Spherical law of cosines

Average Error: 16.5 → 3.9

Time: 59.7s

Precision: 64

Math

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

↓

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

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r5651749 = phi1;
        double r5651750 = sin(r5651749);
        double r5651751 = phi2;
        double r5651752 = sin(r5651751);
        double r5651753 = r5651750 * r5651752;
        double r5651754 = cos(r5651749);
        double r5651755 = cos(r5651751);
        double r5651756 = r5651754 * r5651755;
        double r5651757 = lambda1;
        double r5651758 = lambda2;
        double r5651759 = r5651757 - r5651758;
        double r5651760 = cos(r5651759);
        double r5651761 = r5651756 * r5651760;
        double r5651762 = r5651753 + r5651761;
        double r5651763 = acos(r5651762);
        double r5651764 = R;
        double r5651765 = r5651763 * r5651764;
        return r5651765;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r5651766 = atan2(1.0, 0.0);
        double r5651767 = 2.0;
        double r5651768 = r5651766 / r5651767;
        double r5651769 = phi1;
        double r5651770 = sin(r5651769);
        double r5651771 = phi2;
        double r5651772 = sin(r5651771);
        double r5651773 = cos(r5651769);
        double r5651774 = cos(r5651771);
        double r5651775 = lambda1;
        double r5651776 = cos(r5651775);
        double r5651777 = lambda2;
        double r5651778 = cos(r5651777);
        double r5651779 = sin(r5651775);
        double r5651780 = sin(r5651777);
        double r5651781 = r5651779 * r5651780;
        double r5651782 = fma(r5651776, r5651778, r5651781);
        double r5651783 = r5651774 * r5651782;
        double r5651784 = r5651773 * r5651783;
        double r5651785 = fma(r5651770, r5651772, r5651784);
        double r5651786 = asin(r5651785);
        double r5651787 = r5651768 - r5651786;
        double r5651788 = R;
        double r5651789 = r5651787 * r5651788;
        return r5651789;
}

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 16.5

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

3. Applied cos-diff 3.8

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

5. Applied add-log-exp 3.8

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

6. Simplified 3.8

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

8. Applied acos-asin 3.9

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

9. Applied exp-diff 3.9

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

10. Applied log-div 3.9

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

11. Simplified 3.9

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

12. Simplified 3.9

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

13. Final simplification 3.9

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

Reproduce

herbie shell --seed 2019173 +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))