Average Error: 2.3 → 0.1
Time: 8.3s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \mathbf{if}\;k \leq 6.890075221064955 \cdot 10^{+78}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \frac{{k}^{m}}{\mathsf{hypot}\left(k, 1\right)}}{t_0}}{t_0}\\ \end{array} \]
(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 (sqrt (hypot k (sqrt (fma k 10.0 1.0))))))
   (if (<= k 6.890075221064955e+78)
     (/ (* a (pow k m)) (fma k k (fma k 10.0 1.0)))
     (/ (/ (* a (/ (pow k m) (hypot k 1.0))) 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 = sqrt(hypot(k, sqrt(fma(k, 10.0, 1.0))));
	double tmp;
	if (k <= 6.890075221064955e+78) {
		tmp = (a * pow(k, m)) / fma(k, k, fma(k, 10.0, 1.0));
	} else {
		tmp = ((a * (pow(k, m) / hypot(k, 1.0))) / 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 = sqrt(hypot(k, sqrt(fma(k, 10.0, 1.0))))
	tmp = 0.0
	if (k <= 6.890075221064955e+78)
		tmp = Float64(Float64(a * (k ^ m)) / fma(k, k, fma(k, 10.0, 1.0)));
	else
		tmp = Float64(Float64(Float64(a * Float64((k ^ m) / hypot(k, 1.0))) / 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[N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[k, 6.890075221064955e+78], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(k * k + N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(N[Power[k, m], $MachinePrecision] / N[Sqrt[k ^ 2 + 1.0 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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 := \sqrt{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\
\mathbf{if}\;k \leq 6.890075221064955 \cdot 10^{+78}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{a \cdot \frac{{k}^{m}}{\mathsf{hypot}\left(k, 1\right)}}{t_0}}{t_0}\\


\end{array}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 6.8900752210649551e78

    1. Initial program 0.1

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

    2. Applied egg-rr11.1

      \[\leadsto \color{blue}{\frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]

    3. Taylor expanded in a around 0 21.5

      \[\leadsto \color{blue}{\frac{a \cdot e^{\log k \cdot m}}{10 \cdot k + \left(1 + {k}^{2}\right)}} \]

    4. Simplified0.0

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}} \]

    if 6.8900752210649551e78 < k

    1. Initial program 7.8

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

    2. Applied egg-rr0.1

      \[\leadsto \color{blue}{\frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]

    3. Applied egg-rr0.2

      \[\leadsto \color{blue}{\frac{\frac{a \cdot \frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}{\sqrt{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}}{\sqrt{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}} \]

    4. Taylor expanded in k around 0 0.2

      \[\leadsto \frac{\frac{a \cdot \frac{{k}^{m}}{\mathsf{hypot}\left(k, \color{blue}{1}\right)}}{\sqrt{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}}{\sqrt{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.890075221064955 \cdot 10^{+78}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \frac{{k}^{m}}{\mathsf{hypot}\left(k, 1\right)}}{\sqrt{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}}{\sqrt{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))