Average Error: 1.9 → 0.1
Time: 11.5s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 2.2014896057993437 \cdot 10^{+23}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := e^{m \cdot \log k}\\ \mathsf{fma}\left(\frac{a}{k}, \frac{t_0}{k}, \frac{a \cdot t_0}{{k}^{3}} \cdot -10\right) \end{array}\\ \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 2.2014896057993437 \cdot 10^{+23}:\\
\;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := e^{m \cdot \log k}\\
\mathsf{fma}\left(\frac{a}{k}, \frac{t_0}{k}, \frac{a \cdot t_0}{{k}^{3}} \cdot -10\right)
\end{array}\\


\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 2.2014896057993437e+23)
   (/ a (/ (fma k (+ k 10.0) 1.0) (pow k m)))
   (let* ((t_0 (exp (* m (log k)))))
     (fma (/ a k) (/ t_0 k) (* (/ (* a t_0) (pow k 3.0)) -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 <= 2.2014896057993437e+23) {
		tmp = a / (fma(k, (k + 10.0), 1.0) / pow(k, m));
	} else {
		double t_0 = exp(m * log(k));
		tmp = fma((a / k), (t_0 / k), (((a * t_0) / pow(k, 3.0)) * -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 < 2.20148960579934373e23

    1. Initial program 0.0

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

    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 associate-/r*_binary640.1

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

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

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

    6. Applied sqrt-prod_binary640.1

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

    7. Applied times-frac_binary640.1

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

    8. Applied associate-/l*_binary640.1

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

    9. Simplified0.0

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

    if 2.20148960579934373e23 < k

    1. Initial program 5.5

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

    2. Simplified5.5

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

    3. Applied add-sqr-sqrt_binary645.5

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

    4. Applied associate-/r*_binary645.5

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

    5. Applied *-un-lft-identity_binary645.5

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

    6. Applied add-sqr-sqrt_binary645.5

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

    7. Applied times-frac_binary645.6

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

    8. Applied times-frac_binary645.6

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

    9. Taylor expanded in k around inf 5.5

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

    10. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2014896057993437 \cdot 10^{+23}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{k}, \frac{e^{m \cdot \log k}}{k}, \frac{a \cdot e^{m \cdot \log k}}{{k}^{3}} \cdot -10\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))