Average Error: 16.7 → 4.2
Time: 54.1s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \left(\pi \cdot \frac{1}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right)\]
\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 \left(\pi \cdot \frac{1}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1514500 = phi1;
        double r1514501 = sin(r1514500);
        double r1514502 = phi2;
        double r1514503 = sin(r1514502);
        double r1514504 = r1514501 * r1514503;
        double r1514505 = cos(r1514500);
        double r1514506 = cos(r1514502);
        double r1514507 = r1514505 * r1514506;
        double r1514508 = lambda1;
        double r1514509 = lambda2;
        double r1514510 = r1514508 - r1514509;
        double r1514511 = cos(r1514510);
        double r1514512 = r1514507 * r1514511;
        double r1514513 = r1514504 + r1514512;
        double r1514514 = acos(r1514513);
        double r1514515 = R;
        double r1514516 = r1514514 * r1514515;
        return r1514516;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1514517 = R;
        double r1514518 = atan2(1.0, 0.0);
        double r1514519 = 0.5;
        double r1514520 = r1514518 * r1514519;
        double r1514521 = phi2;
        double r1514522 = sin(r1514521);
        double r1514523 = phi1;
        double r1514524 = sin(r1514523);
        double r1514525 = cos(r1514523);
        double r1514526 = cos(r1514521);
        double r1514527 = r1514525 * r1514526;
        double r1514528 = lambda1;
        double r1514529 = sin(r1514528);
        double r1514530 = lambda2;
        double r1514531 = sin(r1514530);
        double r1514532 = cos(r1514530);
        double r1514533 = cos(r1514528);
        double r1514534 = r1514532 * r1514533;
        double r1514535 = fma(r1514529, r1514531, r1514534);
        double r1514536 = r1514527 * r1514535;
        double r1514537 = fma(r1514522, r1514524, r1514536);
        double r1514538 = asin(r1514537);
        double r1514539 = r1514520 - r1514538;
        double r1514540 = r1514517 * r1514539;
        return r1514540;
}

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 16.7

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

  2. Simplified16.7

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

  4. Applied cos-diff4.1

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

  6. Applied add-log-exp4.1

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

  7. Simplified4.1

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

  9. Applied acos-asin4.2

    \[\leadsto R \cdot \log \left(e^{\color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}}\right)\]

  10. Applied exp-diff4.2

    \[\leadsto R \cdot \log \color{blue}{\left(\frac{e^{\frac{\pi}{2}}}{e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}}\right)}\]

  11. Applied log-div4.2

    \[\leadsto R \cdot \color{blue}{\left(\log \left(e^{\frac{\pi}{2}}\right) - \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right)\right)}\]

  12. Simplified4.2

    \[\leadsto R \cdot \left(\color{blue}{\frac{\pi}{2}} - \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right)\right)\]

  13. Simplified4.2

    \[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right)}\right)\]

  14. Taylor expanded around 0 4.2

    \[\leadsto R \cdot \color{blue}{\left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)}\]

  15. Final simplification4.2

    \[\leadsto R \cdot \left(\pi \cdot \frac{1}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019158 +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))