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Wii VC Round-to-Zero: Difference between revisions
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:<math>\frac{127}{256} \times \frac{1}{4096}\ \frac{\mathrm{units}}{\mathrm{frames}} \times \left(30\ \frac{\mathrm{frames}}{\mathrm{s}} \times 60\ \frac{\mathrm{s}}{\mathrm{min}} \times 60\ \frac{\mathrm{min}}{\mathrm{h}}\right) \approx 13.0806\ \frac{\mathrm{units}}{\mathrm{h}}</math> | :<math>\frac{127}{256} \times \frac{1}{4096}\ \frac{\mathrm{units}}{\mathrm{frames}} \times \left(30\ \frac{\mathrm{frames}}{\mathrm{s}} \times 60\ \frac{\mathrm{s}}{\mathrm{min}} \times 60\ \frac{\mathrm{min}}{\mathrm{h}}\right) \approx 13.0806\ \frac{\mathrm{units}}{\mathrm{h}}</math> | ||
The floating-point numbers are distributed non-uniformly; the gap between floats doubles after each [[wikipedia:Power of two|power of two]], so their number line is denser closer to zero. Because the platform drifts toward zero, the error factor (initially <math display="inline">\frac{1}{4096}</math>) is halved after certain thresholds, and so the platform's net speed is also halved. First, above <var>y</var> = −2048, it drops to about 6.54 units per hour. It continues to halve in net speed as it passes through <var>y</var> = −1024, <var>y</var> = −512, and all other <math display="inline">y = 2^n</math> boundaries, until it stops around <var>y</var> = 0. | The floating-point numbers are distributed non-uniformly; the gap between floats doubles after each [[wikipedia:Power of two|power of two]], so their number line is denser closer to zero. Because the platform drifts toward zero, the error factor (initially <math display="inline">\frac{1}{4096}</math>) is halved after certain thresholds, and so the platform's net speed is also halved. First, above <var>y</var> = −2048, it drops to about 6.54 units per hour. It continues to halve in net speed as it passes through <var>y</var> = −1024, <var>y</var> = −512, and all other <math display="inline">\left|y\right| = 2^n</math> boundaries, until it stops around <var>y</var> = 0. | ||
Each time the platform oscillates through one of these boundaries, its behavior | Each time the platform oscillates through one of these boundaries, its behavior is more complicated. Throughout its cycle, the platform is sometimes above <math display="inline">\left|y\right| = 2^n</math> and sometimes below. The overall speed decrease as the platform's average height passes through the boundary is represented by the [[wikipedia:Differential equation|differential equation]] | ||
:<math>v_y = \frac{dy}{dt} = \arccos\left(y\right)</math> | :<math>v_y = \frac{dy}{dt} = \arccos\left(y\right)</math> | ||
The platform suddenly slows down quickly, then at a slower rate before it decelerates faster again, and finally, suddenly the platform's acceleration completely stops, the speed settling at a constant, half of what it started with. | The platform suddenly slows down quickly, then at a slower rate before it decelerates faster again, and finally, suddenly the platform's acceleration completely stops, the speed settling at a constant, half of what it started with. | ||