Average Error: 1.5 → 1.3
Time: 13.1s
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 r45079 = 1.0;
        double r45080 = 2.0;
        double r45081 = r45079 / r45080;
        double r45082 = l;
        double r45083 = r45080 * r45082;
        double r45084 = Om;
        double r45085 = r45083 / r45084;
        double r45086 = pow(r45085, r45080);
        double r45087 = kx;
        double r45088 = sin(r45087);
        double r45089 = pow(r45088, r45080);
        double r45090 = ky;
        double r45091 = sin(r45090);
        double r45092 = pow(r45091, r45080);
        double r45093 = r45089 + r45092;
        double r45094 = r45086 * r45093;
        double r45095 = r45079 + r45094;
        double r45096 = sqrt(r45095);
        double r45097 = r45079 / r45096;
        double r45098 = r45079 + r45097;
        double r45099 = r45081 * r45098;
        double r45100 = sqrt(r45099);
        return r45100;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r45101 = 1.0;
        double r45102 = 2.0;
        double r45103 = r45101 / r45102;
        double r45104 = l;
        double r45105 = r45102 * r45104;
        double r45106 = Om;
        double r45107 = r45105 / r45106;
        double r45108 = 2.0;
        double r45109 = r45102 / r45108;
        double r45110 = pow(r45107, r45109);
        double r45111 = kx;
        double r45112 = sin(r45111);
        double r45113 = pow(r45112, r45102);
        double r45114 = ky;
        double r45115 = sin(r45114);
        double r45116 = pow(r45115, r45102);
        double r45117 = r45113 + r45116;
        double r45118 = r45110 * r45117;
        double r45119 = r45110 * r45118;
        double r45120 = r45101 + r45119;
        double r45121 = sqrt(r45120);
        double r45122 = r45101 / r45121;
        double r45123 = r45101 + r45122;
        double r45124 = r45103 * r45123;
        double r45125 = sqrt(r45124);
        return r45125;
}

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 2019351 
(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))))))))))