\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ t_1 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -1 \cdot 10^{-108}:\\ \;\;\;\;{\left(\frac{1}{t_1} + \frac{k}{a} \cdot \frac{k}{{k}^{m}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \end{array} \]

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

↓

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot k (sqrt (fma k 10.0 1.0)))) (t_1 (* (pow k m) a)))
   (if (<= m -1e-108)
     (pow (+ (/ 1.0 t_1) (* (/ k a) (/ k (pow k m)))) -1.0)
     (/ (/ t_1 t_0) t_0))))

double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

↓

double code(double a, double k, double m) {
	double t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double t_1 = pow(k, m) * a;
	double tmp;
	if (m <= -1e-108) {
		tmp = pow(((1.0 / t_1) + ((k / a) * (k / pow(k, m)))), -1.0);
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

↓

function code(a, k, m)
	t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	t_1 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -1e-108)
		tmp = Float64(Float64(1.0 / t_1) + Float64(Float64(k / a) * Float64(k / (k ^ m)))) ^ -1.0;
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

code[a_, k_, m_] := N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -1e-108], N[Power[N[(N[(1.0 / t$95$1), $MachinePrecision] + N[(N[(k / a), $MachinePrecision] * N[(k / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}

↓

\begin{array}{l}
t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\
t_1 := {k}^{m} \cdot a\\
\mathbf{if}\;m \leq -1 \cdot 10^{-108}:\\
\;\;\;\;{\left(\frac{1}{t_1} + \frac{k}{a} \cdot \frac{k}{{k}^{m}}\right)}^{-1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\


\end{array}

Derivation

Split input into 2 regimes
if m < -1.00000000000000004e-108
1. Initial program 1.2
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Applied egg-rr1.3
  \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}{a \cdot {k}^{m}}\right)}^{-1}} \]
3. Taylor expanded in k around 0 23.9
  \[\leadsto {\color{blue}{\left(10 \cdot \frac{k}{e^{\log k \cdot m} \cdot a} + \left(\frac{{k}^{2}}{e^{\log k \cdot m} \cdot a} + \frac{1}{e^{\log k \cdot m} \cdot a}\right)\right)}}^{-1} \]
4. Simplified0.3
  \[\leadsto {\color{blue}{\left(\frac{1}{{k}^{m} \cdot a} + \frac{k}{{k}^{m}} \cdot \left(\frac{10}{a} + \frac{k}{a}\right)\right)}}^{-1} \]
5. Taylor expanded in k around inf 24.4
  \[\leadsto {\left(\frac{1}{{k}^{m} \cdot a} + \color{blue}{\frac{{k}^{2}}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}}\right)}^{-1} \]
6. Simplified0.6
  \[\leadsto {\left(\frac{1}{{k}^{m} \cdot a} + \color{blue}{\frac{k}{a} \cdot \frac{k}{{k}^{m}}}\right)}^{-1} \]
if -1.00000000000000004e-108 < m
1. Initial program 2.5
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Applied egg-rr2.7
  \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}{a \cdot {k}^{m}}\right)}^{-1}} \]
3. Applied egg-rr0.1
  \[\leadsto \color{blue}{\frac{\frac{a \cdot {k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]
Recombined 2 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{-108}:\\ \;\;\;\;{\left(\frac{1}{{k}^{m} \cdot a} + \frac{k}{a} \cdot \frac{k}{{k}^{m}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \end{array} \]

Error

Derivation

`if m < -1.00000000000000004e-108`

`if -1.00000000000000004e-108 < m`

Reproduce