Average Error: 1.5 → 1.3
Time: 17.4s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}\]
\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 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r35175 = 1.0;
        double r35176 = 2.0;
        double r35177 = r35175 / r35176;
        double r35178 = l;
        double r35179 = r35176 * r35178;
        double r35180 = Om;
        double r35181 = r35179 / r35180;
        double r35182 = pow(r35181, r35176);
        double r35183 = kx;
        double r35184 = sin(r35183);
        double r35185 = pow(r35184, r35176);
        double r35186 = ky;
        double r35187 = sin(r35186);
        double r35188 = pow(r35187, r35176);
        double r35189 = r35185 + r35188;
        double r35190 = r35182 * r35189;
        double r35191 = r35175 + r35190;
        double r35192 = sqrt(r35191);
        double r35193 = r35175 / r35192;
        double r35194 = r35175 + r35193;
        double r35195 = r35177 * r35194;
        double r35196 = sqrt(r35195);
        return r35196;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r35197 = 1.0;
        double r35198 = 2.0;
        double r35199 = r35197 / r35198;
        double r35200 = l;
        double r35201 = r35198 * r35200;
        double r35202 = Om;
        double r35203 = r35201 / r35202;
        double r35204 = 2.0;
        double r35205 = r35198 / r35204;
        double r35206 = pow(r35203, r35205);
        double r35207 = kx;
        double r35208 = sin(r35207);
        double r35209 = pow(r35208, r35198);
        double r35210 = ky;
        double r35211 = sin(r35210);
        double r35212 = pow(r35211, r35198);
        double r35213 = r35209 + r35212;
        double r35214 = r35206 * r35213;
        double r35215 = r35206 * r35214;
        double r35216 = r35197 + r35215;
        double r35217 = sqrt(r35216);
        double r35218 = r35197 / r35217;
        double r35219 = r35197 + r35218;
        double r35220 = r35199 * r35219;
        double r35221 = sqrt(r35220);
        return r35221;
}

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 1.5

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

  3. Applied sqr-pow1.5

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

  4. Applied associate-*l*1.3

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

  5. Final simplification1.3

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

Reproduce

herbie shell --seed 2019297 
(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))))))))))