Average Error: 16.9 → 4.1
Time: 53.5s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \frac{\mathsf{fma}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right), \cos \lambda_2 \cdot \cos \lambda_1, \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}{\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)}\right) \cdot R\]
\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(\sin \phi_2 \cdot \sin \phi_1 + \frac{\mathsf{fma}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right), \cos \lambda_2 \cdot \cos \lambda_1, \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}{\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)}\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r636136 = phi1;
        double r636137 = sin(r636136);
        double r636138 = phi2;
        double r636139 = sin(r636138);
        double r636140 = r636137 * r636139;
        double r636141 = cos(r636136);
        double r636142 = cos(r636138);
        double r636143 = r636141 * r636142;
        double r636144 = lambda1;
        double r636145 = lambda2;
        double r636146 = r636144 - r636145;
        double r636147 = cos(r636146);
        double r636148 = r636143 * r636147;
        double r636149 = r636140 + r636148;
        double r636150 = acos(r636149);
        double r636151 = R;
        double r636152 = r636150 * r636151;
        return r636152;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r636153 = phi2;
        double r636154 = sin(r636153);
        double r636155 = phi1;
        double r636156 = sin(r636155);
        double r636157 = r636154 * r636156;
        double r636158 = lambda2;
        double r636159 = cos(r636158);
        double r636160 = lambda1;
        double r636161 = cos(r636160);
        double r636162 = r636159 * r636161;
        double r636163 = r636162 * r636162;
        double r636164 = sin(r636158);
        double r636165 = sin(r636160);
        double r636166 = r636164 * r636165;
        double r636167 = r636166 * r636166;
        double r636168 = r636167 * r636166;
        double r636169 = fma(r636163, r636162, r636168);
        double r636170 = cos(r636155);
        double r636171 = cos(r636153);
        double r636172 = r636170 * r636171;
        double r636173 = r636169 * r636172;
        double r636174 = r636166 * r636162;
        double r636175 = r636167 - r636174;
        double r636176 = r636163 + r636175;
        double r636177 = r636173 / r636176;
        double r636178 = r636157 + r636177;
        double r636179 = acos(r636178);
        double r636180 = R;
        double r636181 = r636179 * r636180;
        return r636181;
}

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. Using strategy rm

  3. Applied cos-diff4.1

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

  5. Applied flip3-+4.1

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\frac{{\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \cdot R\]

  6. Applied associate-*r/4.1

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\frac{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left({\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}\right)}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \cdot R\]

  7. Simplified4.1

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

  8. Final simplification4.1

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

Reproduce

herbie shell --seed 2019153 +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))