Average Error: 24.5 → 24.5
Time: 49.1s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[\left(R \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right) + \left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)}\right)\right) \cdot \left(\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right)\right)} - \left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right) \cdot 2\]
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)
↓
\left(R \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right) + \left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)}\right)\right) \cdot \left(\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right)\right)} - \left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right) \cdot 2
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2004094 = R;
        double r2004095 = 2.0;
        double r2004096 = phi1;
        double r2004097 = phi2;
        double r2004098 = r2004096 - r2004097;
        double r2004099 = r2004098 / r2004095;
        double r2004100 = sin(r2004099);
        double r2004101 = pow(r2004100, r2004095);
        double r2004102 = cos(r2004096);
        double r2004103 = cos(r2004097);
        double r2004104 = r2004102 * r2004103;
        double r2004105 = lambda1;
        double r2004106 = lambda2;
        double r2004107 = r2004105 - r2004106;
        double r2004108 = r2004107 / r2004095;
        double r2004109 = sin(r2004108);
        double r2004110 = r2004104 * r2004109;
        double r2004111 = r2004110 * r2004109;
        double r2004112 = r2004101 + r2004111;
        double r2004113 = sqrt(r2004112);
        double r2004114 = 1.0;
        double r2004115 = r2004114 - r2004112;
        double r2004116 = sqrt(r2004115);
        double r2004117 = atan2(r2004113, r2004116);
        double r2004118 = r2004095 * r2004117;
        double r2004119 = r2004094 * r2004118;
        return r2004119;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2004120 = R;
        double r2004121 = phi1;
        double r2004122 = phi2;
        double r2004123 = r2004121 - r2004122;
        double r2004124 = 2.0;
        double r2004125 = r2004123 / r2004124;
        double r2004126 = sin(r2004125);
        double r2004127 = r2004126 * r2004126;
        double r2004128 = cos(r2004122);
        double r2004129 = lambda1;
        double r2004130 = lambda2;
        double r2004131 = r2004129 - r2004130;
        double r2004132 = r2004131 / r2004124;
        double r2004133 = sin(r2004132);
        double r2004134 = r2004128 * r2004133;
        double r2004135 = cos(r2004121);
        double r2004136 = r2004134 * r2004135;
        double r2004137 = r2004136 * r2004133;
        double r2004138 = r2004127 + r2004137;
        double r2004139 = sqrt(r2004138);
        double r2004140 = cos(r2004125);
        double r2004141 = cbrt(r2004140);
        double r2004142 = r2004141 * r2004141;
        double r2004143 = r2004141 * r2004142;
        double r2004144 = r2004140 * r2004140;
        double r2004145 = r2004143 * r2004144;
        double r2004146 = cbrt(r2004145);
        double r2004147 = r2004140 * r2004146;
        double r2004148 = r2004147 - r2004137;
        double r2004149 = sqrt(r2004148);
        double r2004150 = atan2(r2004139, r2004149);
        double r2004151 = r2004120 * r2004150;
        double r2004152 = r2004151 * r2004124;
        return r2004152;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 24.5

    \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

  2. Simplified24.5

    \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}} \cdot R\right) \cdot 2}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube24.5

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right)}} - \left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}} \cdot R\right) \cdot 2\]
  5. Using strategy rm

  6. Applied add-cube-cbrt24.5

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}} - \left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}} \cdot R\right) \cdot 2\]

  7. Final simplification24.5

    \[\leadsto \left(R \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right) + \left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 - \phi_2}{2}\right)}\right)\right) \cdot \left(\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right)\right)} - \left(\left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right) \cdot 2\]

Reproduce

herbie shell --seed 2019152 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  (* R (* 2 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))) (sqrt (- 1 (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))))))))