Average Error: 2.1 → 0.2
Time: 12.2s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 4.9377061128398556 \cdot 10^{+58}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathsf{fma}\left(99, \frac{t_0}{{k}^{4}}, \mathsf{fma}\left(\frac{t_0}{{k}^{3}}, -10, \frac{{k}^{m}}{k \cdot \frac{k}{a}}\right)\right) \end{array}\\ \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 4.9377061128398556 \cdot 10^{+58}:\\
\;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := a \cdot {k}^{m}\\
\mathsf{fma}\left(99, \frac{t_0}{{k}^{4}}, \mathsf{fma}\left(\frac{t_0}{{k}^{3}}, -10, \frac{{k}^{m}}{k \cdot \frac{k}{a}}\right)\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 4.9377061128398556e+58)
   (/ a (/ (fma k (+ k 10.0) 1.0) (pow k m)))
   (let* ((t_0 (* a (pow k m))))
     (fma
      99.0
      (/ t_0 (pow k 4.0))
      (fma (/ t_0 (pow k 3.0)) -10.0 (/ (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 <= 4.9377061128398556e+58) {
		tmp = a / (fma(k, (k + 10.0), 1.0) / pow(k, m));
	} else {
		double t_0 = a * pow(k, m);
		tmp = fma(99.0, (t_0 / pow(k, 4.0)), fma((t_0 / pow(k, 3.0)), -10.0, (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 < 4.9377061128398556e58

    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 clear-num_binary640.2

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

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

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

    5. Applied times-frac_binary640.2

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

    6. Applied associate-/r*_binary640.1

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

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

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

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

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

    9. Applied times-frac_binary640.1

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

    10. Applied associate-/r/_binary640.0

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

    11. Applied times-frac_binary640.0

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

    if 4.9377061128398556e58 < k

    1. Initial program 7.0

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

    2. Simplified7.0

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

    3. Applied clear-num_binary647.2

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

    4. Taylor expanded in k around inf 7.0

      \[\leadsto \color{blue}{\left(\frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}} + 99 \cdot \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{4}}\right) - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{3}}} \]

    5. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(99, \frac{a \cdot {k}^{m}}{{k}^{4}}, \mathsf{fma}\left(\frac{a \cdot {k}^{m}}{{k}^{3}}, -10, \frac{{k}^{m}}{k \cdot \frac{k}{a}}\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.9377061128398556 \cdot 10^{+58}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(99, \frac{a \cdot {k}^{m}}{{k}^{4}}, \mathsf{fma}\left(\frac{a \cdot {k}^{m}}{{k}^{3}}, -10, \frac{{k}^{m}}{k \cdot \frac{k}{a}}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022103 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))