Average Error: 1.9 → 1.9
Time: 6.3s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\frac{\left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right) \cdot {\left(\sqrt[3]{k}\right)}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{\left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right) \cdot {\left(\sqrt[3]{k}\right)}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
↓
(FPCore (a k m)
 :precision binary64
 (/
  (* (* a (pow (* (cbrt k) (cbrt k)) m)) (pow (cbrt k) m))
  (+ (+ 1.0 (* k 10.0)) (* k k))))
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 ((a * pow((cbrt(k) * cbrt(k)), m)) * pow(cbrt(k), m)) / ((1.0 + (k * 10.0)) + (k * k));
}

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. Using strategy rm

  3. Applied add-log-exp_binary6411.3

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

  4. Applied add-log-exp_binary6411.3

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

  5. Applied sum-log_binary6411.3

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

  6. Applied log-prod_binary6411.3

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

  7. Simplified11.3

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

  8. Simplified1.9

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

  10. Applied add-cube-cbrt_binary641.9

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

  11. Applied unpow-prod-down_binary641.9

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

  12. Applied associate-*r*_binary641.9

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

  13. Final simplification1.9

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

Reproduce

herbie shell --seed 2021211 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))