Average Error: 0.2 → 0.2
Time: 40.6s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}\right)\right)}^{3}}{\left(\left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}\right)\right) + \left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}\right)\right) \cdot \cos delta\right) + \cos delta \cdot \cos delta}}\]
\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{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}\right)\right)}^{3}}{\left(\left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}\right)\right) + \left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}\right)\right) \cdot \cos delta\right) + \cos delta \cdot \cos delta}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r2105952 = lambda1;
        double r2105953 = theta;
        double r2105954 = sin(r2105953);
        double r2105955 = delta;
        double r2105956 = sin(r2105955);
        double r2105957 = r2105954 * r2105956;
        double r2105958 = phi1;
        double r2105959 = cos(r2105958);
        double r2105960 = r2105957 * r2105959;
        double r2105961 = cos(r2105955);
        double r2105962 = sin(r2105958);
        double r2105963 = r2105962 * r2105961;
        double r2105964 = r2105959 * r2105956;
        double r2105965 = cos(r2105953);
        double r2105966 = r2105964 * r2105965;
        double r2105967 = r2105963 + r2105966;
        double r2105968 = asin(r2105967);
        double r2105969 = sin(r2105968);
        double r2105970 = r2105962 * r2105969;
        double r2105971 = r2105961 - r2105970;
        double r2105972 = atan2(r2105960, r2105971);
        double r2105973 = r2105952 + r2105972;
        return r2105973;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r2105974 = lambda1;
        double r2105975 = phi1;
        double r2105976 = cos(r2105975);
        double r2105977 = delta;
        double r2105978 = sin(r2105977);
        double r2105979 = theta;
        double r2105980 = sin(r2105979);
        double r2105981 = r2105978 * r2105980;
        double r2105982 = r2105976 * r2105981;
        double r2105983 = cos(r2105977);
        double r2105984 = 3.0;
        double r2105985 = pow(r2105983, r2105984);
        double r2105986 = sin(r2105975);
        double r2105987 = r2105983 * r2105986;
        double r2105988 = cos(r2105979);
        double r2105989 = r2105976 * r2105978;
        double r2105990 = r2105988 * r2105989;
        double r2105991 = r2105987 + r2105990;
        double r2105992 = asin(r2105991);
        double r2105993 = r2105992 * r2105992;
        double r2105994 = r2105992 * r2105993;
        double r2105995 = cbrt(r2105994);
        double r2105996 = sin(r2105995);
        double r2105997 = r2105986 * r2105996;
        double r2105998 = pow(r2105997, r2105984);
        double r2105999 = r2105985 - r2105998;
        double r2106000 = r2105997 * r2105997;
        double r2106001 = r2105997 * r2105983;
        double r2106002 = r2106000 + r2106001;
        double r2106003 = r2105983 * r2105983;
        double r2106004 = r2106002 + r2106003;
        double r2106005 = r2105999 / r2106004;
        double r2106006 = atan2(r2105982, r2106005);
        double r2106007 = r2105974 + r2106006;
        return r2106007;
}

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

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

  5. Applied flip3--0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))