Average Error: 36.5 → 0.1
Time: 52.0s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]
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)}
↓
\mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r4777120 = R;
        double r4777121 = lambda1;
        double r4777122 = lambda2;
        double r4777123 = r4777121 - r4777122;
        double r4777124 = phi1;
        double r4777125 = phi2;
        double r4777126 = r4777124 + r4777125;
        double r4777127 = 2.0;
        double r4777128 = r4777126 / r4777127;
        double r4777129 = cos(r4777128);
        double r4777130 = r4777123 * r4777129;
        double r4777131 = r4777130 * r4777130;
        double r4777132 = r4777124 - r4777125;
        double r4777133 = r4777132 * r4777132;
        double r4777134 = r4777131 + r4777133;
        double r4777135 = sqrt(r4777134);
        double r4777136 = r4777120 * r4777135;
        return r4777136;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r4777137 = phi2;
        double r4777138 = 0.5;
        double r4777139 = r4777137 * r4777138;
        double r4777140 = cos(r4777139);
        double r4777141 = phi1;
        double r4777142 = r4777141 * r4777138;
        double r4777143 = cos(r4777142);
        double r4777144 = r4777140 * r4777143;
        double r4777145 = sin(r4777142);
        double r4777146 = sin(r4777139);
        double r4777147 = r4777145 * r4777146;
        double r4777148 = r4777147 * r4777147;
        double r4777149 = r4777148 * r4777147;
        double r4777150 = cbrt(r4777149);
        double r4777151 = r4777144 - r4777150;
        double r4777152 = lambda1;
        double r4777153 = lambda2;
        double r4777154 = r4777152 - r4777153;
        double r4777155 = r4777151 * r4777154;
        double r4777156 = r4777141 - r4777137;
        double r4777157 = hypot(r4777155, r4777156);
        double r4777158 = R;
        double r4777159 = r4777157 * r4777158;
        return r4777159;
}

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 36.5

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

  2. Simplified3.7

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

  3. Taylor expanded around inf 3.7

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]
  4. Using strategy rm

  5. Applied distribute-rgt-in3.7

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2} + \phi_2 \cdot \frac{1}{2}\right)}, \phi_1 - \phi_2\right) \cdot R\]

  6. Applied cos-sum0.1

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]
  7. Using strategy rm

  8. Applied add-cbrt-cube0.1

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\sqrt[3]{\left(\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}\right), \phi_1 - \phi_2\right) \cdot R\]

  9. Final simplification0.1

    \[\leadsto \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))