Average Error: 37.4 → 0.1
Time: 29.4s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_1, \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_2\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right) - \mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \lambda_1 \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right), \left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_2\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]
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(\mathsf{fma}\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_1, \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_2\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right) - \mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \lambda_1 \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right), \left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_2\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2876457 = R;
        double r2876458 = lambda1;
        double r2876459 = lambda2;
        double r2876460 = r2876458 - r2876459;
        double r2876461 = phi1;
        double r2876462 = phi2;
        double r2876463 = r2876461 + r2876462;
        double r2876464 = 2.0;
        double r2876465 = r2876463 / r2876464;
        double r2876466 = cos(r2876465);
        double r2876467 = r2876460 * r2876466;
        double r2876468 = r2876467 * r2876467;
        double r2876469 = r2876461 - r2876462;
        double r2876470 = r2876469 * r2876469;
        double r2876471 = r2876468 + r2876470;
        double r2876472 = sqrt(r2876471);
        double r2876473 = r2876457 * r2876472;
        return r2876473;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2876474 = phi2;
        double r2876475 = 0.5;
        double r2876476 = r2876474 * r2876475;
        double r2876477 = cos(r2876476);
        double r2876478 = phi1;
        double r2876479 = r2876478 * r2876475;
        double r2876480 = cos(r2876479);
        double r2876481 = lambda1;
        double r2876482 = r2876480 * r2876481;
        double r2876483 = sin(r2876479);
        double r2876484 = lambda2;
        double r2876485 = r2876483 * r2876484;
        double r2876486 = sin(r2876476);
        double r2876487 = r2876485 * r2876486;
        double r2876488 = r2876481 * r2876483;
        double r2876489 = r2876480 * r2876484;
        double r2876490 = r2876489 * r2876477;
        double r2876491 = fma(r2876486, r2876488, r2876490);
        double r2876492 = r2876487 - r2876491;
        double r2876493 = fma(r2876477, r2876482, r2876492);
        double r2876494 = r2876478 - r2876474;
        double r2876495 = hypot(r2876493, r2876494);
        double r2876496 = R;
        double r2876497 = r2876495 * r2876496;
        return r2876497;
}

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 37.4

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

  2. Simplified3.9

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

  3. Taylor expanded around -inf 3.9

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

  4. Simplified3.9

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

  6. Applied distribute-lft-in3.9

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

  7. Applied cos-sum0.1

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

  8. Taylor expanded around inf 0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019138 +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))))))