Spherical law of cosines

Average Error: 17.1 → 3.8

Time: 15.3s

Precision: binary64

Math

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

↓

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return ((double) (((double) acos(((double) (((double) (((double) sin(phi1)) * ((double) sin(phi2)))) + ((double) (((double) (((double) cos(phi1)) * ((double) cos(phi2)))) * ((double) cos(((double) (lambda1 - lambda2)))))))))) * R));
}

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return ((double) (((double) acos(((double) (((double) (((double) sin(phi1)) * ((double) sin(phi2)))) + ((double) (((double) (((double) cos(lambda1)) * ((double) (((double) cos(lambda2)) * ((double) (((double) cos(phi1)) * ((double) cos(phi2)))))))) + ((double) (((double) cbrt(((double) (((double) cos(phi1)) * ((double) (((double) (((double) cos(phi2)) * ((double) sin(lambda2)))) * ((double) sin(lambda1)))))))) * ((double) (((double) cbrt(((double) (((double) cos(phi1)) * ((double) (((double) (((double) cos(phi2)) * ((double) sin(lambda2)))) * ((double) sin(lambda1)))))))) * ((double) cbrt(((double) (((double) cos(phi1)) * ((double) (((double) (((double) cos(phi2)) * ((double) sin(lambda2)))) * ((double) sin(lambda1)))))))))))))))))) * R));
}

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 17.1

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

3. Applied cos-diff 3.7

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

4. Applied distribute-lft-in 3.7

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

5. Simplified 3.7

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

6. Simplified 3.7

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

8. Applied add-cube-cbrt 3.8

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

9. Simplified 3.8

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

10. Simplified 3.8

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

11. Final simplification 3.8

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

Reproduce

herbie shell --seed 2020196 
(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))