Average Error: 0.4 → 0.5
Time: 9.1s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{\sqrt{1}}{\left|\sqrt[3]{k}\right|} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt[3]{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{\sqrt{1}}{\left|\sqrt[3]{k}\right|} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt[3]{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r120249 = 1.0;
        double r120250 = k;
        double r120251 = sqrt(r120250);
        double r120252 = r120249 / r120251;
        double r120253 = 2.0;
        double r120254 = atan2(1.0, 0.0);
        double r120255 = r120253 * r120254;
        double r120256 = n;
        double r120257 = r120255 * r120256;
        double r120258 = r120249 - r120250;
        double r120259 = r120258 / r120253;
        double r120260 = pow(r120257, r120259);
        double r120261 = r120252 * r120260;
        return r120261;
}
↓
double f(double k, double n) {
        double r120262 = 1.0;
        double r120263 = k;
        double r120264 = sqrt(r120263);
        double r120265 = r120262 / r120264;
        double r120266 = sqrt(r120265);
        double r120267 = sqrt(r120262);
        double r120268 = cbrt(r120263);
        double r120269 = fabs(r120268);
        double r120270 = r120267 / r120269;
        double r120271 = sqrt(r120268);
        double r120272 = r120267 / r120271;
        double r120273 = r120270 * r120272;
        double r120274 = sqrt(r120273);
        double r120275 = 2.0;
        double r120276 = atan2(1.0, 0.0);
        double r120277 = r120275 * r120276;
        double r120278 = n;
        double r120279 = r120277 * r120278;
        double r120280 = r120262 - r120263;
        double r120281 = r120280 / r120275;
        double r120282 = pow(r120279, r120281);
        double r120283 = r120274 * r120282;
        double r120284 = r120266 * r120283;
        return r120284;
}

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.5

    \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\sqrt{k}}} \cdot \sqrt{\frac{1}{\sqrt{k}}}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  4. Applied associate-*l*0.5

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.5

    \[\leadsto \sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right) \cdot \sqrt[3]{k}}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]

  7. Applied sqrt-prod0.5

    \[\leadsto \sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{1}{\color{blue}{\sqrt{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \sqrt{\sqrt[3]{k}}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]

  8. Applied add-sqr-sqrt0.5

    \[\leadsto \sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\sqrt{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \sqrt{\sqrt[3]{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]

  9. Applied times-frac0.5

    \[\leadsto \sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\color{blue}{\frac{\sqrt{1}}{\sqrt{\sqrt[3]{k} \cdot \sqrt[3]{k}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt[3]{k}}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]

  10. Simplified0.5

    \[\leadsto \sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\color{blue}{\frac{\sqrt{1}}{\left|\sqrt[3]{k}\right|}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt[3]{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]

  11. Final simplification0.5

    \[\leadsto \sqrt{\frac{1}{\sqrt{k}}} \cdot \left(\sqrt{\frac{\sqrt{1}}{\left|\sqrt[3]{k}\right|} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt[3]{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))