Average Error: 17.3 → 4.1
Time: 20.1s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\right) \cdot R\]
\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
↓
\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25145 = phi1;
        double r25146 = sin(r25145);
        double r25147 = phi2;
        double r25148 = sin(r25147);
        double r25149 = r25146 * r25148;
        double r25150 = cos(r25145);
        double r25151 = cos(r25147);
        double r25152 = r25150 * r25151;
        double r25153 = lambda1;
        double r25154 = lambda2;
        double r25155 = r25153 - r25154;
        double r25156 = cos(r25155);
        double r25157 = r25152 * r25156;
        double r25158 = r25149 + r25157;
        double r25159 = acos(r25158);
        double r25160 = R;
        double r25161 = r25159 * r25160;
        return r25161;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25162 = phi1;
        double r25163 = sin(r25162);
        double r25164 = phi2;
        double r25165 = sin(r25164);
        double r25166 = cos(r25162);
        double r25167 = cos(r25164);
        double r25168 = r25166 * r25167;
        double r25169 = lambda1;
        double r25170 = sin(r25169);
        double r25171 = lambda2;
        double r25172 = sin(r25171);
        double r25173 = cos(r25169);
        double r25174 = cos(r25171);
        double r25175 = r25173 * r25174;
        double r25176 = fma(r25170, r25172, r25175);
        double r25177 = r25168 * r25176;
        double r25178 = expm1(r25177);
        double r25179 = log1p(r25178);
        double r25180 = fma(r25163, r25165, r25179);
        double r25181 = acos(r25180);
        double r25182 = R;
        double r25183 = r25181 * r25182;
        return r25183;
}

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 17.3

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

  2. Simplified17.3

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

  4. Applied cos-diff4.0

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

  5. Simplified4.0

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

  7. Applied log1p-expm1-u4.1

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

  8. Simplified4.1

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

  9. Final simplification4.1

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

Reproduce

herbie shell --seed 2020047 +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))