Average Error: 16.8 → 3.8
Time: 46.8s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right)\right)} \cdot R\]
\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^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1421017 = phi1;
        double r1421018 = sin(r1421017);
        double r1421019 = phi2;
        double r1421020 = sin(r1421019);
        double r1421021 = r1421018 * r1421020;
        double r1421022 = cos(r1421017);
        double r1421023 = cos(r1421019);
        double r1421024 = r1421022 * r1421023;
        double r1421025 = lambda1;
        double r1421026 = lambda2;
        double r1421027 = r1421025 - r1421026;
        double r1421028 = cos(r1421027);
        double r1421029 = r1421024 * r1421028;
        double r1421030 = r1421021 + r1421029;
        double r1421031 = acos(r1421030);
        double r1421032 = R;
        double r1421033 = r1421031 * r1421032;
        return r1421033;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1421034 = phi1;
        double r1421035 = sin(r1421034);
        double r1421036 = phi2;
        double r1421037 = sin(r1421036);
        double r1421038 = lambda1;
        double r1421039 = cos(r1421038);
        double r1421040 = lambda2;
        double r1421041 = cos(r1421040);
        double r1421042 = sin(r1421038);
        double r1421043 = sin(r1421040);
        double r1421044 = r1421042 * r1421043;
        double r1421045 = fma(r1421039, r1421041, r1421044);
        double r1421046 = cos(r1421034);
        double r1421047 = cos(r1421036);
        double r1421048 = r1421046 * r1421047;
        double r1421049 = r1421045 * r1421048;
        double r1421050 = fma(r1421035, r1421037, r1421049);
        double r1421051 = acos(r1421050);
        double r1421052 = log(r1421051);
        double r1421053 = expm1(r1421052);
        double r1421054 = log1p(r1421053);
        double r1421055 = exp(r1421054);
        double r1421056 = R;
        double r1421057 = r1421055 * r1421056;
        return r1421057;
}

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. Using strategy rm

  3. Applied cos-diff3.8

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

  5. Applied add-exp-log3.8

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

  6. Simplified3.8

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

  8. Applied add-log-exp3.8

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

  10. Applied log1p-expm1-u3.8

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

  11. Simplified3.8

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

  12. Final simplification3.8

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

Reproduce

herbie shell --seed 2019171 +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))