Average Error: 16.6 → 3.9
Time: 38.2s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\left(\sqrt[3]{\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)} + \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_2 \cdot \sin \phi_1\right)\]
\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(\left(\sqrt[3]{\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)} + \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_2 \cdot \sin \phi_1\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1356475 = phi1;
        double r1356476 = sin(r1356475);
        double r1356477 = phi2;
        double r1356478 = sin(r1356477);
        double r1356479 = r1356476 * r1356478;
        double r1356480 = cos(r1356475);
        double r1356481 = cos(r1356477);
        double r1356482 = r1356480 * r1356481;
        double r1356483 = lambda1;
        double r1356484 = lambda2;
        double r1356485 = r1356483 - r1356484;
        double r1356486 = cos(r1356485);
        double r1356487 = r1356482 * r1356486;
        double r1356488 = r1356479 + r1356487;
        double r1356489 = acos(r1356488);
        double r1356490 = R;
        double r1356491 = r1356489 * r1356490;
        return r1356491;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1356492 = R;
        double r1356493 = lambda2;
        double r1356494 = sin(r1356493);
        double r1356495 = lambda1;
        double r1356496 = sin(r1356495);
        double r1356497 = r1356494 * r1356496;
        double r1356498 = r1356497 * r1356497;
        double r1356499 = r1356497 * r1356498;
        double r1356500 = cbrt(r1356499);
        double r1356501 = cos(r1356493);
        double r1356502 = cos(r1356495);
        double r1356503 = r1356501 * r1356502;
        double r1356504 = r1356500 + r1356503;
        double r1356505 = phi1;
        double r1356506 = cos(r1356505);
        double r1356507 = phi2;
        double r1356508 = cos(r1356507);
        double r1356509 = r1356506 * r1356508;
        double r1356510 = r1356504 * r1356509;
        double r1356511 = sin(r1356507);
        double r1356512 = sin(r1356505);
        double r1356513 = r1356511 * r1356512;
        double r1356514 = r1356510 + r1356513;
        double r1356515 = acos(r1356514);
        double r1356516 = r1356492 * r1356515;
        return r1356516;
}

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 16.6

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
  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-cbrt-cube3.9

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

  6. Applied add-cbrt-cube3.9

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

  7. Applied cbrt-unprod3.9

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

  8. Simplified3.9

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

  9. Final simplification3.9

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

Reproduce

herbie shell --seed 2019179 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))