Average Error: 24.6 → 24.6
Time: 30.5s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
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)
↓
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 \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\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)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r60096 = R;
        double r60097 = 2.0;
        double r60098 = phi1;
        double r60099 = phi2;
        double r60100 = r60098 - r60099;
        double r60101 = r60100 / r60097;
        double r60102 = sin(r60101);
        double r60103 = pow(r60102, r60097);
        double r60104 = cos(r60098);
        double r60105 = cos(r60099);
        double r60106 = r60104 * r60105;
        double r60107 = lambda1;
        double r60108 = lambda2;
        double r60109 = r60107 - r60108;
        double r60110 = r60109 / r60097;
        double r60111 = sin(r60110);
        double r60112 = r60106 * r60111;
        double r60113 = r60112 * r60111;
        double r60114 = r60103 + r60113;
        double r60115 = sqrt(r60114);
        double r60116 = 1.0;
        double r60117 = r60116 - r60114;
        double r60118 = sqrt(r60117);
        double r60119 = atan2(r60115, r60118);
        double r60120 = r60097 * r60119;
        double r60121 = r60096 * r60120;
        return r60121;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r60122 = R;
        double r60123 = 2.0;
        double r60124 = phi1;
        double r60125 = phi2;
        double r60126 = r60124 - r60125;
        double r60127 = r60126 / r60123;
        double r60128 = sin(r60127);
        double r60129 = pow(r60128, r60123);
        double r60130 = cos(r60124);
        double r60131 = cos(r60125);
        double r60132 = r60130 * r60131;
        double r60133 = lambda1;
        double r60134 = lambda2;
        double r60135 = r60133 - r60134;
        double r60136 = r60135 / r60123;
        double r60137 = sin(r60136);
        double r60138 = r60132 * r60137;
        double r60139 = expm1(r60137);
        double r60140 = log1p(r60139);
        double r60141 = r60138 * r60140;
        double r60142 = r60129 + r60141;
        double r60143 = sqrt(r60142);
        double r60144 = 1.0;
        double r60145 = r60138 * r60137;
        double r60146 = r60129 + r60145;
        double r60147 = r60144 - r60146;
        double r60148 = sqrt(r60147);
        double r60149 = atan2(r60143, r60148);
        double r60150 = r60123 * r60149;
        double r60151 = r60122 * r60150;
        return r60151;
}

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 24.6

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

  3. Applied log1p-expm1-u24.6

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

  4. Final simplification24.6

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* 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))))))))))