Average Error: 1.9 → 1.9
Time: 5.2s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} t_0 := \sqrt{1 + k \cdot \left(k + 10\right)}\\ \frac{\frac{a}{t_0} \cdot {k}^{m}}{t_0} \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
t_0 := \sqrt{1 + k \cdot \left(k + 10\right)}\\
\frac{\frac{a}{t_0} \cdot {k}^{m}}{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 (sqrt (+ 1.0 (* k (+ k 10.0))))))
   (/ (* (/ a t_0) (pow k m)) 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 = sqrt(1.0 + (k * (k + 10.0)));
	return ((a / t_0) * pow(k, m)) / t_0;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.9

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

  2. Simplified1.9

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

  4. Applied add-sqr-sqrt_binary641.9

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

  5. Applied associate-/r*_binary641.9

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

  6. Simplified1.9

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

  7. Final simplification1.9

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

Reproduce

herbie shell --seed 2021204 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))