Average Error: 24.6 → 24.6
Time: 55.0s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}\right)\]
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{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2684038 = R;
        double r2684039 = 2.0;
        double r2684040 = phi1;
        double r2684041 = phi2;
        double r2684042 = r2684040 - r2684041;
        double r2684043 = r2684042 / r2684039;
        double r2684044 = sin(r2684043);
        double r2684045 = pow(r2684044, r2684039);
        double r2684046 = cos(r2684040);
        double r2684047 = cos(r2684041);
        double r2684048 = r2684046 * r2684047;
        double r2684049 = lambda1;
        double r2684050 = lambda2;
        double r2684051 = r2684049 - r2684050;
        double r2684052 = r2684051 / r2684039;
        double r2684053 = sin(r2684052);
        double r2684054 = r2684048 * r2684053;
        double r2684055 = r2684054 * r2684053;
        double r2684056 = r2684045 + r2684055;
        double r2684057 = sqrt(r2684056);
        double r2684058 = 1.0;
        double r2684059 = r2684058 - r2684056;
        double r2684060 = sqrt(r2684059);
        double r2684061 = atan2(r2684057, r2684060);
        double r2684062 = r2684039 * r2684061;
        double r2684063 = r2684038 * r2684062;
        return r2684063;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2684064 = R;
        double r2684065 = 2.0;
        double r2684066 = lambda1;
        double r2684067 = lambda2;
        double r2684068 = r2684066 - r2684067;
        double r2684069 = r2684068 / r2684065;
        double r2684070 = sin(r2684069);
        double r2684071 = phi2;
        double r2684072 = cos(r2684071);
        double r2684073 = phi1;
        double r2684074 = cos(r2684073);
        double r2684075 = r2684072 * r2684074;
        double r2684076 = r2684075 * r2684070;
        double r2684077 = r2684070 * r2684076;
        double r2684078 = r2684073 - r2684071;
        double r2684079 = r2684078 / r2684065;
        double r2684080 = sin(r2684079);
        double r2684081 = pow(r2684080, r2684065);
        double r2684082 = r2684077 + r2684081;
        double r2684083 = sqrt(r2684082);
        double r2684084 = 1.0;
        double r2684085 = log1p(r2684070);
        double r2684086 = expm1(r2684085);
        double r2684087 = r2684086 * r2684076;
        double r2684088 = r2684087 + r2684081;
        double r2684089 = r2684084 - r2684088;
        double r2684090 = sqrt(r2684089);
        double r2684091 = atan2(r2684083, r2684090);
        double r2684092 = r2684065 * r2684091;
        double r2684093 = r2684064 * r2684092;
        return r2684093;
}

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 expm1-log1p-u24.6

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

  4. Final simplification24.6

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(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))))))))))