Average Error: 1.7 → 1.7
Time: 24.8s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\sqrt[3]{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1} \cdot \sqrt[3]{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}} \cdot \sqrt{\sqrt[3]{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}}} + \frac{1}{2}}\]
\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{\sqrt[3]{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1} \cdot \sqrt[3]{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}} \cdot \sqrt{\sqrt[3]{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r838064 = 1.0;
        double r838065 = 2.0;
        double r838066 = r838064 / r838065;
        double r838067 = l;
        double r838068 = r838065 * r838067;
        double r838069 = Om;
        double r838070 = r838068 / r838069;
        double r838071 = pow(r838070, r838065);
        double r838072 = kx;
        double r838073 = sin(r838072);
        double r838074 = pow(r838073, r838065);
        double r838075 = ky;
        double r838076 = sin(r838075);
        double r838077 = pow(r838076, r838065);
        double r838078 = r838074 + r838077;
        double r838079 = r838071 * r838078;
        double r838080 = r838064 + r838079;
        double r838081 = sqrt(r838080);
        double r838082 = r838064 / r838081;
        double r838083 = r838064 + r838082;
        double r838084 = r838066 * r838083;
        double r838085 = sqrt(r838084);
        return r838085;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r838086 = 0.5;
        double r838087 = ky;
        double r838088 = sin(r838087);
        double r838089 = r838088 * r838088;
        double r838090 = kx;
        double r838091 = sin(r838090);
        double r838092 = r838091 * r838091;
        double r838093 = r838089 + r838092;
        double r838094 = 2.0;
        double r838095 = l;
        double r838096 = r838094 * r838095;
        double r838097 = Om;
        double r838098 = r838096 / r838097;
        double r838099 = r838098 * r838098;
        double r838100 = r838093 * r838099;
        double r838101 = 1.0;
        double r838102 = r838100 + r838101;
        double r838103 = cbrt(r838102);
        double r838104 = r838103 * r838103;
        double r838105 = sqrt(r838104);
        double r838106 = sqrt(r838103);
        double r838107 = r838105 * r838106;
        double r838108 = r838086 / r838107;
        double r838109 = r838108 + r838086;
        double r838110 = sqrt(r838109);
        return r838110;
}

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. Simplified1.7

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

  4. Applied add-cube-cbrt1.7

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

  5. Applied sqrt-prod1.7

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

  6. Final simplification1.7

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

Reproduce

herbie shell --seed 2019154 
(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))))))))))