/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.dfp;
/** Mathematical routines for use with {@link Dfp}.
* The constants are defined in {@link DfpField}
* @since 2.2
*/
public class DfpMath {
/** Name for traps triggered by pow. */
private static final String POW_TRAP = "pow";
/**
* Private Constructor.
*/
private DfpMath() {
}
/** Breaks a string representation up into two dfp's.
* <p>The two dfp are such that the sum of them is equivalent
* to the input string, but has higher precision than using a
* single dfp. This is useful for improving accuracy of
* exponentiation and critical multiplies.
* @param field field to which the Dfp must belong
* @param a string representation to split
* @return an array of two {@link Dfp} which sum is a
*/
protected static Dfp[] split(final DfpField field, final String a) {
Dfp result[] = new Dfp[2];
char[] buf;
boolean leading = true;
int sp = 0;
int sig = 0;
buf = new char[a.length()];
for (int i = 0; i < buf.length; i++) {
buf[i] = a.charAt(i);
if (buf[i] >= '1' && buf[i] <= '9') {
leading = false;
}
if (buf[i] == '.') {
sig += (400 - sig) % 4;
leading = false;
}
if (sig == (field.getRadixDigits() / 2) * 4) {
sp = i;
break;
}
if (buf[i] >= '0' && buf[i] <= '9' && !leading) {
sig ++;
}
}
result[0] = field.newDfp(new String(buf, 0, sp));
for (int i = 0; i < buf.length; i++) {
buf[i] = a.charAt(i);
if (buf[i] >= '0' && buf[i] <= '9' && i < sp) {
buf[i] = '0';
}
}
result[1] = field.newDfp(new String(buf));
return result;
}
/** Splits a {@link Dfp} into 2 {@link Dfp}'s such that their sum is equal to the input {@link Dfp}.
* @param a number to split
* @return two elements array containing the split number
*/
protected static Dfp[] split(final Dfp a) {
final Dfp[] result = new Dfp[2];
final Dfp shift = a.multiply(a.power10K(a.getRadixDigits() / 2));
result[0] = a.add(shift).subtract(shift);
result[1] = a.subtract(result[0]);
return result;
}
/** Multiply two numbers that are split in to two pieces that are
* meant to be added together.
* Use binomial multiplication so ab = a0 b0 + a0 b1 + a1 b0 + a1 b1
* Store the first term in result0, the rest in result1
* @param a first factor of the multiplication, in split form
* @param b second factor of the multiplication, in split form
* @return a × b, in split form
*/
protected static Dfp[] splitMult(final Dfp[] a, final Dfp[] b) {
final Dfp[] result = new Dfp[2];
result[1] = a[0].getZero();
result[0] = a[0].multiply(b[0]);
/* If result[0] is infinite or zero, don't compute result[1].
* Attempting to do so may produce NaNs.
*/
if (result[0].classify() == Dfp.INFINITE || result[0].equals(result[1])) {
return result;
}
result[1] = a[0].multiply(b[1]).add(a[1].multiply(b[0])).add(a[1].multiply(b[1]));
return result;
}
/** Divide two numbers that are split in to two pieces that are meant to be added together.
* Inverse of split multiply above:
* (a+b) / (c+d) = (a/c) + ( (bc-ad)/(c**2+cd) )
* @param a dividend, in split form
* @param b divisor, in split form
* @return a / b, in split form
*/
protected static Dfp[] splitDiv(final Dfp[] a, final Dfp[] b) {
final Dfp[] result;
result = new Dfp[2];
result[0] = a[0].divide(b[0]);
result[1] = a[1].multiply(b[0]).subtract(a[0].multiply(b[1]));
result[1] = result[1].divide(b[0].multiply(b[0]).add(b[0].multiply(b[1])));
return result;
}
/** Raise a split base to the a power.
* @param base number to raise
* @param a power
* @return base<sup>a</sup>
*/
protected static Dfp splitPow(final Dfp[] base, int a) {
boolean invert = false;
Dfp[] r = new Dfp[2];
Dfp[] result = new Dfp[2];
result[0] = base[0].getOne();
result[1] = base[0].getZero();
if (a == 0) {
// Special case a = 0
return result[0].add(result[1]);
}
if (a < 0) {
// If a is less than zero
invert = true;
a = -a;
}
// Exponentiate by successive squaring
do {
r[0] = new Dfp(base[0]);
r[1] = new Dfp(base[1]);
int trial = 1;
int prevtrial;
while (true) {
prevtrial = trial;
trial *= 2;
if (trial > a) {
break;
}
r = splitMult(r, r);
}
trial = prevtrial;
a -= trial;
result = splitMult(result, r);
} while (a >= 1);
result[0] = result[0].add(result[1]);
if (invert) {
result[0] = base[0].getOne().divide(result[0]);
}
return result[0];
}
/** Raises base to the power a by successive squaring.
* @param base number to raise
* @param a power
* @return base<sup>a</sup>
*/
public static Dfp pow(Dfp base, int a)
{
boolean invert = false;
Dfp result = base.getOne();
if (a == 0) {
// Special case
return result;
}
if (a < 0) {
invert = true;
a = -a;
}
// Exponentiate by successive squaring
do {
Dfp r = new Dfp(base);
Dfp prevr;
int trial = 1;
int prevtrial;
do {
prevr = new Dfp(r);
prevtrial = trial;
r = r.multiply(r);
trial *= 2;
} while (a>trial);
r = prevr;
trial = prevtrial;
a -= trial;
result = result.multiply(r);
} while (a >= 1);
if (invert) {
result = base.getOne().divide(result);
}
return base.newInstance(result);
}
/** Computes e to the given power.
* a is broken into two parts, such that a = n+m where n is an integer.
* We use pow() to compute e<sup>n</sup> and a Taylor series to compute
* e<sup>m</sup>. We return e*<sup>n</sup> × e<sup>m</sup>
* @param a power at which e should be raised
* @return e<sup>a</sup>
*/
public static Dfp exp(final Dfp a) {
final Dfp inta = a.rint();
final Dfp fraca = a.subtract(inta);
final int ia = inta.intValue();
if (ia > 2147483646) {
// return +Infinity
return a.newInstance((byte)1, Dfp.INFINITE);
}
if (ia < -2147483646) {
// return 0;
return a.newInstance();
}
final Dfp einta = splitPow(a.getField().getESplit(), ia);
final Dfp efraca = expInternal(fraca);
return einta.multiply(efraca);
}
/** Computes e to the given power.
* Where -1 < a < 1. Use the classic Taylor series. 1 + x**2/2! + x**3/3! + x**4/4! ...
* @param a power at which e should be raised
* @return e<sup>a</sup>
*/
protected static Dfp expInternal(final Dfp a) {
Dfp y = a.getOne();
Dfp x = a.getOne();
Dfp fact = a.getOne();
Dfp py = new Dfp(y);
for (int i = 1; i < 90; i++) {
x = x.multiply(a);
fact = fact.divide(i);
y = y.add(x.multiply(fact));
if (y.equals(py)) {
break;
}
py = new Dfp(y);
}
return y;
}
/** Returns the natural logarithm of a.
* a is first split into three parts such that a = (10000^h)(2^j)k.
* ln(a) is computed by ln(a) = ln(5)*h + ln(2)*(h+j) + ln(k)
* k is in the range 2/3 < k <4/3 and is passed on to a series expansion.
* @param a number from which logarithm is requested
* @return log(a)
*/
public static Dfp log(Dfp a) {
int lr;
Dfp x;
int ix;
int p2 = 0;
// Check the arguments somewhat here
if (a.equals(a.getZero()) || a.lessThan(a.getZero()) || a.isNaN()) {
// negative, zero or NaN
a.getField().setIEEEFlagsBits(DfpField.FLAG_INVALID);
return a.dotrap(DfpField.FLAG_INVALID, "ln", a, a.newInstance((byte)1, Dfp.QNAN));
}
if (a.classify() == Dfp.INFINITE) {
return a;
}
x = new Dfp(a);
lr = x.log10K();
x = x.divide(pow(a.newInstance(10000), lr)); /* This puts x in the range 0-10000 */
ix = x.floor().intValue();
while (ix > 2) {
ix >>= 1;
p2++;
}
Dfp[] spx = split(x);
Dfp[] spy = new Dfp[2];
spy[0] = pow(a.getTwo(), p2); // use spy[0] temporarily as a divisor
spx[0] = spx[0].divide(spy[0]);
spx[1] = spx[1].divide(spy[0]);
spy[0] = a.newInstance("1.33333"); // Use spy[0] for comparison
while (spx[0].add(spx[1]).greaterThan(spy[0])) {
spx[0] = spx[0].divide(2);
spx[1] = spx[1].divide(2);
p2++;
}
// X is now in the range of 2/3 < x < 4/3
Dfp[] spz = logInternal(spx);
spx[0] = a.newInstance(new StringBuilder().append(p2+4*lr).toString());
spx[1] = a.getZero();
spy = splitMult(a.getField().getLn2Split(), spx);
spz[0] = spz[0].add(spy[0]);
spz[1] = spz[1].add(spy[1]);
spx[0] = a.newInstance(new StringBuilder().append(4*lr).toString());
spx[1] = a.getZero();
spy = splitMult(a.getField().getLn5Split(), spx);
spz[0] = spz[0].add(spy[0]);
spz[1] = spz[1].add(spy[1]);
return a.newInstance(spz[0].add(spz[1]));
}
/** Computes the natural log of a number between 0 and 2.
* Let f(x) = ln(x),
*
* We know that f'(x) = 1/x, thus from Taylor's theorum we have:
*
* ----- n+1 n
* f(x) = \ (-1) (x - 1)
* / ---------------- for 1 <= n <= infinity
* ----- n
*
* or
* 2 3 4
* (x-1) (x-1) (x-1)
* ln(x) = (x-1) - ----- + ------ - ------ + ...
* 2 3 4
*
* alternatively,
*
* 2 3 4
* x x x
* ln(x+1) = x - - + - - - + ...
* 2 3 4
*
* This series can be used to compute ln(x), but it converges too slowly.
*
* If we substitute -x for x above, we get
*
* 2 3 4
* x x x
* ln(1-x) = -x - - - - - - + ...
* 2 3 4
*
* Note that all terms are now negative. Because the even powered ones
* absorbed the sign. Now, subtract the series above from the previous
* one to get ln(x+1) - ln(1-x). Note the even terms cancel out leaving
* only the odd ones
*
* 3 5 7
* 2x 2x 2x
* ln(x+1) - ln(x-1) = 2x + --- + --- + ---- + ...
* 3 5 7
*
* By the property of logarithms that ln(a) - ln(b) = ln (a/b) we have:
*
* 3 5 7
* x+1 / x x x \
* ln ----- = 2 * | x + ---- + ---- + ---- + ... |
* x-1 \ 3 5 7 /
*
* But now we want to find ln(a), so we need to find the value of x
* such that a = (x+1)/(x-1). This is easily solved to find that
* x = (a-1)/(a+1).
* @param a number from which logarithm is requested, in split form
* @return log(a)
*/
protected static Dfp[] logInternal(final Dfp a[]) {
/* Now we want to compute x = (a-1)/(a+1) but this is prone to
* loss of precision. So instead, compute x = (a/4 - 1/4) / (a/4 + 1/4)
*/
Dfp t = a[0].divide(4).add(a[1].divide(4));
Dfp x = t.add(a[0].newInstance("-0.25")).divide(t.add(a[0].newInstance("0.25")));
Dfp y = new Dfp(x);
Dfp num = new Dfp(x);
Dfp py = new Dfp(y);
int den = 1;
for (int i = 0; i < 10000; i++) {
num = num.multiply(x);
num = num.multiply(x);
den += 2;
t = num.divide(den);
y = y.add(t);
if (y.equals(py)) {
break;
}
py = new Dfp(y);
}
y = y.multiply(a[0].getTwo());
return split(y);
}
/** Computes x to the y power.<p>
*
* Uses the following method:<p>
*
* <ol>
* <li> Set u = rint(y), v = y-u
* <li> Compute a = v * ln(x)
* <li> Compute b = rint( a/ln(2) )
* <li> Compute c = a - b*ln(2)
* <li> x<sup>y</sup> = x<sup>u</sup> * 2<sup>b</sup> * e<sup>c</sup>
* </ol>
* if |y| > 1e8, then we compute by exp(y*ln(x)) <p>
*
* <b>Special Cases</b><p>
* <ul>
* <li> if y is 0.0 or -0.0 then result is 1.0
* <li> if y is 1.0 then result is x
* <li> if y is NaN then result is NaN
* <li> if x is NaN and y is not zero then result is NaN
* <li> if |x| > 1.0 and y is +Infinity then result is +Infinity
* <li> if |x| < 1.0 and y is -Infinity then result is +Infinity
* <li> if |x| > 1.0 and y is -Infinity then result is +0
* <li> if |x| < 1.0 and y is +Infinity then result is +0
* <li> if |x| = 1.0 and y is +/-Infinity then result is NaN
* <li> if x = +0 and y > 0 then result is +0
* <li> if x = +Inf and y < 0 then result is +0
* <li> if x = +0 and y < 0 then result is +Inf
* <li> if x = +Inf and y > 0 then result is +Inf
* <li> if x = -0 and y > 0, finite, not odd integer then result is +0
* <li> if x = -0 and y < 0, finite, and odd integer then result is -Inf
* <li> if x = -Inf and y > 0, finite, and odd integer then result is -Inf
* <li> if x = -0 and y < 0, not finite odd integer then result is +Inf
* <li> if x = -Inf and y > 0, not finite odd integer then result is +Inf
* <li> if x < 0 and y > 0, finite, and odd integer then result is -(|x|<sup>y</sup>)
* <li> if x < 0 and y > 0, finite, and not integer then result is NaN
* </ul>
* @param x base to be raised
* @param y power to which base should be raised
* @return x<sup>y</sup>
*/
public static Dfp pow(Dfp x, final Dfp y) {
// make sure we don't mix number with different precision
if (x.getField().getRadixDigits() != y.getField().getRadixDigits()) {
x.getField().setIEEEFlagsBits(DfpField.FLAG_INVALID);
final Dfp result = x.newInstance(x.getZero());
result.nans = Dfp.QNAN;
return x.dotrap(DfpField.FLAG_INVALID, POW_TRAP, x, result);
}
final Dfp zero = x.getZero();
final Dfp one = x.getOne();
final Dfp two = x.getTwo();
boolean invert = false;
int ui;
/* Check for special cases */
if (y.equals(zero)) {
return x.newInstance(one);
}
if (y.equals(one)) {
if (x.isNaN()) {
// Test for NaNs
x.getField().setIEEEFlagsBits(DfpField.FLAG_INVALID);
return x.dotrap(DfpField.FLAG_INVALID, POW_TRAP, x, x);
}
return x;
}
if (x.isNaN() || y.isNaN()) {
// Test for NaNs
x.getField().setIEEEFlagsBits(DfpField.FLAG_INVALID);
return x.dotrap(DfpField.FLAG_INVALID, POW_TRAP, x, x.newInstance((byte)1, Dfp.QNAN));
}
// X == 0
if (x.equals(zero)) {
if (Dfp.copysign(one, x).greaterThan(zero)) {
// X == +0
if (y.greaterThan(zero)) {
return x.newInstance(zero);
} else {
return x.newInstance(x.newInstance((byte)1, Dfp.INFINITE));
}
} else {
// X == -0
if (y.classify() == Dfp.FINITE && y.rint().equals(y) && !y.remainder(two).equals(zero)) {
// If y is odd integer
if (y.greaterThan(zero)) {
return x.newInstance(zero.negate());
} else {
return x.newInstance(x.newInstance((byte)-1, Dfp.INFINITE));
}
} else {
// Y is not odd integer
if (y.greaterThan(zero)) {
return x.newInstance(zero);
} else {
return x.newInstance(x.newInstance((byte)1, Dfp.INFINITE));
}
}
}
}
if (x.lessThan(zero)) {
// Make x positive, but keep track of it
x = x.negate();
invert = true;
}
if (x.greaterThan(one) && y.classify() == Dfp.INFINITE) {
if (y.greaterThan(zero)) {
return y;
} else {
return x.newInstance(zero);
}
}
if (x.lessThan(one) && y.classify() == Dfp.INFINITE) {
if (y.greaterThan(zero)) {
return x.newInstance(zero);
} else {
return x.newInstance(Dfp.copysign(y, one));
}
}
if (x.equals(one) && y.classify() == Dfp.INFINITE) {
x.getField().setIEEEFlagsBits(DfpField.FLAG_INVALID);
return x.dotrap(DfpField.FLAG_INVALID, POW_TRAP, x, x.newInstance((byte)1, Dfp.QNAN));
}
if (x.classify() == Dfp.INFINITE) {
// x = +/- inf
if (invert) {
// negative infinity
if (y.classify() == Dfp.FINITE && y.rint().equals(y) && !y.remainder(two).equals(zero)) {
// If y is odd integer
if (y.greaterThan(zero)) {
return x.newInstance(x.newInstance((byte)-1, Dfp.INFINITE));
} else {
return x.newInstance(zero.negate());
}
} else {
// Y is not odd integer
if (y.greaterThan(zero)) {
return x.newInstance(x.newInstance((byte)1, Dfp.INFINITE));
} else {
return x.newInstance(zero);
}
}
} else {
// positive infinity
if (y.greaterThan(zero)) {
return x;
} else {
return x.newInstance(zero);
}
}
}
if (invert && !y.rint().equals(y)) {
x.getField().setIEEEFlagsBits(DfpField.FLAG_INVALID);
return x.dotrap(DfpField.FLAG_INVALID, POW_TRAP, x, x.newInstance((byte)1, Dfp.QNAN));
}
// End special cases
Dfp r;
if (y.lessThan(x.newInstance(100000000)) && y.greaterThan(x.newInstance(-100000000))) {
final Dfp u = y.rint();
ui = u.intValue();
final Dfp v = y.subtract(u);
if (v.unequal(zero)) {
final Dfp a = v.multiply(log(x));
final Dfp b = a.divide(x.getField().getLn2()).rint();
final Dfp c = a.subtract(b.multiply(x.getField().getLn2()));
r = splitPow(split(x), ui);
r = r.multiply(pow(two, b.intValue()));
r = r.multiply(exp(c));
} else {
r = splitPow(split(x), ui);
}
} else {
// very large exponent. |y| > 1e8
r = exp(log(x).multiply(y));
}
if (invert && y.rint().equals(y) && !y.remainder(two).equals(zero)) {
// if y is odd integer
r = r.negate();
}
return x.newInstance(r);
}
/** Computes sin(a) Used when 0 < a < pi/4.
* Uses the classic Taylor series. x - x**3/3! + x**5/5! ...
* @param a number from which sine is desired, in split form
* @return sin(a)
*/
protected static Dfp sinInternal(Dfp a[]) {
Dfp c = a[0].add(a[1]);
Dfp y = c;
c = c.multiply(c);
Dfp x = y;
Dfp fact = a[0].getOne();
Dfp py = new Dfp(y);
for (int i = 3; i < 90; i += 2) {
x = x.multiply(c);
x = x.negate();
fact = fact.divide((i-1)*i); // 1 over fact
y = y.add(x.multiply(fact));
if (y.equals(py)) {
break;
}
py = new Dfp(y);
}
return y;
}
/** Computes cos(a) Used when 0 < a < pi/4.
* Uses the classic Taylor series for cosine. 1 - x**2/2! + x**4/4! ...
* @param a number from which cosine is desired, in split form
* @return cos(a)
*/
protected static Dfp cosInternal(Dfp a[]) {
final Dfp one = a[0].getOne();
Dfp x = one;
Dfp y = one;
Dfp c = a[0].add(a[1]);
c = c.multiply(c);
Dfp fact = one;
Dfp py = new Dfp(y);
for (int i = 2; i < 90; i += 2) {
x = x.multiply(c);
x = x.negate();
fact = fact.divide((i - 1) * i); // 1 over fact
y = y.add(x.multiply(fact));
if (y.equals(py)) {
break;
}
py = new Dfp(y);
}
return y;
}
/** computes the sine of the argument.
* @param a number from which sine is desired
* @return sin(a)
*/
public static Dfp sin(final Dfp a) {
final Dfp pi = a.getField().getPi();
final Dfp zero = a.getField().getZero();
boolean neg = false;
/* First reduce the argument to the range of +/- PI */
Dfp x = a.remainder(pi.multiply(2));
/* if x < 0 then apply identity sin(-x) = -sin(x) */
/* This puts x in the range 0 < x < PI */
if (x.lessThan(zero)) {
x = x.negate();
neg = true;
}
/* Since sine(x) = sine(pi - x) we can reduce the range to
* 0 < x < pi/2
*/
if (x.greaterThan(pi.divide(2))) {
x = pi.subtract(x);
}
Dfp y;
if (x.lessThan(pi.divide(4))) {
y = sinInternal(split(x));
} else {
final Dfp c[] = new Dfp[2];
final Dfp[] piSplit = a.getField().getPiSplit();
c[0] = piSplit[0].divide(2).subtract(x);
c[1] = piSplit[1].divide(2);
y = cosInternal(c);
}
if (neg) {
y = y.negate();
}
return a.newInstance(y);
}
/** computes the cosine of the argument.
* @param a number from which cosine is desired
* @return cos(a)
*/
public static Dfp cos(Dfp a) {
final Dfp pi = a.getField().getPi();
final Dfp zero = a.getField().getZero();
boolean neg = false;
/* First reduce the argument to the range of +/- PI */
Dfp x = a.remainder(pi.multiply(2));
/* if x < 0 then apply identity cos(-x) = cos(x) */
/* This puts x in the range 0 < x < PI */
if (x.lessThan(zero)) {
x = x.negate();
}
/* Since cos(x) = -cos(pi - x) we can reduce the range to
* 0 < x < pi/2
*/
if (x.greaterThan(pi.divide(2))) {
x = pi.subtract(x);
neg = true;
}
Dfp y;
if (x.lessThan(pi.divide(4))) {
Dfp c[] = new Dfp[2];
c[0] = x;
c[1] = zero;
y = cosInternal(c);
} else {
final Dfp c[] = new Dfp[2];
final Dfp[] piSplit = a.getField().getPiSplit();
c[0] = piSplit[0].divide(2).subtract(x);
c[1] = piSplit[1].divide(2);
y = sinInternal(c);
}
if (neg) {
y = y.negate();
}
return a.newInstance(y);
}
/** computes the tangent of the argument.
* @param a number from which tangent is desired
* @return tan(a)
*/
public static Dfp tan(final Dfp a) {
return sin(a).divide(cos(a));
}
/** computes the arc-tangent of the argument.
* @param a number from which arc-tangent is desired
* @return atan(a)
*/
protected static Dfp atanInternal(final Dfp a) {
Dfp y = new Dfp(a);
Dfp x = new Dfp(y);
Dfp py = new Dfp(y);
for (int i = 3; i < 90; i += 2) {
x = x.multiply(a);
x = x.multiply(a);
x = x.negate();
y = y.add(x.divide(i));
if (y.equals(py)) {
break;
}
py = new Dfp(y);
}
return y;
}
/** computes the arc tangent of the argument
*
* Uses the typical taylor series
*
* but may reduce arguments using the following identity
* tan(x+y) = (tan(x) + tan(y)) / (1 - tan(x)*tan(y))
*
* since tan(PI/8) = sqrt(2)-1,
*
* atan(x) = atan( (x - sqrt(2) + 1) / (1+x*sqrt(2) - x) + PI/8.0
* @param a number from which arc-tangent is desired
* @return atan(a)
*/
public static Dfp atan(final Dfp a) {
final Dfp zero = a.getField().getZero();
final Dfp one = a.getField().getOne();
final Dfp[] sqr2Split = a.getField().getSqr2Split();
final Dfp[] piSplit = a.getField().getPiSplit();
boolean recp = false;
boolean neg = false;
boolean sub = false;
final Dfp ty = sqr2Split[0].subtract(one).add(sqr2Split[1]);
Dfp x = new Dfp(a);
if (x.lessThan(zero)) {
neg = true;
x = x.negate();
}
if (x.greaterThan(one)) {
recp = true;
x = one.divide(x);
}
if (x.greaterThan(ty)) {
Dfp sty[] = new Dfp[2];
sub = true;
sty[0] = sqr2Split[0].subtract(one);
sty[1] = sqr2Split[1];
Dfp[] xs = split(x);
Dfp[] ds = splitMult(xs, sty);
ds[0] = ds[0].add(one);
xs[0] = xs[0].subtract(sty[0]);
xs[1] = xs[1].subtract(sty[1]);
xs = splitDiv(xs, ds);
x = xs[0].add(xs[1]);
//x = x.subtract(ty).divide(dfp.one.add(x.multiply(ty)));
}
Dfp y = atanInternal(x);
if (sub) {
y = y.add(piSplit[0].divide(8)).add(piSplit[1].divide(8));
}
if (recp) {
y = piSplit[0].divide(2).subtract(y).add(piSplit[1].divide(2));
}
if (neg) {
y = y.negate();
}
return a.newInstance(y);
}
/** computes the arc-sine of the argument.
* @param a number from which arc-sine is desired
* @return asin(a)
*/
public static Dfp asin(final Dfp a) {
return atan(a.divide(a.getOne().subtract(a.multiply(a)).sqrt()));
}
/** computes the arc-cosine of the argument.
* @param a number from which arc-cosine is desired
* @return acos(a)
*/
public static Dfp acos(Dfp a) {
Dfp result;
boolean negative = false;
if (a.lessThan(a.getZero())) {
negative = true;
}
a = Dfp.copysign(a, a.getOne()); // absolute value
result = atan(a.getOne().subtract(a.multiply(a)).sqrt().divide(a));
if (negative) {
result = a.getField().getPi().subtract(result);
}
return a.newInstance(result);
}
}