Average Error: 1.7 → 1.7
Time: 10.4s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \log \left(e^{\frac{1}{\sqrt[3]{{\left(\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}^{3}}}}\right)\right)}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\frac{1}{2} \cdot \left(1 + \log \left(e^{\frac{1}{\sqrt[3]{{\left(\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}^{3}}}}\right)\right)}
double f(double l, double Om, double kx, double ky) {
        double r80200 = 1.0;
        double r80201 = 2.0;
        double r80202 = r80200 / r80201;
        double r80203 = l;
        double r80204 = r80201 * r80203;
        double r80205 = Om;
        double r80206 = r80204 / r80205;
        double r80207 = pow(r80206, r80201);
        double r80208 = kx;
        double r80209 = sin(r80208);
        double r80210 = pow(r80209, r80201);
        double r80211 = ky;
        double r80212 = sin(r80211);
        double r80213 = pow(r80212, r80201);
        double r80214 = r80210 + r80213;
        double r80215 = r80207 * r80214;
        double r80216 = r80200 + r80215;
        double r80217 = sqrt(r80216);
        double r80218 = r80200 / r80217;
        double r80219 = r80200 + r80218;
        double r80220 = r80202 * r80219;
        double r80221 = sqrt(r80220);
        return r80221;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r80222 = 1.0;
        double r80223 = 2.0;
        double r80224 = r80222 / r80223;
        double r80225 = l;
        double r80226 = r80223 * r80225;
        double r80227 = Om;
        double r80228 = r80226 / r80227;
        double r80229 = pow(r80228, r80223);
        double r80230 = kx;
        double r80231 = sin(r80230);
        double r80232 = pow(r80231, r80223);
        double r80233 = ky;
        double r80234 = sin(r80233);
        double r80235 = pow(r80234, r80223);
        double r80236 = r80232 + r80235;
        double r80237 = r80229 * r80236;
        double r80238 = r80222 + r80237;
        double r80239 = sqrt(r80238);
        double r80240 = 3.0;
        double r80241 = pow(r80239, r80240);
        double r80242 = cbrt(r80241);
        double r80243 = r80222 / r80242;
        double r80244 = exp(r80243);
        double r80245 = log(r80244);
        double r80246 = r80222 + r80245;
        double r80247 = r80224 * r80246;
        double r80248 = sqrt(r80247);
        return r80248;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.7

    \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube1.7

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\sqrt[3]{\left(\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)} \cdot \sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]

  4. Simplified1.7

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt[3]{\color{blue}{{\left(\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}^{3}}}}\right)}\]
  5. Using strategy rm

  6. Applied add-log-exp1.7

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\log \left(e^{\frac{1}{\sqrt[3]{{\left(\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}^{3}}}}\right)}\right)}\]

  7. Final simplification1.7

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \log \left(e^{\frac{1}{\sqrt[3]{{\left(\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}^{3}}}}\right)\right)}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))