Average Error: 16.9 → 4.0
Time: 49.2s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\left(\sqrt[3]{\sin \lambda_2} \cdot \left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right)\right) + \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_2 \cdot \sin \phi_1\right)\]
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R
↓
R \cdot \cos^{-1} \left(\left(\sqrt[3]{\sin \lambda_2} \cdot \left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right)\right) + \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_2 \cdot \sin \phi_1\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1159046 = phi1;
        double r1159047 = sin(r1159046);
        double r1159048 = phi2;
        double r1159049 = sin(r1159048);
        double r1159050 = r1159047 * r1159049;
        double r1159051 = cos(r1159046);
        double r1159052 = cos(r1159048);
        double r1159053 = r1159051 * r1159052;
        double r1159054 = lambda1;
        double r1159055 = lambda2;
        double r1159056 = r1159054 - r1159055;
        double r1159057 = cos(r1159056);
        double r1159058 = r1159053 * r1159057;
        double r1159059 = r1159050 + r1159058;
        double r1159060 = acos(r1159059);
        double r1159061 = R;
        double r1159062 = r1159060 * r1159061;
        return r1159062;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1159063 = R;
        double r1159064 = lambda2;
        double r1159065 = sin(r1159064);
        double r1159066 = cbrt(r1159065);
        double r1159067 = lambda1;
        double r1159068 = sin(r1159067);
        double r1159069 = r1159066 * r1159066;
        double r1159070 = r1159068 * r1159069;
        double r1159071 = r1159066 * r1159070;
        double r1159072 = cos(r1159064);
        double r1159073 = cos(r1159067);
        double r1159074 = r1159072 * r1159073;
        double r1159075 = r1159071 + r1159074;
        double r1159076 = phi1;
        double r1159077 = cos(r1159076);
        double r1159078 = phi2;
        double r1159079 = cos(r1159078);
        double r1159080 = r1159077 * r1159079;
        double r1159081 = r1159075 * r1159080;
        double r1159082 = sin(r1159078);
        double r1159083 = sin(r1159076);
        double r1159084 = r1159082 * r1159083;
        double r1159085 = r1159081 + r1159084;
        double r1159086 = acos(r1159085);
        double r1159087 = r1159063 * r1159086;
        return r1159087;
}

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 16.9

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

  3. Applied cos-diff4.0

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R\]
  4. Using strategy rm

  5. Applied add-cube-cbrt4.0

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

  6. Applied associate-*r*4.0

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

  7. Final simplification4.0

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

Reproduce

herbie shell --seed 2019135 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))