Average Error: 16.9 → 4.0
Time: 48.8s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1, \log \left(e^{\sin \phi_2 \cdot \sin \phi_1}\right)\right)\right) \cdot R\]
\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(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1, \log \left(e^{\sin \phi_2 \cdot \sin \phi_1}\right)\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r844610 = phi1;
        double r844611 = sin(r844610);
        double r844612 = phi2;
        double r844613 = sin(r844612);
        double r844614 = r844611 * r844613;
        double r844615 = cos(r844610);
        double r844616 = cos(r844612);
        double r844617 = r844615 * r844616;
        double r844618 = lambda1;
        double r844619 = lambda2;
        double r844620 = r844618 - r844619;
        double r844621 = cos(r844620);
        double r844622 = r844617 * r844621;
        double r844623 = r844614 + r844622;
        double r844624 = acos(r844623);
        double r844625 = R;
        double r844626 = r844624 * r844625;
        return r844626;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r844627 = phi2;
        double r844628 = cos(r844627);
        double r844629 = phi1;
        double r844630 = cos(r844629);
        double r844631 = r844628 * r844630;
        double r844632 = lambda1;
        double r844633 = sin(r844632);
        double r844634 = lambda2;
        double r844635 = sin(r844634);
        double r844636 = r844633 * r844635;
        double r844637 = cos(r844634);
        double r844638 = cos(r844632);
        double r844639 = r844637 * r844638;
        double r844640 = r844636 + r844639;
        double r844641 = sin(r844627);
        double r844642 = sin(r844629);
        double r844643 = r844641 * r844642;
        double r844644 = exp(r844643);
        double r844645 = log(r844644);
        double r844646 = fma(r844631, r844640, r844645);
        double r844647 = acos(r844646);
        double r844648 = R;
        double r844649 = r844647 * r844648;
        return r844649;
}

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 16.9

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

  2. Simplified16.9

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

  4. Applied cos-diff4.0

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

  6. Applied add-log-exp4.0

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

  7. Final simplification4.0

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

Reproduce

herbie shell --seed 2019146 +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))