Average Error: 37.2 → 0.1
Time: 1.2m
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot (\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right)\right) + \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \left(-\sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right))_*\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*\]
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)}
↓
R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot (\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right)\right) + \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \left(-\sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right))_*\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r10510981 = R;
        double r10510982 = lambda1;
        double r10510983 = lambda2;
        double r10510984 = r10510982 - r10510983;
        double r10510985 = phi1;
        double r10510986 = phi2;
        double r10510987 = r10510985 + r10510986;
        double r10510988 = 2.0;
        double r10510989 = r10510987 / r10510988;
        double r10510990 = cos(r10510989);
        double r10510991 = r10510984 * r10510990;
        double r10510992 = r10510991 * r10510991;
        double r10510993 = r10510985 - r10510986;
        double r10510994 = r10510993 * r10510993;
        double r10510995 = r10510992 + r10510994;
        double r10510996 = sqrt(r10510995);
        double r10510997 = r10510981 * r10510996;
        return r10510997;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r10510998 = R;
        double r10510999 = lambda1;
        double r10511000 = lambda2;
        double r10511001 = r10510999 - r10511000;
        double r10511002 = phi1;
        double r10511003 = 0.5;
        double r10511004 = r10511002 * r10511003;
        double r10511005 = cos(r10511004);
        double r10511006 = phi2;
        double r10511007 = r10511006 * r10511003;
        double r10511008 = cos(r10511007);
        double r10511009 = sin(r10511007);
        double r10511010 = sin(r10511004);
        double r10511011 = -r10511010;
        double r10511012 = r10511009 * r10511011;
        double r10511013 = fma(r10511005, r10511008, r10511012);
        double r10511014 = r10511001 * r10511013;
        double r10511015 = r10511002 - r10511006;
        double r10511016 = hypot(r10511014, r10511015);
        double r10511017 = r10510998 * r10511016;
        return r10511017;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 37.2

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

  2. Simplified3.8

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

  3. Taylor expanded around inf 3.8

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

  5. Applied distribute-rgt-in3.8

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

  6. Applied cos-sum0.1

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

  8. Applied fma-neg0.1

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

  9. Final simplification0.1

    \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot (\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right)\right) + \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \left(-\sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right))_*\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*\]

Reproduce

herbie shell --seed 2019107 +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))))))