... exist1
Gradual underflow preserves more mathematical identities. For example, $x - y = 0$ imples $x = y$ with gradual underflow. The former can hold without the latter with flush-to-zero. The performance impact is not obvious. Many "modern" processors do not accelerate arithmetic involving subnormal numbers, so gradual underflow seems slow. The additional identities give more room for programmers and compilers to optimize, however. A longer answer involves preserving relative error bounds in the face of underflow. Gradual underflow keeps simple codes robust. 
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... 2
I often present this section at the very end of the first semester 
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... error3
$\varphi$ in Kincaid and Chenney. 
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... consult4
The book by Reinhardt, Varga, et al. and the book on Optimization by Nocedal and Wright are good starting places for the basics and a survey, respectively 
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... method5
LINK multigrid 
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