Average Error: 16.6 → 3.6
Time: 37.1s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\]
\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
↓
R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r24690 = phi1;
        double r24691 = sin(r24690);
        double r24692 = phi2;
        double r24693 = sin(r24692);
        double r24694 = r24691 * r24693;
        double r24695 = cos(r24690);
        double r24696 = cos(r24692);
        double r24697 = r24695 * r24696;
        double r24698 = lambda1;
        double r24699 = lambda2;
        double r24700 = r24698 - r24699;
        double r24701 = cos(r24700);
        double r24702 = r24697 * r24701;
        double r24703 = r24694 + r24702;
        double r24704 = acos(r24703);
        double r24705 = R;
        double r24706 = r24704 * r24705;
        return r24706;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r24707 = R;
        double r24708 = atan2(1.0, 0.0);
        double r24709 = 2.0;
        double r24710 = r24708 / r24709;
        double r24711 = phi1;
        double r24712 = sin(r24711);
        double r24713 = phi2;
        double r24714 = sin(r24713);
        double r24715 = cos(r24711);
        double r24716 = cos(r24713);
        double r24717 = r24715 * r24716;
        double r24718 = lambda1;
        double r24719 = sin(r24718);
        double r24720 = lambda2;
        double r24721 = sin(r24720);
        double r24722 = cos(r24718);
        double r24723 = cos(r24720);
        double r24724 = r24722 * r24723;
        double r24725 = fma(r24719, r24721, r24724);
        double r24726 = r24717 * r24725;
        double r24727 = fma(r24712, r24714, r24726);
        double r24728 = asin(r24727);
        double r24729 = r24710 - r24728;
        double r24730 = r24707 * r24729;
        return r24730;
}

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

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

  4. Applied cos-diff3.5

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

  6. Applied acos-asin3.6

    \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)}\]

  7. Simplified3.6

    \[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right)\]

  8. Final simplification3.6

    \[\leadsto R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))