Average Error: 2.1 → 2.1
Time: 4.0s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\sqrt[3]{{k}^{m}} \cdot \left(a \cdot \left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right)\right)}{1 + k \cdot \left(k + 10\right)}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{\sqrt[3]{{k}^{m}} \cdot \left(a \cdot \left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right)\right)}{1 + k \cdot \left(k + 10\right)}
(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
↓
(FPCore (a k m)
 :precision binary64
 (/
  (* (cbrt (pow k m)) (* a (* (cbrt (pow k m)) (cbrt (pow k m)))))
  (+ 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) {
	return (cbrt(pow(k, m)) * (a * (cbrt(pow(k, m)) * cbrt(pow(k, m))))) / (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 2.1

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

  2. Simplified2.1

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

  4. Applied add-cube-cbrt_binary642.1

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

  5. Applied associate-*r*_binary642.1

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

  6. Final simplification2.1

    \[\leadsto \frac{\sqrt[3]{{k}^{m}} \cdot \left(a \cdot \left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right)\right)}{1 + k \cdot \left(k + 10\right)}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))