Spherical law of cosines

Average Error: 16.7 → 4.0

Time: 46.4s

Precision: 64

Math

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

↓

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

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r465921 = phi1;
        double r465922 = sin(r465921);
        double r465923 = phi2;
        double r465924 = sin(r465923);
        double r465925 = r465922 * r465924;
        double r465926 = cos(r465921);
        double r465927 = cos(r465923);
        double r465928 = r465926 * r465927;
        double r465929 = lambda1;
        double r465930 = lambda2;
        double r465931 = r465929 - r465930;
        double r465932 = cos(r465931);
        double r465933 = r465928 * r465932;
        double r465934 = r465925 + r465933;
        double r465935 = acos(r465934);
        double r465936 = R;
        double r465937 = r465935 * r465936;
        return r465937;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r465938 = R;
        double r465939 = atan2(1.0, 0.0);
        double r465940 = 2.0;
        double r465941 = r465939 / r465940;
        double r465942 = phi2;
        double r465943 = sin(r465942);
        double r465944 = phi1;
        double r465945 = sin(r465944);
        double r465946 = lambda2;
        double r465947 = sin(r465946);
        double r465948 = lambda1;
        double r465949 = sin(r465948);
        double r465950 = cos(r465948);
        double r465951 = cos(r465946);
        double r465952 = r465950 * r465951;
        double r465953 = fma(r465947, r465949, r465952);
        double r465954 = exp(r465953);
        double r465955 = log(r465954);
        double r465956 = cos(r465942);
        double r465957 = r465955 * r465956;
        double r465958 = cos(r465944);
        double r465959 = r465957 * r465958;
        double r465960 = fma(r465943, r465945, r465959);
        double r465961 = asin(r465960);
        double r465962 = r465941 - r465961;
        double r465963 = r465938 * r465962;
        return r465963;
}

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. 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 3.9

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

6. Simplified 3.8

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

8. Applied acos-asin 3.9

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

9. Applied exp-diff 3.9

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

10. Applied log-div 3.9

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

11. Simplified 3.9

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

12. Simplified 3.9

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

14. Applied add-log-exp 4.0

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

15. Final simplification 4.0

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

Reproduce

herbie shell --seed 2019155 +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))