Average Error: 16.9 → 3.9
Time: 13.6s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R\]
\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
↓
\log \left(e^{\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
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r23289 = phi1;
        double r23290 = sin(r23289);
        double r23291 = phi2;
        double r23292 = sin(r23291);
        double r23293 = r23290 * r23292;
        double r23294 = cos(r23289);
        double r23295 = cos(r23291);
        double r23296 = r23294 * r23295;
        double r23297 = lambda1;
        double r23298 = lambda2;
        double r23299 = r23297 - r23298;
        double r23300 = cos(r23299);
        double r23301 = r23296 * r23300;
        double r23302 = r23293 + r23301;
        double r23303 = acos(r23302);
        double r23304 = R;
        double r23305 = r23303 * r23304;
        return r23305;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r23306 = phi1;
        double r23307 = sin(r23306);
        double r23308 = phi2;
        double r23309 = sin(r23308);
        double r23310 = r23307 * r23309;
        double r23311 = cos(r23306);
        double r23312 = cos(r23308);
        double r23313 = r23311 * r23312;
        double r23314 = lambda1;
        double r23315 = cos(r23314);
        double r23316 = lambda2;
        double r23317 = cos(r23316);
        double r23318 = r23315 * r23317;
        double r23319 = sin(r23314);
        double r23320 = sin(r23316);
        double r23321 = r23319 * r23320;
        double r23322 = r23318 + r23321;
        double r23323 = r23313 * r23322;
        double r23324 = r23310 + r23323;
        double r23325 = acos(r23324);
        double r23326 = exp(r23325);
        double r23327 = log(r23326);
        double r23328 = R;
        double r23329 = r23327 * r23328;
        return r23329;
}

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-diff3.9

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

  5. Applied add-log-exp3.9

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

  6. Final simplification3.9

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

Reproduce

herbie shell --seed 2020065 
(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))