Average Error: 2.1 → 0.1
Time: 6.0s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 10000000000:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-10}{k}\right) \cdot \left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\\ \end{array} \]
(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
↓
(FPCore (a k m)
 :precision binary64
 (if (<= k 10000000000.0)
   (* (/ (pow k m) (fma k (+ k 10.0) 1.0)) a)
   (* (+ 1.0 (/ -10.0 k)) (* (/ a k) (/ (pow k m) k)))))
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 tmp;
	if (k <= 10000000000.0) {
		tmp = (pow(k, m) / fma(k, (k + 10.0), 1.0)) * a;
	} else {
		tmp = (1.0 + (-10.0 / k)) * ((a / k) * (pow(k, m) / k));
	}
	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)
	tmp = 0.0
	if (k <= 10000000000.0)
		tmp = Float64(Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)) * a);
	else
		tmp = Float64(Float64(1.0 + Float64(-10.0 / k)) * Float64(Float64(a / k) * Float64((k ^ m) / k)));
	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_] := If[LessEqual[k, 10000000000.0], N[(N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(1.0 + N[(-10.0 / k), $MachinePrecision]), $MachinePrecision] * N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 10000000000:\\
\;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot a\\

\mathbf{else}:\\
\;\;\;\;\left(1 + \frac{-10}{k}\right) \cdot \left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes

  2. if k < 1e10

    1. Initial program 0.1

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

    2. Simplified0.0

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

    3. Applied egg-rr0.1

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

    4. Applied egg-rr0.2

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

    5. Applied egg-rr0.0

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

    if 1e10 < k

    1. Initial program 5.7

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

    2. Simplified5.7

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

    3. Applied egg-rr5.7

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

    4. Applied egg-rr5.9

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

    5. Taylor expanded in k around inf 5.7

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}} + -10 \cdot \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{3}}} \]

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10000000000:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-10}{k}\right) \cdot \left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022202 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))