Spherical law of cosines

Average Error: 17.1 → 4.2

Time: 18.4s

Precision: binary64

\[[phi1, phi2]=\mathsf{sort}([phi1, phi2])\]

Math

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

↓

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (fma
    (cos phi2)
    (*
     (cos phi1)
     (fma (sin lambda2) (sin lambda1) (* (cos lambda2) (cos lambda1))))
    (* (sin phi1) (sin phi2))))
  (* (log E) R)))

C

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

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(fma(cos(phi2), (cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1)))), (sin(phi1) * sin(phi2)))) * (log((double) M_E) * 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. Applied cos-diff_binary64 4.2

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

3. Applied add-cbrt-cube_binary64 4.2

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

4. Simplified 4.2

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

5. Applied add-log-exp_binary64 4.3

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

6. Simplified 4.2

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

7. Applied *-un-lft-identity_binary64 4.2

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

8. Applied exp-prod_binary64 4.3

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

9. Applied log-pow_binary64 4.2

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

10. Applied associate-*l*_binary64 4.2

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

11. Final simplification 4.2

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

Reproduce

herbie shell --seed 2021307 
(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))