Average Error: 5.3 → 0.2
Time: 12.7s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 9.832168212624921 \cdot 10^{-24}:\\ \;\;\;\;t_0 \cdot \left(1 - k \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{k}{t_0}\\ \frac{1}{\mathsf{fma}\left(k, t_1, \mathsf{fma}\left(10, t_1, \frac{1}{t_0}\right)\right)} \end{array}\\ \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
t_0 := a \cdot {k}^{m}\\
\mathbf{if}\;k \leq 9.832168212624921 \cdot 10^{-24}:\\
\;\;\;\;t_0 \cdot \left(1 - k \cdot 10\right)\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_1 := \frac{k}{t_0}\\
\frac{1}{\mathsf{fma}\left(k, t_1, \mathsf{fma}\left(10, t_1, \frac{1}{t_0}\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
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 9.832168212624921e-24)
     (* t_0 (- 1.0 (* k 10.0)))
     (let* ((t_1 (/ k t_0))) (/ 1.0 (fma k t_1 (fma 10.0 t_1 (/ 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 = a * pow(k, m);
	double tmp;
	if (k <= 9.832168212624921e-24) {
		tmp = t_0 * (1.0 - (k * 10.0));
	} else {
		double t_1 = k / t_0;
		tmp = 1.0 / fma(k, t_1, fma(10.0, t_1, (1.0 / 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 k < 9.8321682126249208e-24

    1. Initial program 1.3

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

    2. Applied clear-num_binary641.3

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

    3. Simplified1.3

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

    4. Taylor expanded in k around 0 36.6

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

    5. Simplified0.2

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

    if 9.8321682126249208e-24 < k

    1. Initial program 11.4

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

    2. Applied clear-num_binary6411.5

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

    3. Simplified11.5

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

    4. Taylor expanded in k around 0 11.5

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

    5. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.832168212624921 \cdot 10^{-24}:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 - k \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k}{a \cdot {k}^{m}}, \mathsf{fma}\left(10, \frac{k}{a \cdot {k}^{m}}, \frac{1}{a \cdot {k}^{m}}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))