Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 1.1m

Precision: 64

• unused variable phi2

Math

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

↓

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

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4418952 = lambda1;
        double r4418953 = theta;
        double r4418954 = sin(r4418953);
        double r4418955 = delta;
        double r4418956 = sin(r4418955);
        double r4418957 = r4418954 * r4418956;
        double r4418958 = phi1;
        double r4418959 = cos(r4418958);
        double r4418960 = r4418957 * r4418959;
        double r4418961 = cos(r4418955);
        double r4418962 = sin(r4418958);
        double r4418963 = r4418962 * r4418961;
        double r4418964 = r4418959 * r4418956;
        double r4418965 = cos(r4418953);
        double r4418966 = r4418964 * r4418965;
        double r4418967 = r4418963 + r4418966;
        double r4418968 = asin(r4418967);
        double r4418969 = sin(r4418968);
        double r4418970 = r4418962 * r4418969;
        double r4418971 = r4418961 - r4418970;
        double r4418972 = atan2(r4418960, r4418971);
        double r4418973 = r4418952 + r4418972;
        return r4418973;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4418974 = phi1;
        double r4418975 = cos(r4418974);
        double r4418976 = delta;
        double r4418977 = sin(r4418976);
        double r4418978 = r4418975 * r4418977;
        double r4418979 = theta;
        double r4418980 = sin(r4418979);
        double r4418981 = r4418978 * r4418980;
        double r4418982 = cos(r4418976);
        double r4418983 = 3.0;
        double r4418984 = pow(r4418982, r4418983);
        double r4418985 = sin(r4418974);
        double r4418986 = r4418985 * r4418985;
        double r4418987 = r4418985 * r4418986;
        double r4418988 = cos(r4418979);
        double r4418989 = r4418988 * r4418978;
        double r4418990 = r4418982 * r4418985;
        double r4418991 = r4418989 + r4418990;
        double r4418992 = asin(r4418991);
        double r4418993 = sin(r4418992);
        double r4418994 = r4418993 * r4418993;
        double r4418995 = r4418993 * r4418994;
        double r4418996 = r4418987 * r4418995;
        double r4418997 = r4418984 - r4418996;
        double r4418998 = r4418982 * r4418982;
        double r4418999 = r4418993 * r4418985;
        double r4419000 = r4418999 * r4418999;
        double r4419001 = r4418982 * r4418999;
        double r4419002 = r4419000 + r4419001;
        double r4419003 = r4418998 + r4419002;
        double r4419004 = r4418997 / r4419003;
        double r4419005 = atan2(r4418981, r4419004);
        double r4419006 = lambda1;
        double r4419007 = r4419005 + r4419006;
        return r4419007;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

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. Simplified 0.2

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

4. Applied flip3-- 0.2

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

6. Applied add-cbrt-cube 0.2

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

7. Applied add-cbrt-cube 0.2

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

8. Applied cbrt-unprod 0.2

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

9. Applied rem-cube-cbrt 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019200 
(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))))))))))