Average Error: 0.2 → 0.2
Time: 14.6s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \left(e^{\cos delta - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}\right)}\]
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \left(e^{\cos delta - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r99986 = lambda1;
        double r99987 = theta;
        double r99988 = sin(r99987);
        double r99989 = delta;
        double r99990 = sin(r99989);
        double r99991 = r99988 * r99990;
        double r99992 = phi1;
        double r99993 = cos(r99992);
        double r99994 = r99991 * r99993;
        double r99995 = cos(r99989);
        double r99996 = sin(r99992);
        double r99997 = r99996 * r99995;
        double r99998 = r99993 * r99990;
        double r99999 = cos(r99987);
        double r100000 = r99998 * r99999;
        double r100001 = r99997 + r100000;
        double r100002 = asin(r100001);
        double r100003 = sin(r100002);
        double r100004 = r99996 * r100003;
        double r100005 = r99995 - r100004;
        double r100006 = atan2(r99994, r100005);
        double r100007 = r99986 + r100006;
        return r100007;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r100008 = lambda1;
        double r100009 = theta;
        double r100010 = sin(r100009);
        double r100011 = delta;
        double r100012 = sin(r100011);
        double r100013 = r100010 * r100012;
        double r100014 = phi1;
        double r100015 = cos(r100014);
        double r100016 = r100013 * r100015;
        double r100017 = cos(r100011);
        double r100018 = sin(r100014);
        double r100019 = r100018 * r100017;
        double r100020 = r100015 * r100012;
        double r100021 = cos(r100009);
        double r100022 = r100020 * r100021;
        double r100023 = r100019 + r100022;
        double r100024 = asin(r100023);
        double r100025 = sin(r100024);
        double r100026 = r100018 * r100025;
        double r100027 = expm1(r100026);
        double r100028 = log1p(r100027);
        double r100029 = r100017 - r100028;
        double r100030 = exp(r100029);
        double r100031 = log(r100030);
        double r100032 = atan2(r100016, r100031);
        double r100033 = r100008 + r100032;
        return r100033;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  3. Applied add-log-exp0.2

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

  4. Applied add-log-exp0.2

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

  5. Applied diff-log0.2

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

  6. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \color{blue}{\left(e^{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}}\]
  7. Using strategy rm

  8. Applied log1p-expm1-u0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \left(e^{\cos delta - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}}\right)}\]

  9. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \left(e^{\cos delta - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}\right)}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))