mathjs
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Math.js is an extensive math library for JavaScript and Node.js. It features a flexible expression parser and offers an integrated solution to work with numbers, big numbers, complex numbers, units, and matrices.
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JavaScript
'use strict';
var BigNumber = require('../type/BigNumber');
var isNumber = require('./number').isNumber;
var digits = require('./number').digits;
var memoize = require('./function').memoize;
/**
* Test whether value is a BigNumber
* @param {*} value
* @return {Boolean} isBigNumber
*/
exports.isBigNumber = function (value) {
return (value instanceof BigNumber);
};
/*************************************
* Constants *
*************************************/
/**
* Calculate BigNumber e
* @param {Number} precision
* @returns {BigNumber} Returns e
*/
exports.e = memoize(function (precision) {
var Big = BigNumber.constructor({precision: precision});
return new Big(1).exp();
});
/**
* Calculate BigNumber golden ratio, phi = (1+sqrt(5))/2
* @param {Number} precision
* @returns {BigNumber} Returns phi
*/
exports.phi = memoize(function (precision) {
var Big = BigNumber.constructor({precision: precision});
return new Big(1).plus(new Big(5).sqrt()).div(2);
});
/**
* Calculate BigNumber pi.
*
* Uses Machin's formula: pi / 4 = 4 * arctan(1 / 5) - arctan(1 / 239)
* http://milan.milanovic.org/math/english/pi/machin.html
* @param {Number} precision
* @returns {BigNumber} Returns pi
*/
exports.pi = memoize(function (precision) {
// we calculate pi with a few decimal places extra to prevent round off issues
var Big = BigNumber.constructor({precision: precision + 4});
var pi4th = new Big(4).times(arctan_taylor(new Big(1).div(5)))
.minus(arctan_taylor(new Big(1).div(239)));
Big.config({precision: precision});
// the final pi has the requested number of decimals
return new Big(4).times(pi4th);
});
/**
* Calculate BigNumber tau, tau = 2 * pi
* @param {Number} precision
* @returns {BigNumber} Returns tau
*/
exports.tau = memoize(function (precision) {
// we calculate pi at a slightly higher precision than configured to prevent round off errors
// when multiplying by two in the end
var pi = exports.pi(precision + 2);
var Big = BigNumber.constructor({precision: precision});
return new Big(2).times(pi);
});
/*************************************
* Bitwise functions *
*************************************/
/*
* Special Cases:
* N & n = N
* n & 0 = 0
* n & -1 = n
* n & n = n
* I & I = I
* -I & -I = -I
* I & -I = 0
* I & n = n
* I & -n = I
* -I & n = 0
* -I & -n = -I
*
* @param {BigNumber} value
* @param {BigNumber} value
* @return {BigNumber} Result of `x` & `y`, is fully precise
*
*/
exports.and = function(x, y) {
if ((x.isFinite() && !x.isInteger()) || (y.isFinite() && !y.isInteger())) {
throw new Error('Parameters in function bitAnd must be integer numbers');
}
var BigNumber = x.constructor;
if (x.isNaN() || y.isNaN()) {
return new BigNumber(NaN);
}
if (x.isZero() || y.eq(-1) || x.eq(y)) {
return x;
}
if (y.isZero() || x.eq(-1)) {
return y;
}
if (!x.isFinite() || !y.isFinite()) {
if (!x.isFinite() && !y.isFinite()) {
if (x.isNegative() == y.isNegtive()) {
return x;
}
return new BigNumber(0);
}
if (!x.isFinite()) {
if (y.isNegative()) {
return x;
}
if (x.isNegative()) {
return new BigNumber(0);
}
return y;
}
if (!y.isFinite()) {
if (x.isNegative()) {
return y;
}
if (y.isNegative()) {
return new BigNumber(0);
}
return x;
}
}
return bitwise(x, y, function (a, b) { return a & b });
};
/*
* Special Cases:
* n << -n = N
* n << N = N
* N << n = N
* n << 0 = n
* 0 << n = 0
* I << I = N
* I << n = I
* n << I = I
*
* @param {BigNumber} value
* @param {BigNumber} value
* @return {BigNumber} Result of `x` << `y`
*
*/
exports.leftShift = function (x, y) {
if ((x.isFinite() && !x.isInteger()) || (y.isFinite() && !y.isInteger())) {
throw new Error('Parameters in function leftShift must be integer numbers');
}
var BigNumber = x.constructor;
if (x.isNaN() || y.isNaN() || (y.isNegative() && !y.isZero())) {
return new BigNumber(NaN);
}
if (x.isZero() || y.isZero()) {
return x;
}
if (!x.isFinite() && !y.isFinite()) {
return new BigNumber(NaN);
}
// Math.pow(2, y) is fully precise for y < 55, and fast
if (y.lt(55)) {
return x.times(Math.pow(2, y.toNumber()) + '');
}
return x.times(new BigNumber(2).pow(y));
};
/*
* @param {BigNumber} value
* @return {BigNumber} Result of ~`x`, fully precise
*
*/
exports.not = function (x) {
if (x.isFinite() && !x.isInteger()) {
throw new Error('Parameter in function bitNot must be integer numbers');
}
var BigNumber = x.constructor;
var prevPrec = BigNumber.precision;
BigNumber.config({precision: 1E9});
var x = x.plus(BigNumber.ONE);
x.s = -x.s || null;
BigNumber.config({precision: prevPrec});
return x;
};
/*
* Special Cases:
* N | n = N
* n | 0 = n
* n | -1 = -1
* n | n = n
* I | I = I
* -I | -I = -I
* I | -n = -1
* I | -I = -1
* I | n = I
* -I | n = -I
* -I | -n = -n
*
* @param {BigNumber} value
* @param {BigNumber} value
* @return {BigNumber} Result of `x` | `y`, fully precise
*
*/
exports.or = function (x, y) {
if ((x.isFinite() && !x.isInteger()) || (y.isFinite() && !y.isInteger())) {
throw new Error('Parameters in function bitOr must be integer numbers');
}
var BigNumber = x.constructor;
if (x.isNaN() || y.isNaN()) {
return new BigNumber(NaN);
}
var negOne = new BigNumber(-1);
if (x.isZero() || y.eq(negOne) || x.eq(y)) {
return y;
}
if (y.isZero() || x.eq(negOne)) {
return x;
}
if (!x.isFinite() || !y.isFinite()) {
if ((!x.isFinite() && !x.isNegative() && y.isNegative()) ||
(x.isNegative() && !y.isNegative() && !y.isFinite())) {
return negOne;
}
if (x.isNegative() && y.isNegative()) {
return x.isFinite() ? x : y;
}
return x.isFinite() ? y : x;
}
return bitwise(x, y, function (a, b) { return a | b });
};
/*
* Special Cases:
* n >> -n = N
* n >> N = N
* N >> n = N
* I >> I = N
* n >> 0 = n
* I >> n = I
* -I >> n = -I
* -I >> I = -I
* n >> I = I
* -n >> I = -1
* 0 >> n = 0
*
* @param {BigNumber} value
* @param {BigNumber} value
* @return {BigNumber} Result of `x` >> `y`
*
*/
exports.rightShift = function (x, y) {
if ((x.isFinite() && !x.isInteger()) || (y.isFinite() && !y.isInteger())) {
throw new Error('Parameters in function rightArithShift must be integer numbers');
}
var BigNumber = x.constructor;
if (x.isNaN() || y.isNaN() || (y.isNegative() && !y.isZero())) {
return new BigNumber(NaN);
}
if (x.isZero() || y.isZero()) {
return x;
}
if (!y.isFinite()) {
if (x.isNegative()) {
return new BigNumber(-1);
}
if (!x.isFinite()) {
return new BigNumber(NaN);
}
return new BigNumber(0);
}
// Math.pow(2, y) is fully precise for y < 55, and fast
if (y.lt(55)) {
return x.div(Math.pow(2, y.toNumber()) + '').floor();
}
return x.div(new BigNumber(2).pow(y)).floor();
};
/*
* Special Cases:
* N ^ n = N
* n ^ 0 = n
* n ^ n = 0
* n ^ -1 = ~n
* I ^ n = I
* I ^ -n = -I
* I ^ -I = -1
* -I ^ n = -I
* -I ^ -n = I
*
* @param {BigNumber} value
* @param {BigNumber} value
* @return {BigNumber} Result of `x` ^ `y`, fully precise
*
*/
exports.xor = function (x, y) {
if ((x.isFinite() && !x.isInteger()) || (y.isFinite() && !y.isInteger())) {
throw new Error('Parameters in function bitXor must be integer numbers');
}
var BigNumber = x.constructor;
if (x.isNaN() || y.isNaN()) {
return new BigNumber(NaN);
}
if (x.isZero()) {
return y;
}
if (y.isZero()) {
return x;
}
if (x.eq(y)) {
return new BigNumber(0);
}
var negOne = new BigNumber(-1);
if (x.eq(negOne)) {
return exports.not(y);
}
if (y.eq(negOne)) {
return exports.not(x);
}
if (!x.isFinite() || !y.isFinite()) {
if (!x.isFinite() && !y.isFinite()) {
return negOne;
}
return new BigNumber(x.isNegative() == y.isNegative()
? Infinity
: -Infinity);
}
return bitwise(x, y, function (a, b) { return a ^ b });
};
/* Applies bitwise function to numbers. */
function bitwise(x, y, func) {
var BigNumber = x.constructor;
var xBits, yBits;
var xSign = +(x.s < 0);
var ySign = +(y.s < 0);
if (xSign) {
xBits = decCoefficientToBinaryString(exports.not(x));
for (var i = 0; i < xBits.length; ++i) {
xBits[i] ^= 1;
}
} else {
xBits = decCoefficientToBinaryString(x);
}
if (ySign) {
yBits = decCoefficientToBinaryString(exports.not(y));
for (var i = 0; i < yBits.length; ++i) {
yBits[i] ^= 1;
}
} else {
yBits = decCoefficientToBinaryString(y);
}
var minBits, maxBits, minSign;
if (xBits.length <= yBits.length) {
minBits = xBits;
maxBits = yBits;
minSign = xSign;
} else {
minBits = yBits;
maxBits = xBits;
minSign = ySign;
}
var shortLen = minBits.length;
var longLen = maxBits.length;
var expFuncVal = func(xSign, ySign) ^ 1;
var outVal = new BigNumber(expFuncVal ^ 1);
var twoPower = BigNumber.ONE;
var two = new BigNumber(2);
var prevPrec = BigNumber.precision;
BigNumber.config({precision: 1E9});
while (shortLen > 0) {
if (func(minBits[--shortLen], maxBits[--longLen]) == expFuncVal) {
outVal = outVal.plus(twoPower);
}
twoPower = twoPower.times(two);
}
while (longLen > 0) {
if (func(minSign, maxBits[--longLen]) == expFuncVal) {
outVal = outVal.plus(twoPower);
}
twoPower = twoPower.times(two);
}
BigNumber.config({precision: prevPrec});
if (expFuncVal == 0) {
outVal.s = -outVal.s;
}
return outVal;
}
/* Extracted from decimal.js, and edited to specialize. */
function decCoefficientToBinaryString(x) {
// Convert to string
var a = x.c;
var r = a[0] + '';
for (var i = 1; i < a.length; ++i) {
var s = a[i] + '';
for (var z = 7 - s.length; z--; ) {
s = '0' + s;
}
r += s;
}
var j;
for (j = r.length - 1; r.charAt(j) == '0'; --j);
var xe = x.e;
var str = r.slice(0, j + 1 || 1);
var strL = str.length;
if (xe > 0) {
if (++xe > strL) {
// Append zeros.
for (xe -= strL; xe--; str += '0');
} else if (xe < strL) {
str = str.slice(0, xe) + '.' + str.slice(xe);
}
}
// Convert from base 10 (decimal) to base 2
var arr = [0];
for (var i = 0; i < str.length; ) {
for (var arrL = arr.length; arrL--; arr[arrL] *= 10);
arr[0] += str.charAt(i++) << 0; // convert to int
for (var j = 0; j < arr.length; ++j) {
if (arr[j] > 1) {
if (arr[j + 1] == null) {
arr[j + 1] = 0;
}
arr[j + 1] += arr[j] >> 1;
arr[j] &= 1;
}
}
}
return arr.reverse();
}
/*************************************
* Trigonometric functions *
*************************************/
/**
* Calculate the arccosine or arcsecant of x
*
* acos(x) = 2*atan(sqrt(1-x^2)/(1+x))
*
* asec(x) = acos(1/x)
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @param {Boolean} reciprocal is sec
* @returns {BigNumber} arccosine or arcsecant of x
*/
exports.arccos_arcsec = function (x, Big, reciprocal) {
var precision = Big.precision;
if (reciprocal) {
if (x.abs().lt(Big.ONE)) {
throw new Error('asec() only has non-complex values for |x| >= 1.');
}
} else if (x.abs().gt(Big.ONE)) {
throw new Error('acos() only has non-complex values for |x| <= 1.');
}
if (x.eq(-1)) {
return exports.pi(precision);
}
Big.config({precision: precision + 4});
if (reciprocal) {
x = Big.ONE.div(x);
}
var acos = exports.arctan_arccot(Big.ONE.minus(x.times(x)).sqrt()
.div(x.plus(Big.ONE)), Big).times(2);
Big.config({precision: precision});
return acos.toDP(precision - 1);
};
/**
* Calculate the arcsine or arccosecant of x
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @param {Boolean} reciprocal is csc
* @returns {BigNumber} arcsine or arccosecant of x
*/
exports.arcsin_arccsc = function (x, Big, reciprocal) {
if (x.isNaN()) {
return new Big(NaN);
}
var precision = Big.precision;
var absX = x.abs();
if (reciprocal) {
if (absX.lt(Big.ONE)) {
throw new Error('acsc() only has non-complex values for |x| >= 1.');
}
Big.config({precision: precision + 2});
x = Big.ONE.div(x);
Big.config({precision: precision});
absX = x.abs();
} else if (absX.gt(Big.ONE)) {
throw new Error('asin() only has non-complex values for |x| <= 1.');
}
// Get x below 0.58
if (absX.gt(0.8)) {
Big.config({precision: precision + 4});
// arcsin(x) = sign(x)*(Pi/2 - arcsin(sqrt(1 - x^2)))
var sign = x.s;
var halfPi = exports.pi(precision + 4).div(2);
x = halfPi.minus(exports.arcsin_arccsc(Big.ONE.minus(x.times(x)).sqrt(), Big));
x.s = sign;
x.constructor = Big;
Big.config({precision: precision});
return x.toDP(precision - 1);
}
var wasReduced = absX.gt(0.58);
if (wasReduced) {
Big.config({precision: precision + 8});
// arcsin(x) = 2*arcsin(x / (sqrt(2)*sqrt(sqrt(1 - x^2) + 1)))
x = x.div(new Big(2).sqrt().times(Big.ONE.minus(x.times(x)).sqrt()
.plus(Big.ONE).sqrt()));
Big.config({precision: precision});
}
// Avoid overhead of Newton's Method if feasible
var ret = (precision <= 60 || ((x.dp() <= Math.log(precision)) && x.lt(0.05)))
? arcsin_taylor(x, precision)
: arcsin_newton(x, Big);
if (wasReduced) {
return ret.times(2);
}
return ret;
};
/**
* Calculate the arctangent or arccotangent of x
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @param {Boolean} reciprocal is cot
* @returns {BigNumber} arctangent or arccotangent of x
*/
exports.arctan_arccot = function (x, Big, reciprocal) {
if (x.isNaN()) {
return new Big(NaN);
}
if ((!reciprocal && x.isZero()) || (reciprocal && !x.isFinite())) {
return new Big(0);
}
var precision = Big.precision;
if ((!reciprocal && !x.isFinite()) || (reciprocal && x.isZero())) {
var halfPi = exports.pi(precision + 2).div(2).toDP(precision - 1);
halfPi.constructor = Big;
halfPi.s = x.s;
return halfPi;
}
Big.config({precision: precision + 4});
if (reciprocal) {
x = Big.ONE.div(x);
}
var absX = x.abs();
if (absX.lte(0.875)) {
var ret = arctan_taylor(x);
ret.constructor = Big;
Big.config({precision: precision});
return ret.toDP(Big.precision - 1);
}
if (absX.gte(1.143)) {
// arctan(x) = sign(x)*((PI / 2) - arctan(1 / |x|))
var halfPi = exports.pi(precision + 4).div(2);
var ret = halfPi.minus(arctan_taylor(Big.ONE.div(absX)));
ret.s = x.s;
ret.constructor = Big;
Big.config({precision: precision});
return ret.toDP(Big.precision - 1);
}
// arctan(x) = arcsin(x / [sqrt(1 + x^2)])
x = x.div(x.times(x).plus(1).sqrt());
Big.config({precision: precision});
return exports.arcsin_arccsc(x, Big);
};
/**
* Calculate the arctangent of y, x
*
* @param {BigNumber} y
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @returns {BigNumber} arctangent of y, x
*/
exports.arctan2 = function (y, x, Big) {
var precision = Big.precision;
if (x.isZero()) {
if (y.isZero()) {
return new Big(NaN);
}
var halfPi = exports.pi(precision + 2).div(2).toDP(precision - 1);
halfPi.constructor = Big;
halfPi.s = y.s;
return halfPi;
}
Big.config({precision: precision + 2});
var ret = exports.arctan_arccot(y.div(x), Big, false);
if (x.isNegative()) {
var pi = exports.pi(precision + 2);
ret = y.isNegative() ? ret.minus(pi) : ret.plus(pi);
}
ret.constructor = Big;
Big.config({precision: precision});
return ret.toDP(precision - 1);
};
/**
* Calculate the hyperbolic arccosine, arcsine, arcsecant, or arccosecant of x
*
* acosh(x) = ln(x + sqrt(x^2 - 1))
*
* asinh(x) = ln(x + sqrt(x^2 + 1))
*
* asech(x) = acosh(1 / x)
*
* acsch(x) = asinh(1 / x)
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @param {Boolean} mode sine function if true, cosine function if false
* @param {Boolean} reciprocal is sec or csc
* @returns {BigNumber} hyperbolic arccosine, arcsine, arcsecant, or arccosecant of x
*/
exports.acosh_asinh_asech_acsch = function (x, Big, mode, reciprocal) {
if (x.isNaN()) {
return new Big(NaN);
}
if (reciprocal && x.isZero()) {
return new Big(Infinity);
}
if (!mode) {
if (reciprocal) {
if (x.isNegative() || x.gt(Big.ONE)) {
throw new Error('asech() only has non-complex values for 0 <= x <= 1.');
}
} else if (x.lt(Big.ONE)) {
throw new Error('acosh() only has non-complex values for x >= 1.');
}
}
var precision = Big.precision;
Big.config({precision: precision + 4});
var y = new Big(x);
y.constructor = Big;
if (reciprocal) {
y = Big.ONE.div(y);
}
var x2PlusOrMinus = (mode) ? y.times(y).plus(Big.ONE) : y.times(y).minus(Big.ONE);
var ret = y.plus(x2PlusOrMinus.sqrt()).ln();
Big.config({precision: precision});
return new Big(ret.toPrecision(precision));
};
/**
* Calculate the hyperbolic arctangent or arccotangent of x
*
* atanh(x) = ln((1 + x)/(1 - x)) / 2
*
* acoth(x) = atanh(1 / x)
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @param {Boolean} reciprocal is sec or csc
* @returns {BigNumber} hyperbolic arctangent or arccotangent of x
*/
exports.atanh_acoth = function (x, Big, reciprocal) {
if (x.isNaN()) {
return new Big(NaN);
}
var absX = x.abs();
if (absX.eq(Big.ONE)) {
return new Big(x.isNegative() ? -Infinity : Infinity);
}
if (absX.gt(Big.ONE)) {
if (!reciprocal) {
throw new Error('atanh() only has non-complex values for |x| <= 1.');
}
} else if (reciprocal) {
throw new Error('acoth() has complex values for |x| < 1.');
}
if (x.isZero()) {
return new Big(0);
}
var precision = Big.precision;
Big.config({precision: precision + 4});
var y = new Big(x);
y.constructor = Big;
if (reciprocal) {
y = Big.ONE.div(y);
}
var ret = Big.ONE.plus(y).div(Big.ONE.minus(y)).ln().div(2);
Big.config({precision: precision});
return new Big(ret.toPrecision(precision));
};
/**
* Calculate the cosine/sine of x using the multiple angle identity:
*
* cos(4x) = 8[cos(x)^4 - cos(x)^2] + 1
*
* sin(5x) = 16sin(x)^5 - 20sin(x)^3 + 5sin(x)
* http://www.tc.umn.edu/~ringx004/sidebar.html
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @param {Number} mode cosine function if 0, sine function if 1
* @param {Boolean} reciprocal is sec or csc
* @returns {BigNumber} cosine, sine, secant, or cosecant of x
*/
exports.cos_sin_sec_csc = function (x, Big, mode, reciprocal) {
if (x.isNaN() || !x.isFinite()) {
return new Big(NaN);
}
var precision = Big.precision;
// Avoid changing the original value
var y = new Big(x);
// sin(-x) == -sin(x), cos(-x) == cos(x)
var isNeg = y.isNegative();
if (isNeg) {
y.s = -y.s;
}
// Apply ~log(precision) guard bits
var precPlusGuardDigits = precision + (Math.log(precision) | 0) + 3;
Big.config({precision: precPlusGuardDigits});
y = reduceToPeriod(y, precPlusGuardDigits, mode); // Make this destructive
y[0].constructor = Big;
if (y[1]) {
y = y[0];
if (reciprocal && y.isZero()) {
y = new Big(Infinity);
}
Big.config({precision: precision});
return y;
}
var ret;
y = y[0];
if (mode) {
ret = cos_sin_taylor(y.div(3125), mode);
Big.config({precision: Math.min(precPlusGuardDigits, precision + 15)});
var five = new Big(5);
var sixteen = new Big(16);
var twenty = new Big(20);
for (var i = 0; i < 5; ++i) {
var ret2 = ret.times(ret);
var ret3 = ret2.times(ret);
var ret5 = ret3.times(ret2);
ret = sixteen.times(ret5).minus(
twenty.times(ret3)).plus(
five.times(ret));
}
if (isNeg) {
ret.s = -ret.s;
}
} else {
var div_factor, loops;
if (y.abs().lt(Big.ONE)) {
div_factor = 64;
loops = 3;
} else {
div_factor = 256;
loops = 4;
}
ret = cos_sin_taylor(y.div(div_factor), mode);
Big.config({precision: Math.min(precPlusGuardDigits, precision + 8)});
var eight = new Big(8);
for (; loops > 0; --loops) {
var ret2 = ret.times(ret);
var ret4 = ret2.times(ret2);
ret = eight.times(ret4.minus(ret2)).plus(Big.ONE);
}
}
if (reciprocal) {
ret = (ret.e <= -precision)
? new Big(Infinity)
: Big.ONE.div(ret);
}
Big.config({precision: precision});
return ret.toDP(precision - 1);
};
/**
* Calculate the tangent of x
*
* tan(x) = sin(x) / cos(x)
*
* cot(x) = cos(x) / sin(x)
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @param {Boolean} reciprocal is cot
* @returns {BigNumber} tangent or cotangent of x
*/
exports.tan_cot = function (x, Big, reciprocal) {
if (x.isNaN()) {
return new Big(NaN);
}
var precision = Big.precision;
var pi = exports.pi(precision + 2);
var halfPi = pi.div(2).toDP(precision - 1);
pi = pi.toDP(precision - 1);
var y = reduceToPeriod(x, precision, 1)[0];
if (y.abs().eq(pi)) {
return new Big(Infinity);
}
Big.config({precision: precision + 4});
var sin = exports.cos_sin_sec_csc(y, Big, 1, false);
var cos = sinToCos(sin);
sin = sin.toDP(precision);
cos = cos.toDP(precision);
// Make sure sign for cosine is correct
if (y.eq(x)) {
if (y.gt(halfPi)) {
cos.s = -cos.s;
}
} else if (pi.minus(y.abs()).gt(halfPi)) {
cos.s = -cos.s;
}
var tan = (reciprocal) ? cos.div(sin) : sin.div(cos);
Big.config({precision: precision});
return new Big(tan.toPrecision(precision));
};
/**
* Calculate the hyperbolic sine, cosine, secant, or cosecant of x
*
* cosh(x) = (exp(x) + exp(-x)) / 2
* = (e^x + 1/e^x) / 2
*
* sinh(x) = (exp(x) - exp(-x)) / 2
* = (e^x - 1/e^x) / 2
*
* sech(x) = 2 / (exp(x) + exp(-x))
* = 2 / (e^x + 1/e^x)
*
* csch(x) = 2 / (exp(x) - exp(-x))
* = 2 / (e^x - 1/e^x)
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @param {Boolean} mode sinh function if true, cosh function if false
* @param {Boolean} reciprocal is sech or csch
* @returns {BigNumber} hyperbolic cosine, sine, secant. or cosecant of x
*/
exports.cosh_sinh_csch_sech = function (x, Big, mode, reciprocal) {
if (x.isNaN()) {
return new Big(NaN);
}
if (!x.isFinite()) {
if (reciprocal) {
return new Big(0);
}
return new Big((mode) ? x : Infinity);
}
var precision = Big.precision;
Big.config({precision: precision + 4});
var y = new Big(x);
y.constructor = Big;
y = y.exp();
y = (mode) ? y.minus(Big.ONE.div(y)) : y.plus(Big.ONE.div(y));
y = (reciprocal) ? new Big(2).div(y) : y.div(2);
Big.config({precision: precision});
return new Big(y.toPrecision(precision));
};
/**
* Calculate the hyperbolic tangent of x
*
* tanh(x) = (exp(x) + exp(-x)) / (exp(x) - exp(-x))
* = (exp(2x) - 1) / (exp(2x) + 1)
* = (e^x - 1/e^x) / (e^x + 1/e^x)
*
* coth(x) = (exp(x) - exp(-x)) / (exp(x) + exp(-x))
* = (exp(2x) + 1) / (exp(2x) - 1)
* = (e^x + 1/e^x) / (e^x - 1/e^x)
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @param {Boolean} reciprocal is coth
* @returns {BigNumber} hyperbolic tangent or cotangent of x
*/
exports.tanh_coth = function (x, Big, reciprocal) {
if (x.isNaN()) {
return new Big(NaN);
}
if (!x.isFinite()) {
return new Big(x.s);
}
var precision = Big.precision;
Big.config({precision: precision + 4});
var y = new Big(x);
y.constructor = Big;
var posExp = y.exp();
var negExp = Big.ONE.div(posExp);
var ret = posExp.minus(negExp);
ret = (reciprocal) ? posExp.plus(negExp).div(ret) : ret.div(posExp.plus(negExp));
Big.config({precision: precision});
return ret.toDP(precision - 1);
};
/**
* Calculate the arc sine of x using Newton's method
*
* f(x) = sin(x) = N => f(x) = sin(x) - N
* f'(x) = cos(x)
*
* Thus we solve each step as follows:
* x_(i+1) = x_i - (sin(x_i) - N)/cos(x_i)
*
* @param {BigNumber} x
* @param {DecimalFactory} Big current BigNumber constructor
* @returns {BigNumber} arc sine of x
*/
function arcsin_newton(x, Big) {
var oldPrecision = Big.precision;
// Calibration variables, adjusted from MAPM
var tolerance = -(oldPrecision + 4);
var maxp = oldPrecision + 8 - x.e;
var localPrecision = 25 - x.e;
var maxIter = Math.max(Math.log(oldPrecision + 2) * 1.442695 | 0 + 5, 5);
Big.config({precision: localPrecision});
var i = 0;
var curr = new Big(Math.asin(x.toNumber()) + '');
do {
var tmp0 = exports.cos_sin_sec_csc(curr, Big, 1, false);
var tmp1 = sinToCos(tmp0);
if (!tmp0.isZero()) {
tmp0.s = curr.s;
}
var tmp2 = tmp0.minus(x).div(tmp1);
curr = curr.minus(tmp2);
localPrecision = Math.min(2*localPrecision, maxp);
Big.config({precision: localPrecision});
} while ((2*tmp2.e >= tolerance) && !tmp2.isZero() && (++i <= maxIter))
if (i == maxIter) {
throw new Error('asin() failed to converge to the requested accuracy.' +
'Try with a higher precision.');
}
Big.config({precision: oldPrecision});
return curr.toDP(oldPrecision - 1);
}
/**
* Calculate the arc sine of x
*
* arcsin(x) = x + (1/2)*x^3/3 + (3/8)*x^5/5 + (15/48)*x^7/7 ...
* = x + (1/2)*x^2*x^1/3 + [(1*3)/(2*4)]*x^2*x^3/5 + [(1*3*5)/(2*4*6)]*x^2*x^5/7 ...
*
* @param {BigNumber} x
* @param {Number} precision
* @returns {BigNumber} arc sine of x
*/
function arcsin_taylor(x, precision) {
var Big = x.constructor;
Big.config({precision: precision + Math.log(precision) | 0 + 4});
var one = new Big(1);
var y = x;
var yPrev = NaN;
var x2 = x.times(x);
var polyNum = x;
var constNum = new Big(one);
var constDen = new Big(one);
var bigK = new Big(one);
for (var k = 3; !y.equals(yPrev); k += 2) {
polyNum = polyNum.times(x2);
constNum = constNum.times(bigK);
constDen = constDen.times(bigK.plus(one));
yPrev = y;
bigK = new Big(k);
y = y.plus(polyNum.times(constNum).div(bigK.times(constDen)));
}
Big.config({precision: precision});
return y.toDP(precision - 1);
}
/**
* Calculate the arc tangent of x using a Taylor expansion
*
* arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + x^9/9 - ...
* = x - x^2*x^1/3 + x^2*x^3/5 - x^2*x^5/7 + x^2*x^7/9 - ...
*
* @param {BigNumber} x
* @returns {BigNumber} arc tangent of x
*/
function arctan_taylor(x) {
var y = x;
var yPrev = NaN;
var x2 = x.times(x);
var num = x;
var add = true;
for (var k = 3; !y.equals(yPrev); k += 2) {
num = num.times(x2);
yPrev = y;
add = !add;
y = (add) ? y.plus(num.div(k)) : y.minus(num.div(k));
}
return y;
}
/**
* Calculate the cosine or sine of x using Taylor Series.
*
* cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ...
* = 1 - 1*x^2/2! + x^2*x^2/4! - x^2*x^4/6! + x^2*x^6/8! - ...
*
* sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ...
* = x - x^2*x^1/3! + x^2*x^3/5! - x^2*x^5/7! + x^2*x^7/9! - ...
*
* @param {BigNumber} x reduced argument
* @param {Number} mode sine function if 1, cosine function if 0
* @returns {BigNumber} sine or cosine of x
*/
function cos_sin_taylor(x, mode) {
var one = x.constructor.ONE;
var y = x;
var yPrev = NaN;
var x2 = x.times(x);
var num = (mode) ? y : y = one;
var den = one;
var add = true;
for (var k = mode; !y.equals(yPrev); k += 2) {
num = num.times(x2);
den = den.times(k+1).times(k+2);
yPrev = y;
add = !add;
y = (add) ? y.plus(num.div(den)) : y.minus(num.div(den));
}
return y;
}
/**
* Reduce x within a period of pi (0, pi] with guard digits.
*
* @param {BigNumber} x
* @param {Number} precision
* @param {Number} mode
* @returns {Array} [Reduced x, is tau multiple?]
*/
function reduceToPeriod(x, precision, mode) {
var pi = exports.pi(precision + 2);
var tau = exports.tau(precision);
if (x.abs().lte(pi.toDP(x.dp()))) {
return [x, false];
}
var Big = x.constructor;
// Catch if input is tau multiple using pi's precision
if (x.div(pi.toDP(x.dp())).toNumber() % 2 == 0) {
return [new Big(mode ^ 1), true];
}
var y = x.mod(tau);
// Catch if tau multiple with tau's precision
if (y.toDP(x.dp(), 1).isZero()) {
return [new Big(mode ^ 1), true];
}
if (y.gt(pi)) {
if (mode) {
// sin(x + pi) = -sin(x)
y = y.minus(pi);
y.s = -y.s;
} else {
// cos(x) = cos(tau - x)
y = tau.minus(y);
}
}
y.constructor = Big;
return [y, false];
}
/**
* Convert from sine to cosine
*
* |cos(x)| = sqrt(1 - sin(x)^2)
*
* @param {BigNumber} sine of x
* @returns {BigNumber} sine as cosine
*/
function sinToCos(sinVal) {
var Big = sinVal.constructor;
var precision = Big.precision;
Big.config({precision: precision + 2});
var ret = Big.ONE.minus(sinVal.times(sinVal)).sqrt();
Big.config({precision: precision});
return ret.toDP(precision - 1);
}
/************************************
* Format functions *
************************************/
/**
* Convert a number to a formatted string representation.
*
* Syntax:
*
* format(value)
* format(value, options)
* format(value, precision)
* format(value, fn)
*
* Where:
*
* {Number} value The value to be formatted
* {Object} options An object with formatting options. Available options:
* {String} notation
* Number notation. Choose from:
* 'fixed' Always use regular number notation.
* For example '123.40' and '14000000'
* 'exponential' Always use exponential notation.
* For example '1.234e+2' and '1.4e+7'
* 'auto' (default) Regular number notation for numbers
* having an absolute value between
* `lower` and `upper` bounds, and uses
* exponential notation elsewhere.
* Lower bound is included, upper bound
* is excluded.
* For example '123.4' and '1.4e7'.
* {Number} precision A number between 0 and 16 to round
* the digits of the number.
* In case of notations 'exponential' and
* 'auto', `precision` defines the total
* number of significant digits returned
* and is undefined by default.
* In case of notation 'fixed',
* `precision` defines the number of
* significant digits after the decimal
* point, and is 0 by default.
* {Object} exponential An object containing two parameters,
* {Number} lower and {Number} upper,
* used by notation 'auto' to determine
* when to return exponential notation.
* Default values are `lower=1e-3` and
* `upper=1e5`.
* Only applicable for notation `auto`.
* {Function} fn A custom formatting function. Can be used to override the
* built-in notations. Function `fn` is called with `value` as
* parameter and must return a string. Is useful for example to
* format all values inside a matrix in a particular way.
*
* Examples:
*
* format(6.4); // '6.4'
* format(1240000); // '1.24e6'
* format(1/3); // '0.3333333333333333'
* format(1/3, 3); // '0.333'
* format(21385, 2); // '21000'
* format(12.071, {notation: 'fixed'}); // '12'
* format(2.3, {notation: 'fixed', precision: 2}); // '2.30'
* format(52.8, {notation: 'exponential'}); // '5.28e+1'
*
* @param {BigNumber} value
* @param {Object | Function | Number} [options]
* @return {String} str The formatted value
*/
exports.format = function(value, options) {
if (typeof options === 'function') {
// handle format(value, fn)
return options(value);
}
// handle special cases
if (!value.isFinite()) {
return value.isNaN() ? 'NaN' : (value.gt(0) ? 'Infinity' : '-Infinity');
}
// default values for options
var notation = 'auto';
var precision = undefined;
if (options !== undefined) {
// determine notation from options
if (options.notation) {
notation = options.notation;
}
// determine precision from options
if (isNumber(options)) {
precision = options;
}
else if (options.precision) {
precision = options.precision;
}
}
// handle the various notations
switch (notation) {
case 'fixed':
return exports.toFixed(value, precision);
case 'exponential':
return exports.toExponential(value, precision);
case 'auto':
// determine lower and upper bound for exponential notation.
// TODO: implement support for upper and lower to be BigNumbers themselves
var lower = 1e-3;
var upper = 1e5;
if (options && options.exponential) {
if (options.exponential.lower !== undefined) {
lower = options.exponential.lower;
}
if (options.exponential.upper !== undefined) {
upper = options.exponential.upper;
}
}
// adjust the configuration of the BigNumber constructor (yeah, this is quite tricky...)
var oldConfig = {
toExpNeg: value.constructor.toExpNeg,
toExpPos: value.constructor.toExpPos
};
value.constructor.config({
toExpNeg: Math.round(Math.log(lower) / Math.LN10),
toExpPos: Math.round(Math.log(upper) / Math.LN10)
});
// handle special case zero
if (value.isZero()) return '0';
// determine whether or not to output exponential notation
var str;
var abs = value.abs();
if (abs.gte(lower) && abs.lt(upper)) {
// normal number notation
str = value.toSignificantDigits(precision).toFixed();
}
else {
// exponential notation
str = exports.toExponential(value, precision);
}
// remove trailing zeros after the decimal point
return str.replace(/((\.\d*?)(0+))($|e)/, function () {
var digits = arguments[2];
var e = arguments[4];
return (digits !== '.') ? digits + e : e;
});
default:
throw new Error('Unknown notation "' + notation + '". ' +
'Choose "auto", "exponential", or "fixed".');
}
};
/**
* Format a number in exponential notation. Like '1.23e+5', '2.3e+0', '3.500e-3'
* @param {BigNumber} value
* @param {Number} [precision] Number of digits in formatted output.
* If not provided, the maximum available digits
* is used.
* @returns {string} str
*/
exports.toExponential = function(value, precision) {
if (precision !== undefined) {
return value.toExponential(precision - 1); // Note the offset of one
}
else {
return value.toExponential();
}
};
/**
* Format a number with fixed notation.
* @param {BigNumber} value
* @param {Number} [precision=0] Optional number of decimals after the
* decimal point. Zero by default.
*/
exports.toFixed = function(value, precision) {
return value.toFixed(precision || 0);
// Note: the (precision || 0) is needed as the toFixed of BigNumber has an
// undefined default precision instead of 0.
};