Average Error: 16.6 → 3.8
Time: 23.4s
Precision: binary64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \lambda_2 \cdot \left(\cos \lambda_1 \cdot t_0\right) + t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right) \cdot R \end{array} \]
\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
↓
\begin{array}{l}
t_0 := \cos \phi_2 \cdot \cos \phi_1\\
\mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \lambda_2 \cdot \left(\cos \lambda_1 \cdot t_0\right) + t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right) \cdot R
\end{array}

(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
 (let* ((t_0 (* (cos phi2) (cos phi1))))
   (*
    (log1p
     (expm1
      (acos
       (fma
        (sin phi1)
        (sin phi2)
        (+
         (* (cos lambda2) (* (cos lambda1) t_0))
         (* t_0 (* (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) {
	double t_0 = cos(phi2) * cos(phi1);
	return log1p(expm1(acos(fma(sin(phi1), sin(phi2), ((cos(lambda2) * (cos(lambda1) * t_0)) + (t_0 * (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 16.6

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

  2. Simplified16.6

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

  3. Applied cos-diff_binary643.8

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

  4. Applied distribute-rgt-in_binary643.8

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

  5. Simplified3.8

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

  6. Simplified3.8

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

  7. Applied log1p-expm1-u_binary643.8

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \lambda_2 \cdot \left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)} \cdot R \]

  8. Final simplification3.8

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \lambda_2 \cdot \left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right) \cdot R \]

Reproduce

herbie shell --seed 2022095 
(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))