Interval
acos(Interval interval) {
if (interval.isEmpty() || interval.lowerBound() > 1.0 || interval.upperBound() < -1.0) {
return Interval::EMPTY();
} else {
return Interval(
acos(min(interval.upperBound(), 1.0)),
acos(max(interval.lowerBound(), -1.0))
);
}
}
Interval
log(Interval interval) {
if (interval.isEmpty() || interval.upperBound() < 0.0) {
return Interval::EMPTY();
} else if (interval.lowerBound() > 0.0) {
return Interval(log(interval.lowerBound()), log(interval.upperBound()));
} else if (interval.upperBound() > 0.0) {
return Interval(-std::numeric_limits<double>::infinity(), log(interval.upperBound()));
} else {
return -std::numeric_limits<double>::infinity();
}
}
Interval
atan2(Interval yInterval, Interval xInterval) {
if (yInterval.isEmpty() || xInterval.isEmpty()) {
return Interval::EMPTY();
} else if (xInterval.lowerBound() > 0.0) {
return atan(yInterval / xInterval);
} else if (yInterval.lowerBound() > 0.0) {
return atan(-xInterval / yInterval) + M_PI / 2;
} else if (yInterval.upperBound() < 0.0) {
return atan(-xInterval / yInterval) - M_PI / 2;
} else {
return Interval(-M_PI, M_PI);
}
}
Interval
exp(Interval interval) {
if (interval.isEmpty()) {
return Interval::EMPTY();
} else {
return Interval(exp(interval.lowerBound()), exp(interval.upperBound()));
}
}
Interval
tan(Interval interval) {
Interval absolute = abs(interval);
bool hasSingularity = std::fmod(absolute.upperBound() + M_PI / 2, M_PI) <= absolute.width();
if (hasSingularity) {
return Interval::WHOLE();
} else {
return Interval(std::tan(interval.lowerBound()), std::tan(interval.upperBound()));
}
}
Interval
cos(Interval interval) {
Interval absolute = abs(interval);
double width = absolute.width();
bool hasMin = std::fmod(absolute.upperBound() + M_PI, 2 * M_PI) <= width;
bool hasMax = std::fmod(absolute.upperBound(), 2 * M_PI) <= width;
if (hasMin && hasMax) {
return Interval(-1.0, 1.0);
} else {
double cosLower = std::cos(absolute.lowerBound());
double cosUpper = std::cos(absolute.upperBound());
auto sorted = std::minmax(cosLower, cosUpper);
return Interval(hasMin ? -1.0 : sorted.first, hasMax ? 1.0 : sorted.second);
}
}
Interval
atan(Interval interval) {
return Interval(atan(interval.lowerBound()), atan(interval.upperBound()));
}