Average Error: 0.2 → 0.2
Time: 27.9s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{{\left(\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}}\]
\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}{\sqrt[3]{{\left(\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)}^{3}}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r87763 = lambda1;
        double r87764 = theta;
        double r87765 = sin(r87764);
        double r87766 = delta;
        double r87767 = sin(r87766);
        double r87768 = r87765 * r87767;
        double r87769 = phi1;
        double r87770 = cos(r87769);
        double r87771 = r87768 * r87770;
        double r87772 = cos(r87766);
        double r87773 = sin(r87769);
        double r87774 = r87773 * r87772;
        double r87775 = r87770 * r87767;
        double r87776 = cos(r87764);
        double r87777 = r87775 * r87776;
        double r87778 = r87774 + r87777;
        double r87779 = asin(r87778);
        double r87780 = sin(r87779);
        double r87781 = r87773 * r87780;
        double r87782 = r87772 - r87781;
        double r87783 = atan2(r87771, r87782);
        double r87784 = r87763 + r87783;
        return r87784;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r87785 = lambda1;
        double r87786 = theta;
        double r87787 = sin(r87786);
        double r87788 = delta;
        double r87789 = sin(r87788);
        double r87790 = r87787 * r87789;
        double r87791 = phi1;
        double r87792 = cos(r87791);
        double r87793 = r87790 * r87792;
        double r87794 = cos(r87788);
        double r87795 = sin(r87791);
        double r87796 = r87795 * r87794;
        double r87797 = r87792 * r87789;
        double r87798 = cos(r87786);
        double r87799 = r87797 * r87798;
        double r87800 = r87796 + r87799;
        double r87801 = asin(r87800);
        double r87802 = sin(r87801);
        double r87803 = r87795 * r87802;
        double r87804 = r87794 - r87803;
        double r87805 = 3.0;
        double r87806 = pow(r87804, r87805);
        double r87807 = cbrt(r87806);
        double r87808 = atan2(r87793, r87807);
        double r87809 = r87785 + r87808;
        return r87809;
}

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-cbrt-cube0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{{\left(\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}}\]

Reproduce

herbie shell --seed 2019351 
(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))))))))))