Average Error: 1.9 → 1.9
Time: 6.5s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} t_0 := {k}^{\left(\frac{m}{2}\right)}\\ \frac{t_0 \cdot \left(a \cdot t_0\right)}{1 + k \cdot \left(k + 10\right)} \end{array} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
t_0 := {k}^{\left(\frac{m}{2}\right)}\\
\frac{t_0 \cdot \left(a \cdot t_0\right)}{1 + k \cdot \left(k + 10\right)}
\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 (pow k (/ m 2.0))))
   (/ (* t_0 (* a t_0)) (+ 1.0 (* k (+ k 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 t_0 = pow(k, (m / 2.0));
	return (t_0 * (a * t_0)) / (1.0 + (k * (k + 10.0)));
}

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 sqr-pow_binary641.9

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

  5. Applied associate-*r*_binary641.9

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

  6. Simplified1.9

    \[\leadsto \frac{\color{blue}{\left(a \cdot \sqrt{{k}^{m}}\right)} \cdot {k}^{\left(\frac{m}{2}\right)}}{1 + k \cdot \left(k + 10\right)} \]
  7. Using strategy rm

  8. Applied add-sqr-sqrt_binary641.9

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

  9. Simplified1.9

    \[\leadsto \frac{\left(a \cdot \left(\color{blue}{{k}^{\left(\frac{m}{4}\right)}} \cdot \sqrt{\sqrt{{k}^{m}}}\right)\right) \cdot {k}^{\left(\frac{m}{2}\right)}}{1 + k \cdot \left(k + 10\right)} \]

  10. Simplified1.9

    \[\leadsto \frac{\left(a \cdot \left({k}^{\left(\frac{m}{4}\right)} \cdot \color{blue}{{k}^{\left(\frac{m}{4}\right)}}\right)\right) \cdot {k}^{\left(\frac{m}{2}\right)}}{1 + k \cdot \left(k + 10\right)} \]
  11. Using strategy rm

  12. Applied *-un-lft-identity_binary641.9

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

  13. Applied associate-*l*_binary641.9

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

  14. Simplified1.9

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

  15. Final simplification1.9

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

Reproduce

herbie shell --seed 2021205 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))