Average Error: 16.4 → 3.7
Time: 20.3s
Precision: binary64
\[[lambda1, lambda2]=\mathsf{sort}([lambda1, lambda2])\]
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\sin \lambda_2 \cdot \left(\sin \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \cos \lambda_2 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\right)\right) \cdot R\]
\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
↓
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\sin \lambda_2 \cdot \left(\sin \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \cos \lambda_2 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\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
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (+
     (* (sin lambda2) (* (sin lambda1) (* (cos phi1) (cos phi2))))
     (* (cos lambda2) (* (* (cos phi1) (cos phi2)) (cos lambda1))))))
  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 acos((sin(phi1) * sin(phi2)) + ((sin(lambda2) * (sin(lambda1) * (cos(phi1) * cos(phi2)))) + (cos(lambda2) * ((cos(phi1) * cos(phi2)) * cos(lambda1))))) * R;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.4

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

  3. Applied cos-diff_binary643.7

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

  4. Simplified3.7

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

  5. Taylor expanded around 0 3.7

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

  6. Final simplification3.7

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

Reproduce

herbie shell --seed 2021173 
(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))