Average Error: 16.9 → 3.9
Time: 29.6s
Precision: binary64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[e^{\log \log \left(e^{e^{\log \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}}\right)} \cdot R \]
\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
↓
e^{\log \log \left(e^{e^{\log \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}}\right)} \cdot R

(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
 (*
  (exp
   (log
    (log
     (exp
      (exp
       (log
        (acos
         (fma
          (sin phi1)
          (sin phi2)
          (*
           (* (cos phi2) (cos phi1))
           (+
            (* (cos lambda2) (cos lambda1))
            (* (sin lambda1) (sin lambda2))))))))))))
  R))
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 exp(log(log(exp(exp(log(acos(fma(sin(phi1), sin(phi2), ((cos(phi2) * cos(phi1)) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))))))))) * 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 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. Simplified16.9

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

  3. Applied cos-diff_binary643.9

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

  4. Applied distribute-rgt-in_binary643.9

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

  5. Simplified3.9

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

  6. Simplified3.9

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

  7. Applied add-log-exp_binary643.9

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

  8. Applied add-exp-log_binary643.9

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

  9. Applied add-exp-log_binary643.9

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

  10. Simplified3.9

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

  11. Final simplification3.9

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

Reproduce

herbie shell --seed 2021352 
(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))