Average Error: 39.0 → 3.5
Time: 6.5s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]
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(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r57096 = R;
        double r57097 = lambda1;
        double r57098 = lambda2;
        double r57099 = r57097 - r57098;
        double r57100 = phi1;
        double r57101 = phi2;
        double r57102 = r57100 + r57101;
        double r57103 = 2.0;
        double r57104 = r57102 / r57103;
        double r57105 = cos(r57104);
        double r57106 = r57099 * r57105;
        double r57107 = r57106 * r57106;
        double r57108 = r57100 - r57101;
        double r57109 = r57108 * r57108;
        double r57110 = r57107 + r57109;
        double r57111 = sqrt(r57110);
        double r57112 = r57096 * r57111;
        return r57112;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r57113 = lambda1;
        double r57114 = lambda2;
        double r57115 = r57113 - r57114;
        double r57116 = phi1;
        double r57117 = phi2;
        double r57118 = r57116 + r57117;
        double r57119 = 2.0;
        double r57120 = r57118 / r57119;
        double r57121 = cos(r57120);
        double r57122 = expm1(r57121);
        double r57123 = expm1(r57122);
        double r57124 = log1p(r57123);
        double r57125 = log1p(r57124);
        double r57126 = r57115 * r57125;
        double r57127 = r57116 - r57117;
        double r57128 = hypot(r57126, r57127);
        double r57129 = R;
        double r57130 = r57128 * r57129;
        return r57130;
}

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 39.0

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

  2. Simplified3.5

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

  4. Applied log1p-expm1-u3.5

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]
  5. Using strategy rm

  6. Applied log1p-expm1-u3.5

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)}\right), \phi_1 - \phi_2\right) \cdot R\]

  7. Final simplification3.5

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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))))))