Equirectangular approximation to distance on a great circle

Average Error: 39.1 → 29.5

Time: 42.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \le -2.44512800894886 \cdot 10^{146}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 9.2572239940457903 \cdot 10^{150}:\\ \;\;\;\;R \cdot \sqrt{\left({\lambda_1}^{2} \cdot \left({\left(\sin \left(0.5 \cdot \phi_2\right)\right)}^{2} \cdot {\left(\sin \left(0.5 \cdot \phi_1\right)\right)}^{2} + {\left(\cos \left(0.5 \cdot \phi_1\right)\right)}^{2} \cdot {\left(\cos \left(0.5 \cdot \phi_2\right)\right)}^{2}\right) + \left({\lambda_2}^{2} \cdot {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} - 2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot {\lambda_1}^{2}\right)\right)\right) + \lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_1 - \lambda_2\right)\\ \end{array}\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r91536 = R;
        double r91537 = lambda1;
        double r91538 = lambda2;
        double r91539 = r91537 - r91538;
        double r91540 = phi1;
        double r91541 = phi2;
        double r91542 = r91540 + r91541;
        double r91543 = 2.0;
        double r91544 = r91542 / r91543;
        double r91545 = cos(r91544);
        double r91546 = r91539 * r91545;
        double r91547 = r91546 * r91546;
        double r91548 = r91540 - r91541;
        double r91549 = r91548 * r91548;
        double r91550 = r91547 + r91549;
        double r91551 = sqrt(r91550);
        double r91552 = r91536 * r91551;
        return r91552;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r91553 = lambda1;
        double r91554 = lambda2;
        double r91555 = r91553 - r91554;
        double r91556 = -2.44512800894886e+146;
        bool r91557 = r91555 <= r91556;
        double r91558 = R;
        double r91559 = r91554 - r91553;
        double r91560 = r91558 * r91559;
        double r91561 = 9.25722399404579e+150;
        bool r91562 = r91555 <= r91561;
        double r91563 = 2.0;
        double r91564 = pow(r91553, r91563);
        double r91565 = 0.5;
        double r91566 = phi2;
        double r91567 = r91565 * r91566;
        double r91568 = sin(r91567);
        double r91569 = pow(r91568, r91563);
        double r91570 = phi1;
        double r91571 = r91565 * r91570;
        double r91572 = sin(r91571);
        double r91573 = pow(r91572, r91563);
        double r91574 = r91569 * r91573;
        double r91575 = cos(r91571);
        double r91576 = pow(r91575, r91563);
        double r91577 = cos(r91567);
        double r91578 = pow(r91577, r91563);
        double r91579 = r91576 * r91578;
        double r91580 = r91574 + r91579;
        double r91581 = r91564 * r91580;
        double r91582 = pow(r91554, r91563);
        double r91583 = r91566 + r91570;
        double r91584 = r91565 * r91583;
        double r91585 = cos(r91584);
        double r91586 = pow(r91585, r91563);
        double r91587 = r91582 * r91586;
        double r91588 = r91568 * r91564;
        double r91589 = r91577 * r91588;
        double r91590 = r91572 * r91589;
        double r91591 = r91575 * r91590;
        double r91592 = r91586 * r91553;
        double r91593 = r91554 * r91592;
        double r91594 = r91591 + r91593;
        double r91595 = r91563 * r91594;
        double r91596 = r91587 - r91595;
        double r91597 = r91581 + r91596;
        double r91598 = r91570 - r91566;
        double r91599 = r91598 * r91598;
        double r91600 = r91597 + r91599;
        double r91601 = sqrt(r91600);
        double r91602 = r91558 * r91601;
        double r91603 = r91558 * r91555;
        double r91604 = r91562 ? r91602 : r91603;
        double r91605 = r91557 ? r91560 : r91604;
        return r91605;
}

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) < -2.44512800894886e+146

   1. Initial program 62.1

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

   2. Taylor expanded around inf 62.1

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\left({\lambda_2}^{2} \cdot {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} + {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot {\lambda_1}^{2}\right) - 2 \cdot \left(\lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   3. Using strategy rm

   4. Applied distribute-lft-in 62.1

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

   5. Applied cos-sum 62.0

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

   6. Taylor expanded around inf 62.0

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

   7. Simplified 62.0

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

   8. Taylor expanded around 0 40.4

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

   if -2.44512800894886e+146 < (- lambda1 lambda2) < 9.25722399404579e+150

   1. Initial program 23.5

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

   2. Taylor expanded around inf 23.5

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\left({\lambda_2}^{2} \cdot {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} + {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot {\lambda_1}^{2}\right) - 2 \cdot \left(\lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   3. Using strategy rm

   4. Applied distribute-lft-in 23.5

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

   5. Applied cos-sum 23.1

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

   6. Taylor expanded around inf 23.1

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

   7. Simplified 23.1

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

   if 9.25722399404579e+150 < (- lambda1 lambda2)

   1. Initial program 63.3

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

   2. Taylor expanded around inf 63.3

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

   3. Taylor expanded around 0 38.1

      \[\leadsto R \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 29.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \le -2.44512800894886 \cdot 10^{146}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 9.2572239940457903 \cdot 10^{150}:\\ \;\;\;\;R \cdot \sqrt{\left({\lambda_1}^{2} \cdot \left({\left(\sin \left(0.5 \cdot \phi_2\right)\right)}^{2} \cdot {\left(\sin \left(0.5 \cdot \phi_1\right)\right)}^{2} + {\left(\cos \left(0.5 \cdot \phi_1\right)\right)}^{2} \cdot {\left(\cos \left(0.5 \cdot \phi_2\right)\right)}^{2}\right) + \left({\lambda_2}^{2} \cdot {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} - 2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot {\lambda_1}^{2}\right)\right)\right) + \lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_1 - \lambda_2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))