Average Error: 16.8 → 3.7
Time: 44.4s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\log \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)} \cdot R\]
\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
↓
e^{\log \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r23689 = phi1;
        double r23690 = sin(r23689);
        double r23691 = phi2;
        double r23692 = sin(r23691);
        double r23693 = r23690 * r23692;
        double r23694 = cos(r23689);
        double r23695 = cos(r23691);
        double r23696 = r23694 * r23695;
        double r23697 = lambda1;
        double r23698 = lambda2;
        double r23699 = r23697 - r23698;
        double r23700 = cos(r23699);
        double r23701 = r23696 * r23700;
        double r23702 = r23693 + r23701;
        double r23703 = acos(r23702);
        double r23704 = R;
        double r23705 = r23703 * r23704;
        return r23705;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r23706 = atan2(1.0, 0.0);
        double r23707 = 2.0;
        double r23708 = r23706 / r23707;
        double r23709 = phi1;
        double r23710 = sin(r23709);
        double r23711 = phi2;
        double r23712 = sin(r23711);
        double r23713 = cos(r23709);
        double r23714 = cos(r23711);
        double r23715 = r23713 * r23714;
        double r23716 = lambda2;
        double r23717 = cos(r23716);
        double r23718 = lambda1;
        double r23719 = cos(r23718);
        double r23720 = sin(r23718);
        double r23721 = sin(r23716);
        double r23722 = r23720 * r23721;
        double r23723 = fma(r23717, r23719, r23722);
        double r23724 = r23715 * r23723;
        double r23725 = fma(r23710, r23712, r23724);
        double r23726 = asin(r23725);
        double r23727 = r23708 - r23726;
        double r23728 = log(r23727);
        double r23729 = exp(r23728);
        double r23730 = R;
        double r23731 = r23729 * r23730;
        return r23731;
}

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 16.8

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

  2. Simplified16.7

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

  4. Applied sub-neg16.7

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

  5. Applied cos-sum3.6

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

  6. Simplified3.6

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

  8. Applied add-exp-log3.6

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

  9. Simplified3.6

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

  11. Applied acos-asin3.7

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

  12. Final simplification3.7

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(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))