Midpoint on a great circle

Average Error: 0.8 → 0.3

Time: 12.2s

Precision: 64

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\frac{\mathsf{fma}\left({\left(\cos \phi_2\right)}^{3}, {\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}, {\left(\cos \phi_1\right)}^{3}\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), -\cos \phi_1\right)\right)\right), \cos \phi_1 \cdot \cos \phi_1\right)}}\]

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r59562 = lambda1;
        double r59563 = phi2;
        double r59564 = cos(r59563);
        double r59565 = lambda2;
        double r59566 = r59562 - r59565;
        double r59567 = sin(r59566);
        double r59568 = r59564 * r59567;
        double r59569 = phi1;
        double r59570 = cos(r59569);
        double r59571 = cos(r59566);
        double r59572 = r59564 * r59571;
        double r59573 = r59570 + r59572;
        double r59574 = atan2(r59568, r59573);
        double r59575 = r59562 + r59574;
        return r59575;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r59576 = lambda1;
        double r59577 = phi2;
        double r59578 = cos(r59577);
        double r59579 = sin(r59576);
        double r59580 = lambda2;
        double r59581 = cos(r59580);
        double r59582 = r59579 * r59581;
        double r59583 = cos(r59576);
        double r59584 = -r59580;
        double r59585 = sin(r59584);
        double r59586 = r59583 * r59585;
        double r59587 = r59582 + r59586;
        double r59588 = r59578 * r59587;
        double r59589 = 3.0;
        double r59590 = pow(r59578, r59589);
        double r59591 = sin(r59580);
        double r59592 = r59579 * r59591;
        double r59593 = fma(r59583, r59581, r59592);
        double r59594 = pow(r59593, r59589);
        double r59595 = phi1;
        double r59596 = cos(r59595);
        double r59597 = pow(r59596, r59589);
        double r59598 = fma(r59590, r59594, r59597);
        double r59599 = -r59596;
        double r59600 = fma(r59578, r59593, r59599);
        double r59601 = expm1(r59600);
        double r59602 = log1p(r59601);
        double r59603 = r59593 * r59602;
        double r59604 = r59596 * r59596;
        double r59605 = fma(r59578, r59603, r59604);
        double r59606 = r59598 / r59605;
        double r59607 = atan2(r59588, r59606);
        double r59608 = r59576 + r59607;
        return r59608;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 0.8

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
2. Using strategy rm

3. Applied cos-diff 0.8

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

5. Applied sub-neg 0.8

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

6. Applied sin-sum 0.2

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

7. Simplified 0.2

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

9. Applied flip3-+ 0.3

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\color{blue}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\cos \phi_1 \cdot \cos \phi_1 + \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}}}\]

10. Simplified 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\frac{\color{blue}{\mathsf{fma}\left({\left(\cos \phi_2\right)}^{3}, {\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}, {\left(\cos \phi_1\right)}^{3}\right)}}{\cos \phi_1 \cdot \cos \phi_1 + \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}}\]

11. Simplified 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\frac{\mathsf{fma}\left({\left(\cos \phi_2\right)}^{3}, {\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}, {\left(\cos \phi_1\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), -\cos \phi_1\right), \cos \phi_1 \cdot \cos \phi_1\right)}}}\]
12. Using strategy rm

13. Applied log1p-expm1-u 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\frac{\mathsf{fma}\left({\left(\cos \phi_2\right)}^{3}, {\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}, {\left(\cos \phi_1\right)}^{3}\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), -\cos \phi_1\right)\right)\right)}, \cos \phi_1 \cdot \cos \phi_1\right)}}\]

14. Final simplification 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\frac{\mathsf{fma}\left({\left(\cos \phi_2\right)}^{3}, {\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}, {\left(\cos \phi_1\right)}^{3}\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), -\cos \phi_1\right)\right)\right), \cos \phi_1 \cdot \cos \phi_1\right)}}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))