Average Error: 1.7 → 1.5
Time: 3.9m
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\left(\ell \cdot \frac{2}{Om}\right) \cdot \left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right) + 1}} + \frac{1}{2}}\]
\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{\frac{1}{2}}{\sqrt{\left(\ell \cdot \frac{2}{Om}\right) \cdot \left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right) + 1}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r24093165 = 1.0;
        double r24093166 = 2.0;
        double r24093167 = r24093165 / r24093166;
        double r24093168 = l;
        double r24093169 = r24093166 * r24093168;
        double r24093170 = Om;
        double r24093171 = r24093169 / r24093170;
        double r24093172 = pow(r24093171, r24093166);
        double r24093173 = kx;
        double r24093174 = sin(r24093173);
        double r24093175 = pow(r24093174, r24093166);
        double r24093176 = ky;
        double r24093177 = sin(r24093176);
        double r24093178 = pow(r24093177, r24093166);
        double r24093179 = r24093175 + r24093178;
        double r24093180 = r24093172 * r24093179;
        double r24093181 = r24093165 + r24093180;
        double r24093182 = sqrt(r24093181);
        double r24093183 = r24093165 / r24093182;
        double r24093184 = r24093165 + r24093183;
        double r24093185 = r24093167 * r24093184;
        double r24093186 = sqrt(r24093185);
        return r24093186;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r24093187 = 0.5;
        double r24093188 = l;
        double r24093189 = 2.0;
        double r24093190 = Om;
        double r24093191 = r24093189 / r24093190;
        double r24093192 = r24093188 * r24093191;
        double r24093193 = ky;
        double r24093194 = sin(r24093193);
        double r24093195 = r24093194 * r24093194;
        double r24093196 = kx;
        double r24093197 = sin(r24093196);
        double r24093198 = r24093197 * r24093197;
        double r24093199 = r24093195 + r24093198;
        double r24093200 = r24093199 * r24093192;
        double r24093201 = r24093192 * r24093200;
        double r24093202 = 1.0;
        double r24093203 = r24093201 + r24093202;
        double r24093204 = sqrt(r24093203);
        double r24093205 = r24093187 / r24093204;
        double r24093206 = r24093205 + r24093187;
        double r24093207 = sqrt(r24093206);
        return r24093207;
}

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. Simplified1.5

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

  3. Taylor expanded around -inf 1.5

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

  4. Simplified1.5

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

  5. Final simplification1.5

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

Reproduce

herbie shell --seed 2019125 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))