Average Error: 16.5 → 3.6
Time: 31.4s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\log \left(e^{\sin \phi_1 \cdot \sin \phi_2}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\]
\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(\log \left(e^{\sin \phi_1 \cdot \sin \phi_2}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22200 = phi1;
        double r22201 = sin(r22200);
        double r22202 = phi2;
        double r22203 = sin(r22202);
        double r22204 = r22201 * r22203;
        double r22205 = cos(r22200);
        double r22206 = cos(r22202);
        double r22207 = r22205 * r22206;
        double r22208 = lambda1;
        double r22209 = lambda2;
        double r22210 = r22208 - r22209;
        double r22211 = cos(r22210);
        double r22212 = r22207 * r22211;
        double r22213 = r22204 + r22212;
        double r22214 = acos(r22213);
        double r22215 = R;
        double r22216 = r22214 * r22215;
        return r22216;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22217 = R;
        double r22218 = phi1;
        double r22219 = sin(r22218);
        double r22220 = phi2;
        double r22221 = sin(r22220);
        double r22222 = r22219 * r22221;
        double r22223 = exp(r22222);
        double r22224 = log(r22223);
        double r22225 = cos(r22218);
        double r22226 = cos(r22220);
        double r22227 = r22225 * r22226;
        double r22228 = lambda1;
        double r22229 = cos(r22228);
        double r22230 = lambda2;
        double r22231 = cos(r22230);
        double r22232 = r22229 * r22231;
        double r22233 = sin(r22228);
        double r22234 = sin(r22230);
        double r22235 = r22233 * r22234;
        double r22236 = r22232 + r22235;
        double r22237 = r22227 * r22236;
        double r22238 = r22224 + r22237;
        double r22239 = acos(r22238);
        double r22240 = r22217 * r22239;
        return r22240;
}

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 16.5

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

  3. Applied cos-diff3.6

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

  5. Applied add-exp-log3.6

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

  7. Applied add-log-exp3.6

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

  9. Applied add-log-exp3.6

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

  10. Final simplification3.6

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

Reproduce

herbie shell --seed 2019308 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))