Average Error: 16.9 → 3.8
Time: 35.5s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\log \left({\left(e^{\sin \phi_1}\right)}^{\left(\sin \phi_2\right)}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot R\]
\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(\log \left({\left(e^{\sin \phi_1}\right)}^{\left(\sin \phi_2\right)}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r21134 = phi1;
        double r21135 = sin(r21134);
        double r21136 = phi2;
        double r21137 = sin(r21136);
        double r21138 = r21135 * r21137;
        double r21139 = cos(r21134);
        double r21140 = cos(r21136);
        double r21141 = r21139 * r21140;
        double r21142 = lambda1;
        double r21143 = lambda2;
        double r21144 = r21142 - r21143;
        double r21145 = cos(r21144);
        double r21146 = r21141 * r21145;
        double r21147 = r21138 + r21146;
        double r21148 = acos(r21147);
        double r21149 = R;
        double r21150 = r21148 * r21149;
        return r21150;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r21151 = phi1;
        double r21152 = sin(r21151);
        double r21153 = exp(r21152);
        double r21154 = phi2;
        double r21155 = sin(r21154);
        double r21156 = pow(r21153, r21155);
        double r21157 = log(r21156);
        double r21158 = cos(r21151);
        double r21159 = cos(r21154);
        double r21160 = r21158 * r21159;
        double r21161 = lambda1;
        double r21162 = cos(r21161);
        double r21163 = lambda2;
        double r21164 = cos(r21163);
        double r21165 = r21162 * r21164;
        double r21166 = sin(r21161);
        double r21167 = -r21163;
        double r21168 = sin(r21167);
        double r21169 = r21166 * r21168;
        double r21170 = r21165 - r21169;
        double r21171 = r21160 * r21170;
        double r21172 = r21157 + r21171;
        double r21173 = acos(r21172);
        double r21174 = R;
        double r21175 = r21173 * r21174;
        return r21175;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 sub-neg16.9

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

  4. Applied cos-sum3.8

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

  5. Simplified3.8

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

  7. Applied add-log-exp3.8

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

  9. Applied add-log-exp3.8

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

  10. Simplified3.8

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

  11. Final simplification3.8

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

Reproduce

herbie shell --seed 2019305 
(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))