Average Error: 17.0 → 3.7
Time: 47.8s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \log \left(e^{\sin \phi_2 \cdot \sin \phi_1}\right)\right)\right)\]
\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(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \log \left(e^{\sin \phi_2 \cdot \sin \phi_1}\right)\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1467402 = phi1;
        double r1467403 = sin(r1467402);
        double r1467404 = phi2;
        double r1467405 = sin(r1467404);
        double r1467406 = r1467403 * r1467405;
        double r1467407 = cos(r1467402);
        double r1467408 = cos(r1467404);
        double r1467409 = r1467407 * r1467408;
        double r1467410 = lambda1;
        double r1467411 = lambda2;
        double r1467412 = r1467410 - r1467411;
        double r1467413 = cos(r1467412);
        double r1467414 = r1467409 * r1467413;
        double r1467415 = r1467406 + r1467414;
        double r1467416 = acos(r1467415);
        double r1467417 = R;
        double r1467418 = r1467416 * r1467417;
        return r1467418;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1467419 = R;
        double r1467420 = phi1;
        double r1467421 = cos(r1467420);
        double r1467422 = phi2;
        double r1467423 = cos(r1467422);
        double r1467424 = lambda1;
        double r1467425 = sin(r1467424);
        double r1467426 = lambda2;
        double r1467427 = sin(r1467426);
        double r1467428 = cos(r1467424);
        double r1467429 = cos(r1467426);
        double r1467430 = r1467428 * r1467429;
        double r1467431 = fma(r1467425, r1467427, r1467430);
        double r1467432 = r1467423 * r1467431;
        double r1467433 = sin(r1467422);
        double r1467434 = sin(r1467420);
        double r1467435 = r1467433 * r1467434;
        double r1467436 = exp(r1467435);
        double r1467437 = log(r1467436);
        double r1467438 = fma(r1467421, r1467432, r1467437);
        double r1467439 = acos(r1467438);
        double r1467440 = r1467419 * r1467439;
        return r1467440;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 17.0

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

  2. Simplified17.0

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

  4. Applied cos-diff3.7

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

  6. Applied add-log-exp3.7

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

  7. Simplified3.7

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

  8. Taylor expanded around 0 3.7

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

  10. Applied add-log-exp3.7

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

  11. Final simplification3.7

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

Reproduce

herbie shell --seed 2019165 +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))