Rationality means someone acts according to reason, i.e. thinks about cause and effect, adequacy of means and ends, intersubjectivity, a favorable outcome that is understandable for anyone not deluded (i.e. somehow according to the Kantian categorial imperative). It means you are not (immediately) slave to (blind) emotions. There are different and somehow distinguished types of reason (and e.g. postmodernism and critical theory, in their attempt to liberate us, probably have denounced natural forms of reason as alienations („instrumental reason“, „culture industry“ , etc) or inflated our notion about the heterogeneity of reason too much (but that is not as much a problem as anti-postmodernists are inclined to think)). Max Weber distinguishes between wertrational (value-rational) and zweckrational (goal/instrumental-rational), where zweckrational means orientation towards a rational outcome and wertrational means acting and reasoning in a rational way according to values (which, themselves, are not rationally investigated). Other types of action are, says Weber, emotional/affective action and traditional action (which are not rational). Rationality will be the dominant thinking mode of the somehow intelligent person. How much a person can distinguish himself from the downsides of Wertrationalität is a matter of psychology. Whereas strict Wertrationalität makes the stubborn fanatic, complete ideological/emotional unbiasedness is rarely ever there among humans, and, irrespective of what Western or Eastern enlightenment (KantHegelMarxetc vs. TaoZenShankaraetc) propose, there are probably no thoughts that are not based or come in with emotions at any rate.
Hyperrationality means permanently adjusting his worldview and actions to that what the sober rational insight demands. The hyperrational person will have insight and (at least a rational) access to that that is wertrational, zweckrational as well as affectual and traditional, and overview over the grand scheme and over the fabric of society which means that in his understanding of society (i.e. of the great heterogeneity) the hyperrational person will be flexible, fluid, experienced and quick. In order to execute hyperrational understanding over complex problems (i.e. problems to which there are, opposed to complicated problems, no definite solutions) a high crystallised intelligence is necessary (that will be accumulated via a vivid fluid intelligence). Hyperrationality means a higher level of awareness than mere rationality and, at least concerning the intellectual insight, less stubbornness, but does not rule out stubbornness due to emotional reasons. High intelligence means someone is likely to draw correct rational/logical conclusions from assumptions, however this does not mean the assumptions are correct, their selection can be heavily ideologically biased, and that high IQ persons have the same petty political opinions (or petty understandings in many other domains) and use the same weak rationalisations to justify their emotional or tradition-based choices as persons with a very low IQ is quite frequently the case. The probable downside of hyperrationality are detachedness from the living world and missed opportunities, but that need not be the case.
The genius is commonly perceived as an eminently or hyperrational person who seemingly also has access to the irrational (respectively to the abstractions of the irrational and to the aesthetic realm). They develop their rational concepts by asking themselves questions like how it would be if one travels along a ray of light, or they test their hypotheses by putting a blunt needle in their eye or endanger their eyes because of gazing into the sun. While such questions (and actions) are not actually irrational, they are not likely to come to the mind of a person who has a purely rational epistemology and way to look at things. It is difficult to sort out the true nature of that (and probably it is not one thing only), but the genius thinks eminently intuitively as well as counterintuitively and (apparently) paradoxical, and, in a way, ultradialectic, as he throws up many ideas and then tries to illuminate them from all different angles, with not much propensity to favour a specific angle over others (while on the other hand usually being extremely value-oriented concerning a universe that makes sense, which made Newton a theological alchemist and Einstein opposing quantum mechanics, i.e. somehow stubbornly irrational). It is as if the genius can see into an additional dimension that is invisible to others, obviously due to capability of making plethora of (counter/intuitive) associations to any given concept (which, in a genuine way, is no necessary quality of mere intelligence and convergent thinking, but of creativity and divergent thinking, respectively, as Cooijmans calls it, associative horizon). However, it usually turns out that the genius just sees what is so obvious and rational that other people, due to their indoctrinations, don´t see it, because he brings back a very basic rationality to the perspective (that gravity is a force and means curvature is, upon reflection, actually quite obvious). Let us say the basic/dominant way of thinking of the genius is metarationality.