Average Error: 2.1 → 0.2
Time: 8.8s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 3.2317114830070396 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}}{\frac{{k}^{\left(-m\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a \cdot {k}^{m}}}\\ \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 3.2317114830070396 \cdot 10^{+78}:\\
\;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}}{\frac{{k}^{\left(-m\right)}}{a}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{k \cdot \frac{k}{a \cdot {k}^{m}}}\\


\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 3.2317114830070396e+78)
   (/ (/ 1.0 (fma k (+ k 10.0) 1.0)) (/ (pow k (- m)) a))
   (/ 1.0 (* k (/ k (* a (pow k m)))))))
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 <= 3.2317114830070396e+78) {
		tmp = (1.0 / fma(k, (k + 10.0), 1.0)) / (pow(k, -m) / a);
	} else {
		tmp = 1.0 / (k * (k / (a * pow(k, m))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 3.2317114830070396e78

    1. Initial program 0.1

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

    2. Applied clear-num_binary640.2

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

    3. Simplified0.2

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

    4. Applied div-inv_binary640.2

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

    5. Applied associate-/r*_binary640.1

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

    6. Taylor expanded in a around 0 21.4

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

    7. Simplified0.1

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

    if 3.2317114830070396e78 < k

    1. Initial program 7.3

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

    2. Applied clear-num_binary647.4

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

    3. Simplified7.4

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

    4. Taylor expanded in k around inf 7.4

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

    5. Simplified0.6

      \[\leadsto \frac{1}{\color{blue}{k \cdot \frac{k}{a \cdot {k}^{m}}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2317114830070396 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}}{\frac{{k}^{\left(-m\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a \cdot {k}^{m}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022095 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))