Average Error: 16.8 → 3.9
Time: 1.0m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \mathsf{log1p}\left(\frac{e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} - 1}{1 + e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)}}\right)\]
\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 \mathsf{log1p}\left(\frac{e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} - 1}{1 + e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)}}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1000714 = phi1;
        double r1000715 = sin(r1000714);
        double r1000716 = phi2;
        double r1000717 = sin(r1000716);
        double r1000718 = r1000715 * r1000717;
        double r1000719 = cos(r1000714);
        double r1000720 = cos(r1000716);
        double r1000721 = r1000719 * r1000720;
        double r1000722 = lambda1;
        double r1000723 = lambda2;
        double r1000724 = r1000722 - r1000723;
        double r1000725 = cos(r1000724);
        double r1000726 = r1000721 * r1000725;
        double r1000727 = r1000718 + r1000726;
        double r1000728 = acos(r1000727);
        double r1000729 = R;
        double r1000730 = r1000728 * r1000729;
        return r1000730;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1000731 = R;
        double r1000732 = phi2;
        double r1000733 = cos(r1000732);
        double r1000734 = lambda1;
        double r1000735 = sin(r1000734);
        double r1000736 = lambda2;
        double r1000737 = sin(r1000736);
        double r1000738 = cos(r1000734);
        double r1000739 = cos(r1000736);
        double r1000740 = r1000738 * r1000739;
        double r1000741 = fma(r1000735, r1000737, r1000740);
        double r1000742 = r1000733 * r1000741;
        double r1000743 = phi1;
        double r1000744 = cos(r1000743);
        double r1000745 = sin(r1000732);
        double r1000746 = sin(r1000743);
        double r1000747 = r1000745 * r1000746;
        double r1000748 = fma(r1000742, r1000744, r1000747);
        double r1000749 = acos(r1000748);
        double r1000750 = exp(r1000749);
        double r1000751 = r1000750 * r1000750;
        double r1000752 = 1.0;
        double r1000753 = r1000751 - r1000752;
        double r1000754 = r1000752 + r1000750;
        double r1000755 = r1000753 / r1000754;
        double r1000756 = log1p(r1000755);
        double r1000757 = r1000731 * r1000756;
        return r1000757;
}

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 16.8

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

  2. Simplified16.8

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

  4. Applied cos-diff3.8

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

  6. Applied log1p-expm1-u3.8

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

  7. Simplified3.8

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

  9. Applied expm1-udef3.8

    \[\leadsto R \cdot \mathsf{log1p}\left(\color{blue}{e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} - 1}\right)\]
  10. Using strategy rm

  11. Applied flip--3.9

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

  12. Final simplification3.9

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

Reproduce

herbie shell --seed 2019149 +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))