Average Error: 2.2 → 0.1
Time: 9.2s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 9.77492532341824 \cdot 10^{+94}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{m \cdot \log k}}{k}, \frac{a}{k}, \frac{a}{\frac{{k}^{3}}{{k}^{m}}} \cdot -10\right)\\ \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 9.77492532341824 \cdot 10^{+94}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{e^{m \cdot \log k}}{k}, \frac{a}{k}, \frac{a}{\frac{{k}^{3}}{{k}^{m}}} \cdot -10\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 9.77492532341824e+94)
   (/ (* a (pow k m)) (fma k (+ k 10.0) 1.0))
   (fma
    (/ (exp (* m (log k))) k)
    (/ a k)
    (* (/ a (/ (pow k 3.0) (pow k m))) -10.0))))
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 tmp;
	if (k <= 9.77492532341824e+94) {
		tmp = (a * pow(k, m)) / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = fma((exp(m * log(k)) / k), (a / k), ((a / (pow(k, 3.0) / pow(k, m))) * -10.0));
	}
	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 < 9.77492532341824072e94

    1. Initial program 0.1

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

    2. Simplified0.0

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

    if 9.77492532341824072e94 < k

    1. Initial program 8.1

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

    2. Simplified8.1

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

    3. Applied *-un-lft-identity_binary648.1

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

    4. Applied times-frac_binary648.2

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

    5. Simplified8.2

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

    6. Applied add-sqr-sqrt_binary648.2

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

    7. Applied associate-/r*_binary648.2

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

    8. Taylor expanded in k around inf 8.1

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

    9. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{-m \cdot \left(-\log k\right)}}{k}, \frac{a}{k}, \frac{a \cdot e^{-m \cdot \left(-\log k\right)}}{{k}^{3}} \cdot -10\right)} \]

    10. Applied exp-neg_binary640.1

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

    11. Applied associate-*r/_binary640.1

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

    12. Applied associate-/l/_binary640.1

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

    13. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.77492532341824 \cdot 10^{+94}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{m \cdot \log k}}{k}, \frac{a}{k}, \frac{a}{\frac{{k}^{3}}{{k}^{m}}} \cdot -10\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))