Equirectangular approximation to distance on a great circle

Average Error: 39.2 → 39.5

Time: 1.5m

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_2 \le -2.6219847008147082 \cdot 10^{208}:\\ \;\;\;\;R \cdot \left(-\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \le -5.5068609279832755 \cdot 10^{47}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le -343.899314599184834:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le -1.39901496078204538 \cdot 10^{-33}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 1.1505150553105537 \cdot 10^{-291}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \frac{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}{\lambda_1 + \lambda_2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le 3.62734158549870918 \cdot 10^{-162}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 2.0523002437040101 \cdot 10^{31}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le 1.8914867016819411 \cdot 10^{68}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 2.83361654007986611 \cdot 10^{153}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\\ \end{array}\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r394063 = R;
        double r394064 = lambda1;
        double r394065 = lambda2;
        double r394066 = r394064 - r394065;
        double r394067 = phi1;
        double r394068 = phi2;
        double r394069 = r394067 + r394068;
        double r394070 = 2.0;
        double r394071 = r394069 / r394070;
        double r394072 = cos(r394071);
        double r394073 = r394066 * r394072;
        double r394074 = r394073 * r394073;
        double r394075 = r394067 - r394068;
        double r394076 = r394075 * r394075;
        double r394077 = r394074 + r394076;
        double r394078 = sqrt(r394077);
        double r394079 = r394063 * r394078;
        return r394079;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r394080 = lambda2;
        double r394081 = -2.621984700814708e+208;
        bool r394082 = r394080 <= r394081;
        double r394083 = R;
        double r394084 = phi2;
        double r394085 = phi1;
        double r394086 = r394080 * r394085;
        double r394087 = r394084 * r394086;
        double r394088 = 0.5;
        double r394089 = r394084 + r394085;
        double r394090 = r394088 * r394089;
        double r394091 = cos(r394090);
        double r394092 = lambda1;
        double r394093 = 2.0;
        double r394094 = pow(r394092, r394093);
        double r394095 = r394091 * r394094;
        double r394096 = r394087 / r394095;
        double r394097 = r394084 * r394085;
        double r394098 = r394091 * r394092;
        double r394099 = r394097 / r394098;
        double r394100 = r394080 * r394091;
        double r394101 = r394099 + r394100;
        double r394102 = r394096 + r394101;
        double r394103 = -r394102;
        double r394104 = r394083 * r394103;
        double r394105 = -5.5068609279832755e+47;
        bool r394106 = r394080 <= r394105;
        double r394107 = r394084 - r394085;
        double r394108 = r394083 * r394107;
        double r394109 = -343.89931459918483;
        bool r394110 = r394080 <= r394109;
        double r394111 = r394092 - r394080;
        double r394112 = r394085 + r394084;
        double r394113 = 2.0;
        double r394114 = r394112 / r394113;
        double r394115 = cos(r394114);
        double r394116 = r394111 * r394115;
        double r394117 = r394115 * r394116;
        double r394118 = r394111 * r394117;
        double r394119 = r394085 - r394084;
        double r394120 = r394119 * r394119;
        double r394121 = r394118 + r394120;
        double r394122 = sqrt(r394121);
        double r394123 = r394083 * r394122;
        double r394124 = -1.3990149607820454e-33;
        bool r394125 = r394080 <= r394124;
        double r394126 = 1.1505150553105537e-291;
        bool r394127 = r394080 <= r394126;
        double r394128 = r394092 * r394092;
        double r394129 = r394080 * r394080;
        double r394130 = r394128 - r394129;
        double r394131 = r394130 * r394115;
        double r394132 = r394115 * r394131;
        double r394133 = r394092 + r394080;
        double r394134 = r394132 / r394133;
        double r394135 = r394111 * r394134;
        double r394136 = r394135 + r394120;
        double r394137 = sqrt(r394136);
        double r394138 = r394083 * r394137;
        double r394139 = 3.627341585498709e-162;
        bool r394140 = r394080 <= r394139;
        double r394141 = 2.05230024370401e+31;
        bool r394142 = r394080 <= r394141;
        double r394143 = r394111 * r394111;
        double r394144 = r394143 + r394120;
        double r394145 = sqrt(r394144);
        double r394146 = r394083 * r394145;
        double r394147 = 1.891486701681941e+68;
        bool r394148 = r394080 <= r394147;
        double r394149 = 2.833616540079866e+153;
        bool r394150 = r394080 <= r394149;
        double r394151 = r394083 * r394102;
        double r394152 = r394150 ? r394123 : r394151;
        double r394153 = r394148 ? r394108 : r394152;
        double r394154 = r394142 ? r394146 : r394153;
        double r394155 = r394140 ? r394108 : r394154;
        double r394156 = r394127 ? r394138 : r394155;
        double r394157 = r394125 ? r394108 : r394156;
        double r394158 = r394110 ? r394123 : r394157;
        double r394159 = r394106 ? r394108 : r394158;
        double r394160 = r394082 ? r394104 : r394159;
        return r394160;
}

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 6 regimes

2. if lambda2 < -2.621984700814708e+208

   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. Taylor expanded around inf 50.0

      \[\leadsto R \cdot \color{blue}{\left(-\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\right)}\]

   if -2.621984700814708e+208 < lambda2 < -5.5068609279832755e+47 or -343.89931459918483 < lambda2 < -1.3990149607820454e-33 or 1.1505150553105537e-291 < lambda2 < 3.627341585498709e-162 or 2.05230024370401e+31 < lambda2 < 1.891486701681941e+68

   1. Initial program 38.0

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

   2. Taylor expanded around 0 44.7

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]

   if -5.5068609279832755e+47 < lambda2 < -343.89931459918483 or 1.891486701681941e+68 < lambda2 < 2.833616540079866e+153

   1. Initial program 32.9

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

   3. Applied associate-*l* 32.9

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

   if -1.3990149607820454e-33 < lambda2 < 1.1505150553105537e-291

   1. Initial program 31.4

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

   3. Applied associate-*l* 31.4

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

   5. Applied flip-- 31.4

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

   6. Applied associate-*l/ 31.4

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

   7. Applied associate-*r/ 31.4

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

   if 3.627341585498709e-162 < lambda2 < 2.05230024370401e+31

   1. Initial program 30.9

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

   3. Applied associate-*l* 30.9

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

   4. Taylor expanded around 0 33.4

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

   if 2.833616540079866e+153 < lambda2

   1. Initial program 63.8

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

   2. Taylor expanded around -inf 52.6

      \[\leadsto R \cdot \color{blue}{\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)}\]
3. Recombined 6 regimes into one program.

4. Final simplification 39.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \le -2.6219847008147082 \cdot 10^{208}:\\ \;\;\;\;R \cdot \left(-\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \le -5.5068609279832755 \cdot 10^{47}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le -343.899314599184834:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le -1.39901496078204538 \cdot 10^{-33}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 1.1505150553105537 \cdot 10^{-291}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \frac{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}{\lambda_1 + \lambda_2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le 3.62734158549870918 \cdot 10^{-162}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 2.0523002437040101 \cdot 10^{31}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le 1.8914867016819411 \cdot 10^{68}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 2.83361654007986611 \cdot 10^{153}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))))