Average Error: 16.8 → 3.8
Time: 24.5s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\log \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)} \cdot R\]
\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
↓
e^{\log \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25810 = phi1;
        double r25811 = sin(r25810);
        double r25812 = phi2;
        double r25813 = sin(r25812);
        double r25814 = r25811 * r25813;
        double r25815 = cos(r25810);
        double r25816 = cos(r25812);
        double r25817 = r25815 * r25816;
        double r25818 = lambda1;
        double r25819 = lambda2;
        double r25820 = r25818 - r25819;
        double r25821 = cos(r25820);
        double r25822 = r25817 * r25821;
        double r25823 = r25814 + r25822;
        double r25824 = acos(r25823);
        double r25825 = R;
        double r25826 = r25824 * r25825;
        return r25826;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25827 = atan2(1.0, 0.0);
        double r25828 = 2.0;
        double r25829 = r25827 / r25828;
        double r25830 = phi1;
        double r25831 = sin(r25830);
        double r25832 = phi2;
        double r25833 = sin(r25832);
        double r25834 = lambda2;
        double r25835 = cos(r25834);
        double r25836 = lambda1;
        double r25837 = cos(r25836);
        double r25838 = sin(r25836);
        double r25839 = sin(r25834);
        double r25840 = r25838 * r25839;
        double r25841 = fma(r25835, r25837, r25840);
        double r25842 = cos(r25830);
        double r25843 = cos(r25832);
        double r25844 = r25842 * r25843;
        double r25845 = r25841 * r25844;
        double r25846 = fma(r25831, r25833, r25845);
        double r25847 = asin(r25846);
        double r25848 = r25829 - r25847;
        double r25849 = log(r25848);
        double r25850 = exp(r25849);
        double r25851 = R;
        double r25852 = r25850 * r25851;
        return r25852;
}

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

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

  4. Applied sub-neg16.8

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

  5. Applied cos-sum3.7

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

  6. Simplified3.7

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

  8. Applied add-exp-log3.7

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

  9. Simplified3.7

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

  11. Applied acos-asin3.8

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

  12. Final simplification3.8

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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))