Average Error: 16.9 → 4.1
Time: 46.5s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \frac{\cos \phi_2 \cdot \cos \phi_1}{\frac{\left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)}{\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) + \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)}}\right) \cdot R\]
\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(\sin \phi_2 \cdot \sin \phi_1 + \frac{\cos \phi_2 \cdot \cos \phi_1}{\frac{\left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)}{\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) + \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)}}\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r657330 = phi1;
        double r657331 = sin(r657330);
        double r657332 = phi2;
        double r657333 = sin(r657332);
        double r657334 = r657331 * r657333;
        double r657335 = cos(r657330);
        double r657336 = cos(r657332);
        double r657337 = r657335 * r657336;
        double r657338 = lambda1;
        double r657339 = lambda2;
        double r657340 = r657338 - r657339;
        double r657341 = cos(r657340);
        double r657342 = r657337 * r657341;
        double r657343 = r657334 + r657342;
        double r657344 = acos(r657343);
        double r657345 = R;
        double r657346 = r657344 * r657345;
        return r657346;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r657347 = phi2;
        double r657348 = sin(r657347);
        double r657349 = phi1;
        double r657350 = sin(r657349);
        double r657351 = r657348 * r657350;
        double r657352 = cos(r657347);
        double r657353 = cos(r657349);
        double r657354 = r657352 * r657353;
        double r657355 = lambda2;
        double r657356 = sin(r657355);
        double r657357 = lambda1;
        double r657358 = sin(r657357);
        double r657359 = r657356 * r657358;
        double r657360 = r657359 * r657359;
        double r657361 = cos(r657355);
        double r657362 = cos(r657357);
        double r657363 = r657361 * r657362;
        double r657364 = r657363 * r657359;
        double r657365 = r657360 - r657364;
        double r657366 = r657363 * r657363;
        double r657367 = r657365 + r657366;
        double r657368 = r657359 * r657360;
        double r657369 = r657366 * r657363;
        double r657370 = r657368 + r657369;
        double r657371 = r657367 / r657370;
        double r657372 = r657354 / r657371;
        double r657373 = r657351 + r657372;
        double r657374 = acos(r657373);
        double r657375 = R;
        double r657376 = r657374 * r657375;
        return r657376;
}

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-diff4.1

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

  5. Applied flip3-+4.1

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

  6. Applied associate-*r/4.1

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

  7. Simplified4.1

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

  9. Applied associate-/l*4.1

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

  10. Final simplification4.1

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

Reproduce

herbie shell --seed 2019153 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))