Average Error: 16.8 → 3.9
Time: 1.1m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\frac{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}{\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \lambda_2} + \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right) \cdot R\]
\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
↓
\cos^{-1} \left(\frac{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}{\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \lambda_2} + \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1713313 = phi1;
        double r1713314 = sin(r1713313);
        double r1713315 = phi2;
        double r1713316 = sin(r1713315);
        double r1713317 = r1713314 * r1713316;
        double r1713318 = cos(r1713313);
        double r1713319 = cos(r1713315);
        double r1713320 = r1713318 * r1713319;
        double r1713321 = lambda1;
        double r1713322 = lambda2;
        double r1713323 = r1713321 - r1713322;
        double r1713324 = cos(r1713323);
        double r1713325 = r1713320 * r1713324;
        double r1713326 = r1713317 + r1713325;
        double r1713327 = acos(r1713326);
        double r1713328 = R;
        double r1713329 = r1713327 * r1713328;
        return r1713329;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1713330 = lambda1;
        double r1713331 = cos(r1713330);
        double r1713332 = lambda2;
        double r1713333 = cos(r1713332);
        double r1713334 = r1713331 * r1713333;
        double r1713335 = r1713334 * r1713334;
        double r1713336 = sin(r1713330);
        double r1713337 = sin(r1713332);
        double r1713338 = r1713336 * r1713337;
        double r1713339 = r1713338 * r1713338;
        double r1713340 = r1713335 - r1713339;
        double r1713341 = phi1;
        double r1713342 = cos(r1713341);
        double r1713343 = phi2;
        double r1713344 = cos(r1713343);
        double r1713345 = r1713342 * r1713344;
        double r1713346 = r1713340 * r1713345;
        double r1713347 = r1713334 - r1713338;
        double r1713348 = r1713346 / r1713347;
        double r1713349 = sin(r1713343);
        double r1713350 = sin(r1713341);
        double r1713351 = r1713349 * r1713350;
        double r1713352 = expm1(r1713351);
        double r1713353 = log1p(r1713352);
        double r1713354 = r1713348 + r1713353;
        double r1713355 = acos(r1713354);
        double r1713356 = R;
        double r1713357 = r1713355 * r1713356;
        return r1713357;
}

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 16.8

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
  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 log1p-expm1-u3.9

    \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\]
  6. Using strategy rm

  7. Applied flip-+3.9

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

  8. Applied associate-*r/3.9

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

  9. Final simplification3.9

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))