Average Error: 0.2 → 0.2
Time: 13.7s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\left(\sqrt[3]{\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \cdot \sqrt[3]{\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\right)}\]
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\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}{\cos delta - \sin \phi_1 \cdot \sin \left(\left(\sqrt[3]{\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \cdot \sqrt[3]{\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r91036 = lambda1;
        double r91037 = theta;
        double r91038 = sin(r91037);
        double r91039 = delta;
        double r91040 = sin(r91039);
        double r91041 = r91038 * r91040;
        double r91042 = phi1;
        double r91043 = cos(r91042);
        double r91044 = r91041 * r91043;
        double r91045 = cos(r91039);
        double r91046 = sin(r91042);
        double r91047 = r91046 * r91045;
        double r91048 = r91043 * r91040;
        double r91049 = cos(r91037);
        double r91050 = r91048 * r91049;
        double r91051 = r91047 + r91050;
        double r91052 = asin(r91051);
        double r91053 = sin(r91052);
        double r91054 = r91046 * r91053;
        double r91055 = r91045 - r91054;
        double r91056 = atan2(r91044, r91055);
        double r91057 = r91036 + r91056;
        return r91057;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r91058 = lambda1;
        double r91059 = theta;
        double r91060 = sin(r91059);
        double r91061 = delta;
        double r91062 = sin(r91061);
        double r91063 = r91060 * r91062;
        double r91064 = phi1;
        double r91065 = cos(r91064);
        double r91066 = r91063 * r91065;
        double r91067 = cos(r91061);
        double r91068 = sin(r91064);
        double r91069 = r91068 * r91067;
        double r91070 = r91065 * r91062;
        double r91071 = cos(r91059);
        double r91072 = r91070 * r91071;
        double r91073 = r91069 + r91072;
        double r91074 = asin(r91073);
        double r91075 = cbrt(r91074);
        double r91076 = r91075 * r91075;
        double r91077 = r91076 * r91075;
        double r91078 = sin(r91077);
        double r91079 = r91068 * r91078;
        double r91080 = r91067 - r91079;
        double r91081 = atan2(r91066, r91080);
        double r91082 = r91058 + r91081;
        return r91082;
}

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-cube-cbrt0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020059 
(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))))))))))