Average Error: 2.0 → 14.8
Time: 12.2s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} t_0 := \frac{0}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{if}\;m \leq 3.6592210806688855 \cdot 10^{-267}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 222.24157684555098:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
t_0 := \frac{0}{\mathsf{fma}\left(k, k + 10, 1\right)}\\
\mathbf{if}\;m \leq 3.6592210806688855 \cdot 10^{-267}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;m \leq 222.24157684555098:\\
\;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\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
 (let* ((t_0 (/ 0.0 (fma k (+ k 10.0) 1.0))))
   (if (<= m 3.6592210806688855e-267)
     t_0
     (if (<= m 222.24157684555098) (/ a (fma k 10.0 (fma k k 1.0))) t_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 t_0 = 0.0 / fma(k, (k + 10.0), 1.0);
	double tmp;
	if (m <= 3.6592210806688855e-267) {
		tmp = t_0;
	} else if (m <= 222.24157684555098) {
		tmp = a / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = t_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 m < 3.65922108066888551e-267 or 222.24157684555098 < m

    1. Initial program 1.4

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

    2. Simplified1.4

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

    3. Applied egg-rr18.8

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

    4. Taylor expanded in a around 0 17.4

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

    if 3.65922108066888551e-267 < m < 222.24157684555098

    1. Initial program 3.9

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

    2. Simplified3.9

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

    3. Taylor expanded in m around 0 5.4

      \[\leadsto \color{blue}{\frac{a}{10 \cdot k + \left(1 + {k}^{2}\right)}} \]

    4. Simplified5.4

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.6592210806688855 \cdot 10^{-267}:\\ \;\;\;\;\frac{0}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 222.24157684555098:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022127 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))