trigonometry.cpp
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#include <poincare/trigonometry.h>
#include <poincare/hyperbolic_cosine.h>
#include <poincare/symbol.h>
#include <poincare/preferences.h>
#include <poincare/undefined.h>
#include <poincare/rational.h>
#include <poincare/multiplication.h>
#include <poincare/subtraction.h>
#include <poincare/derivative.h>
#include <poincare/decimal.h>
#include <ion.h>
extern "C" {
#include <assert.h>
}
#include <cmath>
#include <float.h>
namespace Poincare {
float Trigonometry::characteristicXRange(const Expression * e, Context & context, Expression::AngleUnit angleUnit) {
assert(e->numberOfOperands() == 1);
const Expression * op = e->operand(0);
int d = op->polynomialDegree('x');
// op is not linear so we cannot not easily find an interesting range
if (d < 0 || d > 1) {
return op->characteristicXRange(context, angleUnit);
}
// The expression e is x-independent
if (d == 0) {
return 0.0f;
}
// e has the form cos/sin/tan(ax+b) so it is periodic of period 2*Pi/a
assert(d == 1);
/* To compute a, the slope of the expression op, we compute the derivative of
* op for any x value. */
Poincare::Approximation<float> x(1.0f);
const Poincare::Expression * args[2] = {op, &x};
Poincare::Derivative derivative(args, true);
float a = derivative.approximateToScalar<float>(context, angleUnit);
float pi = angleUnit == Expression::AngleUnit::Radian ? M_PI : 180.0f;
return 2.0f*pi/std::fabs(a);
}
Expression * Trigonometry::shallowReduceDirectFunction(Expression * e, Context& context, Expression::AngleUnit angleUnit) {
assert(e->type() == Expression::Type::Sine || e->type() == Expression::Type::Cosine || e->type() == Expression::Type::Tangent);
Expression * lookup = Trigonometry::table(e->operand(0), e->type(), context, angleUnit);
if (lookup != nullptr) {
return e->replaceWith(lookup, true);
}
Expression::Type correspondingType = e->type() == Expression::Type::Cosine ? Expression::Type::ArcCosine : (e->type() == Expression::Type::Sine ? Expression::Type::ArcSine : Expression::Type::ArcTangent);
if (e->operand(0)->type() == correspondingType) {
return e->replaceWith(e->editableOperand(0)->editableOperand(0), true);
}
if (e->operand(0)->sign() == Expression::Sign::Negative) {
Expression * op = e->editableOperand(0);
Expression * newOp = op->setSign(Expression::Sign::Positive, context, angleUnit);
newOp->shallowReduce(context, angleUnit);
if (e->type() == Expression::Type::Cosine) {
return e->shallowReduce(context, angleUnit);
} else {
Multiplication * m = new Multiplication(new Rational(-1), e->clone(), false);
m->editableOperand(1)->shallowReduce(context, angleUnit);
return e->replaceWith(m, true)->shallowReduce(context, angleUnit);
}
}
if ((angleUnit == Expression::AngleUnit::Radian && e->operand(0)->type() == Expression::Type::Multiplication && e->operand(0)->numberOfOperands() == 2 && e->operand(0)->operand(1)->type() == Expression::Type::Symbol && static_cast<const Symbol *>(e->operand(0)->operand(1))->name() == Ion::Charset::SmallPi && e->operand(0)->operand(0)->type() == Expression::Type::Rational) || (angleUnit == Expression::AngleUnit::Degree && e->operand(0)->type() == Expression::Type::Rational)) {
Rational * r = angleUnit == Expression::AngleUnit::Radian ? static_cast<Rational *>(e->editableOperand(0)->editableOperand(0)) : static_cast<Rational *>(e->editableOperand(0));
int unaryCoefficient = 1; // store 1 or -1
// Replace argument in [0, Pi/2[ or [0, 90[
Integer divisor = angleUnit == Expression::AngleUnit::Radian ? r->denominator() : Integer::Multiplication(r->denominator(), Integer(90));
Integer dividand = angleUnit == Expression::AngleUnit::Radian ? Integer::Addition(r->numerator(), r->numerator()) : r->numerator();
if (divisor.isLowerThan(dividand)) {
Integer piDivisor = angleUnit == Expression::AngleUnit::Radian ? r->denominator() : Integer::Multiplication(r->denominator(), Integer(180));
IntegerDivision div = Integer::Division(r->numerator(), piDivisor);
dividand = angleUnit == Expression::AngleUnit::Radian ? Integer::Addition(div.remainder, div.remainder) : div.remainder;
if (divisor.isLowerThan(dividand)) {
div.remainder = Integer::Subtraction(piDivisor, div.remainder);
if (e->type() == Expression::Type::Cosine || e->type() == Expression::Type::Tangent) {
unaryCoefficient *= -1;
}
}
Rational * newR = new Rational(div.remainder, r->denominator());
Expression * rationalParent = angleUnit == Expression::AngleUnit::Radian ? e->editableOperand(0) : e;
rationalParent->replaceOperand(r, newR, true);
e->editableOperand(0)->shallowReduce(context, angleUnit);
if (Integer::Division(div.quotient, Integer(2)).remainder.isOne() && e->type() != Expression::Type::Tangent) {
unaryCoefficient *= -1;
}
Expression * simplifiedCosine = e->shallowReduce(context, angleUnit); // recursive
Multiplication * m = new Multiplication(new Rational(unaryCoefficient), simplifiedCosine->clone(), false);
return simplifiedCosine->replaceWith(m, true)->shallowReduce(context, angleUnit);
}
assert(r->sign() == Expression::Sign::Positive);
assert(!divisor.isLowerThan(dividand));
}
return e;
}
bool Trigonometry::ExpressionIsEquivalentToTangent(const Expression * e) {
assert(Expression::Type::Power < Expression::Type::Sine);
if (e->type() == Expression::Type::Multiplication && e->operand(1)->type() == Expression::Type::Sine && e->operand(0)->type() == Expression::Type::Power && e->operand(0)->operand(0)->type() == Expression::Type::Cosine && e->operand(0)->operand(1)->type() == Expression::Type::Rational && static_cast<const Rational *>(e->operand(0)->operand(1))->isMinusOne()) {
return true;
}
return false;
}
Expression * Trigonometry::shallowReduceInverseFunction(Expression * e, Context& context, Expression::AngleUnit angleUnit) {
assert(e->type() == Expression::Type::ArcCosine || e->type() == Expression::Type::ArcSine || e->type() == Expression::Type::ArcTangent);
Expression::Type correspondingType = e->type() == Expression::Type::ArcCosine ? Expression::Type::Cosine : (e->type() == Expression::Type::ArcSine ? Expression::Type::Sine : Expression::Type::Tangent);
float pi = angleUnit == Expression::AngleUnit::Radian ? M_PI : 180;
if (e->operand(0)->type() == correspondingType) {
float trigoOp = e->operand(0)->operand(0)->approximateToScalar<float>(context, angleUnit);
if ((e->type() == Expression::Type::ArcCosine && trigoOp >= 0.0f && trigoOp <= pi) ||
(e->type() == Expression::Type::ArcSine && trigoOp >= -pi/2.0f && trigoOp <= pi/2.0f) ||
(e->type() == Expression::Type::ArcTangent && trigoOp >= -pi/2.0f && trigoOp <= pi/2.0f)) {
return e->replaceWith(e->editableOperand(0)->editableOperand(0), true);
}
}
// Special case for arctan(sin(x)/cos(x))
if (e->type() == Expression::Type::ArcTangent && ExpressionIsEquivalentToTangent(e->operand(0))) {
float trigoOp = e->operand(0)->operand(1)->operand(0)->approximateToScalar<float>(context, angleUnit);
if (trigoOp >= -pi/2.0f && trigoOp <= pi/2.0f) {
return e->replaceWith(e->editableOperand(0)->editableOperand(1)->editableOperand(0), true);
}
}
Expression * lookup = Trigonometry::table(e->operand(0), e->type(), context, angleUnit);
if (lookup != nullptr) {
return e->replaceWith(lookup, true);
}
// arccos(-x) = Pi-arcos(x), arcsin(-x) = -arcsin(x), arctan(-x)=-arctan(x)
if (e->operand(0)->sign() == Expression::Sign::Negative || (e->operand(0)->type() == Expression::Type::Multiplication && e->operand(0)->operand(0)->type() == Expression::Type::Rational && static_cast<const Rational *>(e->operand(0)->operand(0))->isMinusOne())) {
Expression * op = e->editableOperand(0);
if (e->operand(0)->sign() == Expression::Sign::Negative) {
Expression * newOp = op->setSign(Expression::Sign::Positive, context, angleUnit);
newOp->shallowReduce(context, angleUnit);
} else {
((Multiplication *)op)->removeOperand(op->editableOperand(0), true);
op->shallowReduce(context, angleUnit);
}
if (e->type() == Expression::Type::ArcCosine) {
Expression * pi = angleUnit == Expression::AngleUnit::Radian ? static_cast<Expression *>(new Symbol(Ion::Charset::SmallPi)) : static_cast<Expression *>(new Rational(180));
Subtraction * s = new Subtraction(pi, e->clone(), false);
s->editableOperand(1)->shallowReduce(context, angleUnit);
return e->replaceWith(s, true)->shallowReduce(context, angleUnit);
} else {
Multiplication * m = new Multiplication(new Rational(-1), e->clone(), false);
m->editableOperand(1)->shallowReduce(context, angleUnit);
return e->replaceWith(m, true)->shallowReduce(context, angleUnit);
}
}
return e;
}
static_assert('\x8A' == Ion::Charset::SmallPi, "Unicode error");
constexpr const char * cheatTable[Trigonometry::k_numberOfEntries][5] =
{{"-90", "\x8A*(-2)^(-1)", "", "-1", "undef"},
{"-75", "\x8A*(-5)*12^(-1)", "", "(-1)*6^(1/2)*4^(-1)-2^(1/2)*4^(-1)", "-(3^(1/2)+2)"},
{"-72", "\x8A*2*(-5)^(-1)", "", "-(5/8+5^(1/2)/8)^(1/2)", "-(5+2*5^(1/2))^(1/2)"},
{"-135/2", "\x8A*(-3)*8^(-1)", "", "-(2+2^(1/2))^(1/2)*2^(-1)", "-1-2^(1/2)"},
{"-60", "\x8A*(-3)^(-1)", "", "-3^(1/2)*2^(-1)", "-3^(1/2)"},
{"-54", "\x8A*(-3)*10^(-1)", "", "4^(-1)*(-1-5^(1/2))", "-(1+2*5^(-1/2))^(1/2)"},
{"-45", "\x8A*(-4)^(-1)", "", "(-1)*(2^(-1/2))", "-1"},
{"-36", "\x8A*(-5)^(-1)", "", "-(5/8-5^(1/2)/8)^(1/2)", "-(5-2*5^(1/2))^(1/2)"},
{"-30", "\x8A*(-6)^(-1)", "", "-0.5", "-3^(-1/2)"},
{"-45/2", "\x8A*(-8)^(-1)", "", "(2-2^(1/2))^(1/2)*(-2)^(-1)", "1-2^(1/2)"},
{"-18", "\x8A*(-10)^(-1)", "", "4^(-1)*(1-5^(1/2))", "-(1-2*5^(-1/2))^(1/2)"},
{"-15", "\x8A*(-12)^(-1)", "", "-6^(1/2)*4^(-1)+2^(1/2)*4^(-1)", "3^(1/2)-2"},
{"0", "0", "1", "0", "0"},
{"15", "\x8A*12^(-1)", "6^(1/2)*4^(-1)+2^(1/2)*4^(-1)", "6^(1/2)*4^(-1)+2^(1/2)*(-4)^(-1)", "-(3^(1/2)-2)"},
{"18", "\x8A*10^(-1)", "(5/8+5^(1/2)/8)^(1/2)", "4^(-1)*(5^(1/2)-1)", "(1-2*5^(-1/2))^(1/2)"},
{"45/2", "\x8A*8^(-1)", "(2+2^(1/2))^(1/2)*2^(-1)", "(2-2^(1/2))^(1/2)*2^(-1)", "2^(1/2)-1"},
{"30", "\x8A*6^(-1)", "3^(1/2)*2^(-1)", "0.5", "3^(-1/2)"},
{"36", "\x8A*5^(-1)", "(5^(1/2)+1)*4^(-1)", "(5/8-5^(1/2)/8)^(1/2)", "(5-2*5^(1/2))^(1/2)"},
{"45", "\x8A*4^(-1)", "2^(-1/2)", "2^(-1/2)", "1"},
{"54", "\x8A*3*10^(-1)", "(5/8-5^(1/2)/8)^(1/2)", "4^(-1)*(5^(1/2)+1)", "(1+2*5^(-1/2))^(1/2)"},
{"60", "\x8A*3^(-1)", "0.5", "3^(1/2)*2^(-1)", "3^(1/2)"},
{"135/2", "\x8A*3*8^(-1)", "(2-2^(1/2))^(1/2)*2^(-1)", "(2+2^(1/2))^(1/2)*2^(-1)", "1+2^(1/2)"},
{"72", "\x8A*2*5^(-1)", "(5^(1/2)-1)*4^(-1)", "(5/8+5^(1/2)/8)^(1/2)", "(5+2*5^(1/2))^(1/2)"},
{"75", "\x8A*5*12^(-1)", "6^(1/2)*4^(-1)+2^(1/2)*(-4)^(-1)", "6^(1/2)*4^(-1)+2^(1/2)*4^(-1)", "3^(1/2)+2"},
{"90", "\x8A*2^(-1)", "0", "1", "undef"},
{"105", "\x8A*7*12^(-1)", "-6^(1/2)*4^(-1)+2^(1/2)*4^(-1)", "", ""},
{"108", "\x8A*3*5^(-1)", "(1-5^(1/2))*4^(-1)", "", ""},
{"225/2", "\x8A*5*8^(-1)", "(2-2^(1/2))^(1/2)*(-2)^(-1)", "", ""},
{"120", "\x8A*2*3^(-1)", "-0.5", "", ""},
{"126", "\x8A*7*10^(-1)", "-(5*8^(-1)-5^(1/2)*8^(-1))^(1/2)", "", ""},
{"135", "\x8A*3*4^(-1)", "(-1)*(2^(-1/2))", "", ""},
{"144", "\x8A*4*5^(-1)", "(-5^(1/2)-1)*4^(-1)", "", ""},
{"150", "\x8A*5*6^(-1)", "-3^(1/2)*2^(-1)", "", ""},
{"315/2", "\x8A*7*8^(-1)", "-(2+2^(1/2))^(1/2)*2^(-1)", "", ""},
{"162", "\x8A*9*10^(-1)", "-(5*8^(-1)+5^(1/2)*8^(-1))^(1/2)", "", ""},
{"165", "\x8A*11*12^(-1)", "(-1)*6^(1/2)*4^(-1)-2^(1/2)*4^(-1)", "", ""},
{"180", "\x8A", "-1", "0", "0"}};
Expression * Trigonometry::table(const Expression * e, Expression::Type type, Context & context, Expression::AngleUnit angleUnit) {
assert(type == Expression::Type::Sine || type == Expression::Type::Cosine || type == Expression::Type::Tangent || type == Expression::Type::ArcCosine || type == Expression::Type::ArcSine || type == Expression::Type::ArcTangent);
int angleUnitIndex = angleUnit == Expression::AngleUnit::Radian ? 1 : 0;
int trigonometricFunctionIndex = type == Expression::Type::Cosine || type == Expression::Type::ArcCosine ? 2 : (type == Expression::Type::Sine || type == Expression::Type::ArcSine ? 3 : 4);
int inputIndex = type == Expression::Type::ArcCosine || type == Expression::Type::ArcSine || type == Expression::Type::ArcTangent ? trigonometricFunctionIndex : angleUnitIndex;
int outputIndex = type == Expression::Type::ArcCosine || type == Expression::Type::ArcSine || type == Expression::Type::ArcTangent ? angleUnitIndex : trigonometricFunctionIndex;
/* Avoid looping if we can exclude quickly that the e is in the table */
if (inputIndex == 0 && e->type() != Expression::Type::Rational) {
return nullptr;
}
if (inputIndex == 1 && e->type() != Expression::Type::Rational && e->type() != Expression::Type::Multiplication && e->type() != Expression::Type::Symbol) {
return nullptr;
}
if (inputIndex >1 && e->type() != Expression::Type::Rational && e->type() != Expression::Type::Multiplication && e->type() != Expression::Type::Power && e->type() != Expression::Type::Addition) {
return nullptr;
}
for (int i = 0; i < k_numberOfEntries; i++) {
Expression * input = Expression::parse(cheatTable[i][inputIndex]);
if (input == nullptr) {
continue;
}
Expression::Reduce(&input, context, angleUnit);
bool rightInput = input->isIdenticalTo(e);
delete input;
if (rightInput) {
Expression * output = Expression::parse(cheatTable[i][outputIndex]);
if (output == nullptr) {
return nullptr;
}
Expression::Reduce(&output, context, angleUnit);
return output;
}
}
return nullptr;
}
template <typename T>
std::complex<T> Trigonometry::ConvertToRadian(const std::complex<T> c, Expression::AngleUnit angleUnit) {
if (angleUnit == Expression::AngleUnit::Degree) {
return c*std::complex<T>(M_PI/180.0);
}
return c;
}
template <typename T>
std::complex<T> Trigonometry::ConvertRadianToAngleUnit(const std::complex<T> c, Expression::AngleUnit angleUnit) {
if (angleUnit == Expression::AngleUnit::Degree) {
return c*std::complex<T>(180/M_PI);
}
return c;
}
template<typename T>
T Trigonometry::RoundToMeaningfulDigits(T f) {
/* Cheat: openbsd trigonometric functions are numerical implementation and
* thus are approximative.
* The error epsilon is ~1E-7 on float and ~1E-15 on double. In order to
* avoid weird results as acos(1) = 6E-17 or cos(Pi/2) = 4E-17, we keep only
* 15 (or 7) decimals.
* We can't do that for all evaluation as the user can operate on values as
* small as 1E-308 (in double) and most results still be correct. */
T precision = 10*Expression::epsilon<T>();
return std::round(f/precision)*precision;
}
template <typename T>
std::complex<T> Trigonometry::RoundToMeaningfulDigits(const std::complex<T> c) {
return std::complex<T>(RoundToMeaningfulDigits(c.real()), RoundToMeaningfulDigits(c.imag()));
}
template std::complex<float> Trigonometry::ConvertToRadian<float>(std::complex<float>, Expression::AngleUnit);
template std::complex<double> Trigonometry::ConvertToRadian<double>(std::complex<double>, Expression::AngleUnit);
template std::complex<float> Trigonometry::ConvertRadianToAngleUnit<float>(std::complex<float>, Expression::AngleUnit);
template std::complex<double> Trigonometry::ConvertRadianToAngleUnit<double>(std::complex<double>, Expression::AngleUnit);
template std::complex<float> Trigonometry::RoundToMeaningfulDigits<float>(std::complex<float>);
template std::complex<double> Trigonometry::RoundToMeaningfulDigits<double>(std::complex<double>);
}