Average Error: 17.0 → 3.6
Time: 47.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)\right)}\right)\right)\]
\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
↓
R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)\right)}\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1160118 = phi1;
        double r1160119 = sin(r1160118);
        double r1160120 = phi2;
        double r1160121 = sin(r1160120);
        double r1160122 = r1160119 * r1160121;
        double r1160123 = cos(r1160118);
        double r1160124 = cos(r1160120);
        double r1160125 = r1160123 * r1160124;
        double r1160126 = lambda1;
        double r1160127 = lambda2;
        double r1160128 = r1160126 - r1160127;
        double r1160129 = cos(r1160128);
        double r1160130 = r1160125 * r1160129;
        double r1160131 = r1160122 + r1160130;
        double r1160132 = acos(r1160131);
        double r1160133 = R;
        double r1160134 = r1160132 * r1160133;
        return r1160134;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1160135 = R;
        double r1160136 = lambda1;
        double r1160137 = sin(r1160136);
        double r1160138 = lambda2;
        double r1160139 = sin(r1160138);
        double r1160140 = cos(r1160136);
        double r1160141 = cos(r1160138);
        double r1160142 = r1160140 * r1160141;
        double r1160143 = fma(r1160137, r1160139, r1160142);
        double r1160144 = phi1;
        double r1160145 = cos(r1160144);
        double r1160146 = r1160143 * r1160145;
        double r1160147 = phi2;
        double r1160148 = cos(r1160147);
        double r1160149 = sin(r1160147);
        double r1160150 = sin(r1160144);
        double r1160151 = r1160149 * r1160150;
        double r1160152 = fma(r1160146, r1160148, r1160151);
        double r1160153 = acos(r1160152);
        double r1160154 = log(r1160153);
        double r1160155 = expm1(r1160154);
        double r1160156 = log1p(r1160155);
        double r1160157 = exp(r1160156);
        double r1160158 = expm1(r1160157);
        double r1160159 = log1p(r1160158);
        double r1160160 = r1160135 * r1160159;
        return r1160160;
}

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 17.0

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

  2. Simplified17.0

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

  4. Applied cos-diff3.5

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

  6. Applied log1p-expm1-u3.6

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

  7. Simplified3.5

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

  9. Applied add-exp-log3.6

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

  11. Applied log1p-expm1-u3.6

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

  12. Final simplification3.6

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

Reproduce

herbie shell --seed 2019169 +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))