Average Error: 16.5 → 3.7
Time: 23.1s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \left(-\lambda_2\right)} \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}\right)\right) \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}\right)\right) \cdot R\]
\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
↓
\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 - \left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \left(-\lambda_2\right)} \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}\right)\right) \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r23329 = phi1;
        double r23330 = sin(r23329);
        double r23331 = phi2;
        double r23332 = sin(r23331);
        double r23333 = r23330 * r23332;
        double r23334 = cos(r23329);
        double r23335 = cos(r23331);
        double r23336 = r23334 * r23335;
        double r23337 = lambda1;
        double r23338 = lambda2;
        double r23339 = r23337 - r23338;
        double r23340 = cos(r23339);
        double r23341 = r23336 * r23340;
        double r23342 = r23333 + r23341;
        double r23343 = acos(r23342);
        double r23344 = R;
        double r23345 = r23343 * r23344;
        return r23345;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r23346 = phi1;
        double r23347 = sin(r23346);
        double r23348 = phi2;
        double r23349 = sin(r23348);
        double r23350 = r23347 * r23349;
        double r23351 = cos(r23346);
        double r23352 = cos(r23348);
        double r23353 = r23351 * r23352;
        double r23354 = lambda1;
        double r23355 = cos(r23354);
        double r23356 = lambda2;
        double r23357 = cos(r23356);
        double r23358 = r23355 * r23357;
        double r23359 = sin(r23354);
        double r23360 = -r23356;
        double r23361 = sin(r23360);
        double r23362 = cbrt(r23361);
        double r23363 = r23362 * r23362;
        double r23364 = r23359 * r23363;
        double r23365 = r23364 * r23362;
        double r23366 = r23358 - r23365;
        double r23367 = r23353 * r23366;
        double r23368 = r23350 + r23367;
        double r23369 = acos(r23368);
        double r23370 = R;
        double r23371 = r23369 * r23370;
        return r23371;
}

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 sub-neg16.5

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

  4. Applied cos-sum3.7

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

  5. Simplified3.7

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

  7. Applied add-cube-cbrt3.7

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

  8. Applied associate-*r*3.7

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

  9. Final simplification3.7

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

Reproduce

herbie shell --seed 2020042 
(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))