Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 19.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}{\mathsf{fma}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \frac{\pi}{2} + \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \frac{\pi}{4} \cdot \pi\right)}\right), \cos delta\right)\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r90062 = lambda1;
        double r90063 = theta;
        double r90064 = sin(r90063);
        double r90065 = delta;
        double r90066 = sin(r90065);
        double r90067 = r90064 * r90066;
        double r90068 = phi1;
        double r90069 = cos(r90068);
        double r90070 = r90067 * r90069;
        double r90071 = cos(r90065);
        double r90072 = sin(r90068);
        double r90073 = r90072 * r90071;
        double r90074 = r90069 * r90066;
        double r90075 = cos(r90063);
        double r90076 = r90074 * r90075;
        double r90077 = r90073 + r90076;
        double r90078 = asin(r90077);
        double r90079 = sin(r90078);
        double r90080 = r90072 * r90079;
        double r90081 = r90071 - r90080;
        double r90082 = atan2(r90070, r90081);
        double r90083 = r90062 + r90082;
        return r90083;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r90084 = lambda1;
        double r90085 = theta;
        double r90086 = sin(r90085);
        double r90087 = delta;
        double r90088 = sin(r90087);
        double r90089 = r90086 * r90088;
        double r90090 = phi1;
        double r90091 = cos(r90090);
        double r90092 = r90089 * r90091;
        double r90093 = cos(r90087);
        double r90094 = 3.0;
        double r90095 = pow(r90093, r90094);
        double r90096 = sin(r90090);
        double r90097 = r90091 * r90088;
        double r90098 = cos(r90085);
        double r90099 = r90097 * r90098;
        double r90100 = fma(r90096, r90093, r90099);
        double r90101 = asin(r90100);
        double r90102 = sin(r90101);
        double r90103 = r90102 * r90096;
        double r90104 = pow(r90103, r90094);
        double r90105 = r90095 - r90104;
        double r90106 = r90096 * r90102;
        double r90107 = atan2(1.0, 0.0);
        double r90108 = 2.0;
        double r90109 = r90107 / r90108;
        double r90110 = pow(r90109, r90094);
        double r90111 = acos(r90100);
        double r90112 = pow(r90111, r90094);
        double r90113 = r90110 - r90112;
        double r90114 = r90109 + r90111;
        double r90115 = 4.0;
        double r90116 = r90107 / r90115;
        double r90117 = r90116 * r90107;
        double r90118 = fma(r90111, r90114, r90117);
        double r90119 = r90113 / r90118;
        double r90120 = sin(r90119);
        double r90121 = fma(r90096, r90120, r90093);
        double r90122 = r90106 * r90121;
        double r90123 = fma(r90093, r90093, r90122);
        double r90124 = r90105 / r90123;
        double r90125 = atan2(r90092, r90124);
        double r90126 = r90084 + r90125;
        return r90126;
}

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

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

4. Applied flip3-- 0.2

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

5. Simplified 0.2

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

7. Applied asin-acos 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}, \cos delta\right)\right)}}\]
8. Using strategy rm

9. Applied flip3-- 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \color{blue}{\left(\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \frac{\pi}{2} \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}\right)}, \cos delta\right)\right)}}\]

10. Simplified 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \frac{\pi}{2} + \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \frac{\pi}{4} \cdot \pi\right)}}\right), \cos delta\right)\right)}}\]

11. Final simplification 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}{\mathsf{fma}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \frac{\pi}{2} + \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \frac{\pi}{4} \cdot \pi\right)}\right), \cos delta\right)\right)}}\]

Reproduce

herbie shell --seed 2020046 +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))))))))))