Average Error: 2.0 → 0.1
Time: 13.3s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 2.343786894830444 \cdot 10^{+150}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(1, 1 + k \cdot 10, k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := {\left({\left(\frac{1}{k}\right)}^{-0.6666666666666666}\right)}^{m}\\ t_1 := a \cdot {\left({\left(\frac{1}{k}\right)}^{-0.3333333333333333}\right)}^{m}\\ \mathsf{fma}\left(\frac{t_0}{k}, \frac{t_1}{k}, \frac{t_0 \cdot t_1}{{k}^{3}} \cdot -10\right) \end{array}\\ \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 2.343786894830444 \cdot 10^{+150}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(1, 1 + k \cdot 10, k \cdot k\right)}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := {\left({\left(\frac{1}{k}\right)}^{-0.6666666666666666}\right)}^{m}\\
t_1 := a \cdot {\left({\left(\frac{1}{k}\right)}^{-0.3333333333333333}\right)}^{m}\\
\mathsf{fma}\left(\frac{t_0}{k}, \frac{t_1}{k}, \frac{t_0 \cdot t_1}{{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 2.343786894830444e+150)
   (/ (* a (pow k m)) (fma 1.0 (+ 1.0 (* k 10.0)) (* k k)))
   (let* ((t_0 (pow (pow (/ 1.0 k) -0.6666666666666666) m))
          (t_1 (* a (pow (pow (/ 1.0 k) -0.3333333333333333) m))))
     (fma (/ t_0 k) (/ t_1 k) (* (/ (* t_0 t_1) (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 <= 2.343786894830444e+150) {
		tmp = (a * pow(k, m)) / fma(1.0, (1.0 + (k * 10.0)), (k * k));
	} else {
		double t_0 = pow(pow((1.0 / k), -0.6666666666666666), m);
		double t_1 = a * pow(pow((1.0 / k), -0.3333333333333333), m);
		tmp = fma((t_0 / k), (t_1 / k), (((t_0 * t_1) / 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 < 2.3437868948304442e150

    1. Initial program 0.1

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

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

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

    3. Applied fma-def_binary640.1

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

    if 2.3437868948304442e150 < k

    1. Initial program 9.9

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

    2. Applied add-cube-cbrt_binary649.9

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

    3. Applied unpow-prod-down_binary649.9

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

    4. Applied associate-*r*_binary649.9

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

    5. Taylor expanded in k around inf 9.9

      \[\leadsto \color{blue}{\frac{{\left({\left(\frac{1}{k}\right)}^{-0.6666666666666666}\right)}^{m} \cdot \left(a \cdot e^{\log \left({\left(\frac{1}{k}\right)}^{-0.3333333333333333}\right) \cdot m}\right)}{{k}^{2}} - 10 \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{-0.6666666666666666}\right)}^{m} \cdot \left(a \cdot e^{\log \left({\left(\frac{1}{k}\right)}^{-0.3333333333333333}\right) \cdot m}\right)}{{k}^{3}}} \]

    6. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left({\left(\frac{1}{k}\right)}^{-0.6666666666666666}\right)}^{m}}{k}, \frac{a \cdot {\left({\left(\frac{1}{k}\right)}^{-0.3333333333333333}\right)}^{m}}{k}, \frac{{\left({\left(\frac{1}{k}\right)}^{-0.6666666666666666}\right)}^{m} \cdot \left(a \cdot {\left({\left(\frac{1}{k}\right)}^{-0.3333333333333333}\right)}^{m}\right)}{{k}^{3}} \cdot -10\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.343786894830444 \cdot 10^{+150}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(1, 1 + k \cdot 10, k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left({\left(\frac{1}{k}\right)}^{-0.6666666666666666}\right)}^{m}}{k}, \frac{a \cdot {\left({\left(\frac{1}{k}\right)}^{-0.3333333333333333}\right)}^{m}}{k}, \frac{{\left({\left(\frac{1}{k}\right)}^{-0.6666666666666666}\right)}^{m} \cdot \left(a \cdot {\left({\left(\frac{1}{k}\right)}^{-0.3333333333333333}\right)}^{m}\right)}{{k}^{3}} \cdot -10\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))