Average Error: 16.4 → 3.8
Time: 1.7m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\left(\cos \lambda_2\right), \left(\cos \lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), \left(\cos \phi_1\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)\right)\right)\right)\]
\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 \cos^{-1} \left(\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\left(\cos \lambda_2\right), \left(\cos \lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), \left(\cos \phi_1\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)\right)\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3324635 = phi1;
        double r3324636 = sin(r3324635);
        double r3324637 = phi2;
        double r3324638 = sin(r3324637);
        double r3324639 = r3324636 * r3324638;
        double r3324640 = cos(r3324635);
        double r3324641 = cos(r3324637);
        double r3324642 = r3324640 * r3324641;
        double r3324643 = lambda1;
        double r3324644 = lambda2;
        double r3324645 = r3324643 - r3324644;
        double r3324646 = cos(r3324645);
        double r3324647 = r3324642 * r3324646;
        double r3324648 = r3324639 + r3324647;
        double r3324649 = acos(r3324648);
        double r3324650 = R;
        double r3324651 = r3324649 * r3324650;
        return r3324651;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3324652 = R;
        double r3324653 = phi2;
        double r3324654 = cos(r3324653);
        double r3324655 = lambda2;
        double r3324656 = cos(r3324655);
        double r3324657 = lambda1;
        double r3324658 = cos(r3324657);
        double r3324659 = sin(r3324657);
        double r3324660 = sin(r3324655);
        double r3324661 = r3324659 * r3324660;
        double r3324662 = fma(r3324656, r3324658, r3324661);
        double r3324663 = r3324654 * r3324662;
        double r3324664 = phi1;
        double r3324665 = cos(r3324664);
        double r3324666 = sin(r3324653);
        double r3324667 = sin(r3324664);
        double r3324668 = r3324666 * r3324667;
        double r3324669 = fma(r3324663, r3324665, r3324668);
        double r3324670 = log1p(r3324669);
        double r3324671 = expm1(r3324670);
        double r3324672 = acos(r3324671);
        double r3324673 = r3324652 * r3324672;
        return r3324673;
}

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 16.4

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

  2. Simplified16.4

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

  4. Applied cos-diff3.7

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

  6. Applied expm1-log1p-u3.8

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

  7. Simplified3.8

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

  8. Final simplification3.8

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

Reproduce

herbie shell --seed 2019125 +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))