Spherical law of cosines

Average Error: 16.9 → 4.1

Time: 45.0s

Precision: 64

Math

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

↓

\[\cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \frac{\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), \cos \lambda_2 \cdot \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)}\right) \cdot R\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r648753 = phi1;
        double r648754 = sin(r648753);
        double r648755 = phi2;
        double r648756 = sin(r648755);
        double r648757 = r648754 * r648756;
        double r648758 = cos(r648753);
        double r648759 = cos(r648755);
        double r648760 = r648758 * r648759;
        double r648761 = lambda1;
        double r648762 = lambda2;
        double r648763 = r648761 - r648762;
        double r648764 = cos(r648763);
        double r648765 = r648760 * r648764;
        double r648766 = r648757 + r648765;
        double r648767 = acos(r648766);
        double r648768 = R;
        double r648769 = r648767 * r648768;
        return r648769;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r648770 = phi2;
        double r648771 = sin(r648770);
        double r648772 = phi1;
        double r648773 = sin(r648772);
        double r648774 = r648771 * r648773;
        double r648775 = lambda2;
        double r648776 = cos(r648775);
        double r648777 = lambda1;
        double r648778 = cos(r648777);
        double r648779 = r648776 * r648778;
        double r648780 = r648779 * r648779;
        double r648781 = expm1(r648780);
        double r648782 = log1p(r648781);
        double r648783 = sin(r648777);
        double r648784 = sin(r648775);
        double r648785 = r648783 * r648784;
        double r648786 = r648785 * r648785;
        double r648787 = r648785 * r648786;
        double r648788 = fma(r648782, r648779, r648787);
        double r648789 = cos(r648770);
        double r648790 = cos(r648772);
        double r648791 = r648789 * r648790;
        double r648792 = r648788 * r648791;
        double r648793 = r648785 * r648779;
        double r648794 = r648786 - r648793;
        double r648795 = r648780 + r648794;
        double r648796 = r648792 / r648795;
        double r648797 = r648774 + r648796;
        double r648798 = acos(r648797);
        double r648799 = R;
        double r648800 = r648798 * r648799;
        return r648800;
}

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 16.9

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

3. Applied cos-diff 4.1

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

5. Applied flip3-+ 4.1

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

6. Applied associate-*r/ 4.1

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

7. Simplified 4.1

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

9. Applied log1p-expm1-u 4.1

   \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)}, \cos \lambda_2 \cdot \cos \lambda_1, \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R\]

10. Final simplification 4.1

    \[\leadsto \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \frac{\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), \cos \lambda_2 \cdot \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)}\right) \cdot R\]

Reproduce

herbie shell --seed 2019153 +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))