Equirectangular approximation to distance on a great circle

Average Error: 39.0 → 34.1

Time: 10.4s

Precision: 64

Math

\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \le -1.2975233664675797 \cdot 10^{155}:\\ \;\;\;\;R \cdot e^{\log \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right) - \log \left(\frac{-1}{\lambda_1}\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 3.69636763241291605 \cdot 10^{140}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\log \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right) - \log \left(\frac{1}{\lambda_1}\right)}\\ \end{array}\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r75150 = R;
        double r75151 = lambda1;
        double r75152 = lambda2;
        double r75153 = r75151 - r75152;
        double r75154 = phi1;
        double r75155 = phi2;
        double r75156 = r75154 + r75155;
        double r75157 = 2.0;
        double r75158 = r75156 / r75157;
        double r75159 = cos(r75158);
        double r75160 = r75153 * r75159;
        double r75161 = r75160 * r75160;
        double r75162 = r75154 - r75155;
        double r75163 = r75162 * r75162;
        double r75164 = r75161 + r75163;
        double r75165 = sqrt(r75164);
        double r75166 = r75150 * r75165;
        return r75166;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r75167 = lambda1;
        double r75168 = lambda2;
        double r75169 = r75167 - r75168;
        double r75170 = -1.2975233664675797e+155;
        bool r75171 = r75169 <= r75170;
        double r75172 = R;
        double r75173 = 0.5;
        double r75174 = phi2;
        double r75175 = phi1;
        double r75176 = r75174 + r75175;
        double r75177 = r75173 * r75176;
        double r75178 = cos(r75177);
        double r75179 = log(r75178);
        double r75180 = -1.0;
        double r75181 = r75180 / r75167;
        double r75182 = log(r75181);
        double r75183 = r75179 - r75182;
        double r75184 = exp(r75183);
        double r75185 = r75172 * r75184;
        double r75186 = 3.696367632412916e+140;
        bool r75187 = r75169 <= r75186;
        double r75188 = r75175 + r75174;
        double r75189 = 2.0;
        double r75190 = r75188 / r75189;
        double r75191 = cos(r75190);
        double r75192 = r75169 * r75191;
        double r75193 = r75192 * r75192;
        double r75194 = r75175 - r75174;
        double r75195 = r75194 * r75194;
        double r75196 = r75193 + r75195;
        double r75197 = sqrt(r75196);
        double r75198 = r75172 * r75197;
        double r75199 = 1.0;
        double r75200 = r75199 / r75167;
        double r75201 = log(r75200);
        double r75202 = r75179 - r75201;
        double r75203 = exp(r75202);
        double r75204 = r75172 * r75203;
        double r75205 = r75187 ? r75198 : r75204;
        double r75206 = r75171 ? r75185 : r75205;
        return r75206;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Split input into 3 regimes

2. if (- lambda1 lambda2) < -1.2975233664675797e+155

   1. Initial program 64.0

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 64.0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   4. Using strategy rm

   5. Applied add-exp-log 64.0

      \[\leadsto R \cdot \color{blue}{e^{\log \left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}}\]
   6. Using strategy rm

   7. Applied pow1 64.0

      \[\leadsto R \cdot e^{\log \color{blue}{\left({\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}^{1}\right)}}\]

   8. Applied log-pow 64.0

      \[\leadsto R \cdot e^{\color{blue}{1 \cdot \log \left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}}\]

   9. Applied exp-prod 64.0

      \[\leadsto R \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)\right)}}\]

   10. Simplified 64.0

       \[\leadsto R \cdot {\color{blue}{e}}^{\left(\log \left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)\right)}\]

   11. Taylor expanded around -inf 49.4

       \[\leadsto R \cdot \color{blue}{e^{\log \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right) - \log \left(\frac{-1}{\lambda_1}\right)}}\]

   if -1.2975233664675797e+155 < (- lambda1 lambda2) < 3.696367632412916e+140

   1. Initial program 23.9

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   if 3.696367632412916e+140 < (- lambda1 lambda2)

   1. Initial program 61.0

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 61.0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   4. Using strategy rm

   5. Applied add-exp-log 61.2

      \[\leadsto R \cdot \color{blue}{e^{\log \left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}}\]
   6. Using strategy rm

   7. Applied pow1 61.2

      \[\leadsto R \cdot e^{\log \color{blue}{\left({\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}^{1}\right)}}\]

   8. Applied log-pow 61.2

      \[\leadsto R \cdot e^{\color{blue}{1 \cdot \log \left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}}\]

   9. Applied exp-prod 61.2

      \[\leadsto R \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)\right)}}\]

   10. Simplified 61.2

       \[\leadsto R \cdot {\color{blue}{e}}^{\left(\log \left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)\right)}\]

   11. Taylor expanded around inf 50.4

       \[\leadsto R \cdot \color{blue}{e^{\log \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right) - \log \left(\frac{1}{\lambda_1}\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 34.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \le -1.2975233664675797 \cdot 10^{155}:\\ \;\;\;\;R \cdot e^{\log \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right) - \log \left(\frac{-1}{\lambda_1}\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 3.69636763241291605 \cdot 10^{140}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\log \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right) - \log \left(\frac{1}{\lambda_1}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))