class except Java libraries! */ public class Maths { /** Value after which the language switches from scientific to double */ private final static double E_TO_DOUBLE = 1E-3; /** The string of zeros */ private final static String strZEROS = "000000000000000000000000000000000"; /** * Determines how many zeros are to be appended after the decimal digits. * @param SIGNIFICANTS_AFTER Requested significant digits after decimal * @param SEPARATOR Language-specific decimal separator * @param SIGNIFICANT_DIGITS Number of all significant digits * @param NUMBER Rounded number * @return Requested value */ private static byte calculateMissingSignificantZeros( final byte SIGNIFICANTS_AFTER, final char SEPARATOR, final double NUMBER) { final byte AFTER = findSignificantsAfterDecimal(SEPARATOR, NUMBER); final byte ZEROS = (byte) (SIGNIFICANTS_AFTER - ((AFTER == 0) ? 1 : AFTER)); return ((ZEROS >= 0) ? ZEROS : 0); } /** * Calculates the number of all significant digits (without the sign and * the decimal separator). * @param SIGNIFICANTS_AFTER Requested significant digits after decimal * @param SEPARATOR Language-specific decimal separator * @param NUMBER Value where the digits are to be counted * @return Number of significant digits */ private static byte findSignificantDigits(final byte SIGNIFICANTS_AFTER, final char SEPARATOR, final double NUMBER) { if (NUMBER == 0) return 0; else { String mantissa = findMantissa(SEPARATOR, new Double(NUMBER).toString()); if (NUMBER == (long)NUMBER) { mantissa = mantissa.substring(0, mantissa.length() - 1); } mantissa = RetrieveDigits(SEPARATOR, mantissa); // Find the position of the first non-zero digit: short nonZeroAt = 0; for (; (nonZeroAt < mantissa.length()) && (mantissa.charAt(nonZeroAt) == '0'); nonZeroAt++) ; return (byte)mantissa.substring(nonZeroAt).length(); } } /** * Determines the number of significant digits after the decimal separator * knowing the total number of significant digits and the number before the * decimal separator. * @param SIGNIFICANTS_BEFORE Number of significant digits before separator * @param SIGNIFICANT_DIGITS Number of all significant digits * @return Number of significant decimals after the separator */ private static byte findSignificantsAfterDecimal( final byte SIGNIFICANTS_BEFORE, final byte SIGNIFICANT_DIGITS) { final byte AFTER_DECIMAL = (byte)(SIGNIFICANT_DIGITS - SIGNIFICANTS_BEFORE); return (byte)((AFTER_DECIMAL > 0) ? AFTER_DECIMAL : 0); } /** * Finds the significant digits after the decimal separator of a mantissa. * @param SEPARATOR Language-specific decimal separator * @param NUMBER Value to be scrutinised * @return Number of significant zeros after decimal separator. */ public static byte findSignificantsAfterDecimal(final char SEPARATOR, final double NUMBER) { if (NUMBER == 0) return 1; else { String value = (new Double(NUMBER)).toString(); final short SEPARATOR_AT = (short) value.indexOf(SEPARATOR); if (SEPARATOR_AT > -1) { value = value.substring(SEPARATOR_AT + 1); } final short E_AT = (short) value.indexOf(Common.EXPONENT); if (E_AT > 0) value = value.substring(0, E_AT); Long longValue = new Long(value).longValue(); if (Math.abs(NUMBER) < 1) { return (byte) longValue.toString().length(); } else if (longValue == 0) return 0; else { value = "0." + value; return (byte) (value.length() - 2); } } } /** * Determines the number of digits before the decimal point. * @param SEPARATOR Language-specific decimal separator * @param NUMBER Value to be scrutinised * @return Number of digits before the decimal separator */ private static byte findSignificantsBeforeDecimal(final char SEPARATOR, final double NUMBER) { final String VALUE = new Double(NUMBER).toString(); // Return immediately, if result is clear: Special handling at // crossroads of floating point and exponential numbers: if ((NUMBER == 0) || (Math.abs(NUMBER) >= E_TO_DOUBLE) && (Math.abs(NUMBER) < 1)) { return 0; } else if ((Math.abs(NUMBER) > 0) && (Math.abs(NUMBER) < E_TO_DOUBLE)) { return 1; } else { byte significants = 0; // Significant digits to the right of decimal separator: for (byte y = 0; y < VALUE.length(); y++) { if (VALUE.charAt(y) == SEPARATOR) break; else if (VALUE.charAt(y) != Common.DASH) significants++; } return significants; } } /** * Returns the exponent part of the double number. * @param NUMBER Value of which the exponent is of interest * @return Exponent of the number or zero. */ private static short findExponent(final double NUMBER) { return new Short(findExponent((new Double(NUMBER)).toString())); } /** * Finds the exponent of a number. * @param VALUE Value where an exponent is to be searched * @return Exponent, if it exists, or "0". */ private static String findExponent(final String VALUE) { final short E_AT = (short) VALUE.indexOf(Common.EXPONENT); if (E_AT < 0) return Common.ZERO; else return VALUE.substring(E_AT + 1); } /** * Finds the mantissa of a number. * @param SEPARATOR Language-specific decimal separator * @param VALUE Value where the mantissa is to be found * @return Mantissa of the number */ private static String findMantissa(final char SEPARATOR, final String VALUE) { String value = VALUE; final short E_AT = (short) value.indexOf(Common.EXPONENT); if (E_AT > -1) value = value.substring(0, E_AT); return value; } /** * Retrieves the digits of the value without decimal separator or sign. * @param SEPARATOR * @param number Mantissa to be scrutinised * @return The digits only */ private static String RetrieveDigits(final char SEPARATOR, String number) { // Strip off exponent part, if it exists: short eAt = (short)number.indexOf(Common.EXPONENT); if (eAt > -1) number = number.substring(0, eAt); return number.replace((new Character(Common.DASH)).toString(), ""). replace((new Character(SEPARATOR)).toString(), ""); } /** * Rounds a number to the decimal places. * @param SIGNIFICANTS_AFTER Requested significant digits after decimal * @param SEPARATOR Language-specific decimal separator * @param NUMBER Number to be rounded * @return Rounded number to the requested decimal places */ public static double round(final byte SIGNIFICANTS_AFTER, final char SEPARATOR, final double NUMBER) { if (NUMBER == 0) return 0; else { final double CONSTANT = power(10, (short) (FindInsignificantZerosAfterDecimal(SEPARATOR, NUMBER) + SIGNIFICANTS_AFTER)); final short D_EXPONENT = findExponent(NUMBER); short exponent = D_EXPONENT; double value = NUMBER*CONSTANT*Math.pow(10, -exponent); final String EXPONENT_SIGN = (exponent < 0) ? String.valueOf(Common.DASH) : ""; if (exponent != 0) { exponent = (short) Math.abs(exponent); value = round(value); } else value = round(value)/CONSTANT; // Power method cannot be used, as the exponentiated number may // exceed the maximal long value. exponent -= Math.signum(D_EXPONENT)*(findSignificantDigits (SIGNIFICANTS_AFTER, SEPARATOR, value) - 1); if (D_EXPONENT != 0) { String strValue = Double.toString(value); strValue = strValue.substring(0, strValue.indexOf(SEPARATOR)) + Common.EXPONENT + EXPONENT_SIGN + Short.toString(exponent); value = new Double(strValue); } return value; } } /** * @param SEPARATOR Language-specific decimal separator * @param NUMBER * @return */ private static byte FindInsignificantZerosAfterDecimal( final char SEPARATOR, final double NUMBER) { if ((Math.abs(NUMBER) >= 1) || isScientific(NUMBER)) return 0; else { final StringBuilder STRING = new StringBuilder(); STRING.append(NUMBER); STRING.delete(0, STRING.indexOf(new Character(SEPARATOR).toString()) + 1); // Determine what to match: final String REGEX = "[1-9]"; final String[] SPLIT = STRING.toString().split(REGEX); return (SPLIT.length > 0) ? (byte) SPLIT[0].length() : 0; } } /** * Determines, if the number uses a scientific representation. * @param NUMBER * @return true, if it is a scientific number, false otherwise */ private static boolean isScientific(final double NUMBER) { return ((new Double(NUMBER)).toString().indexOf(Common.EXPONENT) > 0); } /** * Rounds a number according to mathematical rules. * @param VALUE * @return */ public static double round(final double VALUE) { return (long)(VALUE + .5); } /** * Calculates the power of the base to the exponent without changing the * least-significant digits of a number. * @param BASIS * @param EXPONENT * @return BASIS to power of EXPONENT */ public static double power(final int BASIS, final short EXPONENT) { return power((short)BASIS, EXPONENT); } /** * Calculates the power of the base to the exponent without changing the * least-significant digits of a number. * @param BASIS * @param EXPONENT * @return BASIS to power of EXPONENT */ public static double power(final short BASIS, final short EXPONENT) { if (BASIS == 0) return (EXPONENT != 0) ? 1 : 0; else { if (EXPONENT == 0) return 1; else { // The Math method power does change the least significant // digits after the decimal separator and is therefore useless. double result = 1; short s = 0; if (EXPONENT > 0) { for (; s < EXPONENT; s++) result *= BASIS; } else if (EXPONENT < 0) { for (s = EXPONENT; s < 0; s++) result /= BASIS; } return result; } } } /** * Rounds to a fixed number of significant digits. * @param SIGNIFICANT_DIGITS Requested number of significant digits * @param SEPARATOR Language-specific decimal separator * @param dNumber Number to be rounded * @return Rounded number */ public static String roundToString(final byte SIGNIFICANT_DIGITS, final char SEPARATOR, double dNumber) { // Number of significants that *are* before the decimal separator: final byte SIGNIFICANTS_BEFORE = findSignificantsBeforeDecimal(SEPARATOR, dNumber); // Number of decimals that *should* be after the decimal separator: final byte SIGNIFICANTS_AFTER = findSignificantsAfterDecimal( SIGNIFICANTS_BEFORE, SIGNIFICANT_DIGITS); // Round to the specified number of digits after decimal separator: final double ROUNDED = Maths.round(SIGNIFICANTS_AFTER, SEPARATOR, dNumber); final String EXPONENT = findExponent((new Double(ROUNDED)).toString()); final String MANTISSA = findMantissa(SEPARATOR, (new Double(ROUNDED)).toString()); final double dMANTISSA = new Double(MANTISSA).doubleValue(); final StringBuilder RESULT = new StringBuilder(MANTISSA); // Determine the significant digits in this number: final byte SIGNIFICANTS = findSignificantDigits(SIGNIFICANTS_AFTER, SEPARATOR, dMANTISSA); // Add lagging zeros, if necessary: if (SIGNIFICANTS <= SIGNIFICANT_DIGITS) { if (SIGNIFICANTS_AFTER != 0) { RESULT.append(strZEROS.substring(0, calculateMissingSignificantZeros(SIGNIFICANTS_AFTER, SEPARATOR, dMANTISSA))); } else { // Cut off the decimal separator & after decimal digits: final short DECIMAL = (short) RESULT.indexOf( new Character(SEPARATOR).toString()); if (DECIMAL > -1) RESULT.setLength(DECIMAL); } } else if (SIGNIFICANTS_BEFORE > SIGNIFICANT_DIGITS) { dNumber /= power(10, (short)(SIGNIFICANTS_BEFORE - SIGNIFICANT_DIGITS)); dNumber = round(dNumber); final short DIGITS = (short)(SIGNIFICANT_DIGITS + ((dNumber < 0) ? 1 : 0)); String D_VALUE = (new Double(dNumber)).toString().substring(0, DIGITS); RESULT.setLength(0); RESULT.append(D_VALUE + strZEROS.substring(0, SIGNIFICANTS_BEFORE - SIGNIFICANT_DIGITS)); } if (new Short(EXPONENT) != 0) { RESULT.append(Common.EXPONENT + EXPONENT); } return RESULT.toString(); } // public static String roundToString(…) /** * Rounds to a fixed number of significant digits. * @param SEPARATOR Language-specific decimal separator * @param SIGNIFICANT_DIGITS Requested number of significant digits * @param value Number to be rounded * @return Rounded number */ public static String roundToString(final char SEPARATOR, final int SIGNIFICANT_DIGITS, float value) { return roundToString((byte)SIGNIFICANT_DIGITS, SEPARATOR, (double)value); } } // class Maths

The test class checks all possibilities:

Code listing 3.22: TestCommon.java package common; import java.util.Vector; /** * Test class for the common functionality * @author Saban */ public class TestCommon { /** * Test for the common functionality * @param args */ public static void main(String[] args) { // Test rounding Vector<Double> values = new Vector<Double>(); values.add(0.0); AddValue(1.4012984643248202e-45, values); AddValue(1.999999757e-5, values); AddValue(1.999999757e-4, values); AddValue(1.999999757e-3, values); AddValue(0.000640589, values); AddValue(0.3396899998188019, values); AddValue(0.34, values); AddValue(7.07, values); AddValue(118.188, values); AddValue(118.2, values); AddValue(123.405009, values); AddValue(30.76994323730469, values); AddValue(130.76994323730469, values); AddValue(540, values); AddValue(12345, values); AddValue(123456, values); AddValue(540911, values); AddValue(9.223372036854776e56, values); final byte SIGNIFICANTS = 5; for (double element : values) { System.out.println(" Maths.Round(" + SIGNIFICANTS + ", '" + Common.PERIOD + "', " + element + ") = " + Maths.roundToString(SIGNIFICANTS, Common.PERIOD, element)); } } // void main(String[] args) /** * Method that adds a negative and a positive value to values. * @param D * @param values */ private static void AddValue(final double D, Vector<Double> values) { values.add(-D); values.add(D); } } // class TestCommon

Output for code listing 3.22 Maths.Round(5, '.', 0.0) = 0.00000 Maths.Round(5, '.', -1.4012984643248202E-45) = -1.4012E-45 Maths.Round(5, '.', 1.4012984643248202E-45) = 1.4013E-45 Maths.Round(5, '.', -1.999999757E-5) = -1.9999E-5 Maths.Round(5, '.', 1.999999757E-5) = 2.0000E-5 Maths.Round(5, '.', -1.999999757E-4) = -1.9999E-4 Maths.Round(5, '.', 1.999999757E-4) = 2.0000E-4 Maths.Round(5, '.', -0.001999999757) = -0.0019999 Maths.Round(5, '.', 0.001999999757) = 0.0020000 Maths.Round(5, '.', -6.40589E-4) = -6.4058E-4 Maths.Round(5, '.', 6.40589E-4) = 6.4059E-4 Maths.Round(5, '.', -0.3396899998188019) = -0.33968 Maths.Round(5, '.', 0.3396899998188019) = 0.33969 Maths.Round(5, '.', -0.34) = -0.33999 Maths.Round(5, '.', 0.34) = 0.34000 Maths.Round(5, '.', -7.07) = -7.0699 Maths.Round(5, '.', 7.07) = 7.0700 Maths.Round(5, '.', -118.188) = -118.18 Maths.Round(5, '.', 118.188) = 118.19 Maths.Round(5, '.', -118.2) = -118.19 Maths.Round(5, '.', 118.2) = 118.20 Maths.Round(5, '.', -123.405009) = -123.40 Maths.Round(5, '.', 123.405009) = 123.41 Maths.Round(5, '.', -30.76994323730469) = -30.769 Maths.Round(5, '.', 30.76994323730469) = 30.770 Maths.Round(5, '.', -130.7699432373047) = -130.76 Maths.Round(5, '.', 130.7699432373047) = 130.77 Maths.Round(5, '.', -540.0) = -539.99 Maths.Round(5, '.', 540.0) = 540.00 Maths.Round(5, '.', -12345.0) = -12344 Maths.Round(5, '.', 12345.0) = 12345 Maths.Round(5, '.', -123456.0) = -123450 Maths.Round(5, '.', 123456.0) = 123460 Maths.Round(5, '.', -540911.0) = -540900 Maths.Round(5, '.', 540911.0) = 540910 Maths.Round(5, '.', -9.223372036854776E56) = -9.2233E56 Maths.Round(5, '.', 9.223372036854776E56) = 9.2234E56

If you are interested in a comparison with C#, take a look at the rounding number example there. If you are interested in a comparison with C++, you can compare this code here with the same example over there.

Notice that in the expression starting with if ((D == 0), I have to use OR instead of the || because of a bug in the source template.