Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add

At each precision, let the representable set, normal range, and unit roundoff be an instance of The standard relative floating-point model and unit roundoff specialized to a finite binary format of integer precision $p\ge 2$ and integer normal-exponent bounds $j_{\min }\le j\le j_{\max }$. Its normal values are $\pm m{2}^{j-p+1}$ for integers $2^{p-1}\le m<2^p$ and $j_{\min }\le j\le j_{\max }$, with $N_{\min }=2^{j_{\min }}$ and $N_{\max }=(2^p-1)2^{j_{\max }-p+1}$. The rounding rule is round-to-nearest, ties-to-even by the parity of the normalized significand. Each precision in a mixed-precision procedure has its own format, range, and valid unit-roundoff bound. On every real exact result that is zero or normal, $\mathrm{fl}$ uses this rule; every other nonzero exact result (subnormal or overflow), NaN, infinity, invalid operation, or division by zero returns range-failure. A real add, subtract, multiply, or divide is one rounded operation. For positive in-range $x$, $\mathrm{fl}(\sqrt{x})$ is correctly rounded to nearest, ties-to-even, and fails unless its result is positive and normal. A real fused multiply-add returns the correctly rounded value of the single exact expression $ab+c$; it is not two operations.

Complex addition and subtraction round real and imaginary components separately. A complex product (a+ib)(c+id) uses four rounded real products ac, bd, ad, bc, then rounded subtraction and addition, for six real operations. Conjugation and sign changes are exact. Scaling and division by a positive real scalar round each complex component separately; no other complex division primitive is used here.

A procedure succeeds only when each exact intermediate is zero or normal and every rounded output is finite; otherwise it returns the first range-failure with its operation or pivot index. For ku<1 write gamma_k=ku/(1-ku). This is a no-underflow success model, not the gradual-underflow model.