Average Error: 2.1 → 0.2
Time: 8.4s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 8.720811547314525 \cdot 10^{+67}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k \cdot \frac{k}{a}}\\ \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 8.720811547314525 \cdot 10^{+67}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\

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


\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 8.720811547314525e+67)
   (/ (* a (pow k m)) (fma k (+ k 10.0) 1.0))
   (/ (pow k m) (* k (/ k a)))))
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 <= 8.720811547314525e+67) {
		tmp = (a * pow(k, m)) / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = pow(k, m) / (k * (k / a));
	}
	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 < 8.7208115473145246e67

    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 add-sqr-sqrt_binary640.1

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

    4. Applied times-frac_binary640.1

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

    5. Applied *-un-lft-identity_binary640.1

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

    6. Applied associate-*l*_binary640.1

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

    7. Simplified0.0

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

    if 8.7208115473145246e67 < k

    1. Initial program 6.9

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

    2. Simplified6.9

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

    3. Applied clear-num_binary647.1

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

    4. Taylor expanded in k around inf 6.9

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

    5. Simplified0.6

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.720811547314525 \cdot 10^{+67}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k \cdot \frac{k}{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022094 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))