Average Error: 0.2 → 0.2
Time: 14.2s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right), \cos delta\right)\right)\right)}}\]
\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}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right), \cos delta\right)\right)\right)}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r96031 = lambda1;
        double r96032 = theta;
        double r96033 = sin(r96032);
        double r96034 = delta;
        double r96035 = sin(r96034);
        double r96036 = r96033 * r96035;
        double r96037 = phi1;
        double r96038 = cos(r96037);
        double r96039 = r96036 * r96038;
        double r96040 = cos(r96034);
        double r96041 = sin(r96037);
        double r96042 = r96041 * r96040;
        double r96043 = r96038 * r96035;
        double r96044 = cos(r96032);
        double r96045 = r96043 * r96044;
        double r96046 = r96042 + r96045;
        double r96047 = asin(r96046);
        double r96048 = sin(r96047);
        double r96049 = r96041 * r96048;
        double r96050 = r96040 - r96049;
        double r96051 = atan2(r96039, r96050);
        double r96052 = r96031 + r96051;
        return r96052;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r96053 = lambda1;
        double r96054 = theta;
        double r96055 = sin(r96054);
        double r96056 = delta;
        double r96057 = sin(r96056);
        double r96058 = r96055 * r96057;
        double r96059 = phi1;
        double r96060 = cos(r96059);
        double r96061 = r96058 * r96060;
        double r96062 = cos(r96056);
        double r96063 = 3.0;
        double r96064 = pow(r96062, r96063);
        double r96065 = sin(r96059);
        double r96066 = pow(r96065, r96063);
        double r96067 = cos(r96054);
        double r96068 = r96060 * r96067;
        double r96069 = r96057 * r96068;
        double r96070 = r96065 * r96062;
        double r96071 = r96069 + r96070;
        double r96072 = asin(r96071);
        double r96073 = sin(r96072);
        double r96074 = pow(r96073, r96063);
        double r96075 = r96066 * r96074;
        double r96076 = r96064 - r96075;
        double r96077 = fma(r96065, r96073, r96062);
        double r96078 = r96073 * r96077;
        double r96079 = r96065 * r96078;
        double r96080 = fma(r96062, r96062, r96079);
        double r96081 = r96076 / r96080;
        double r96082 = atan2(r96061, r96081);
        double r96083 = r96053 + r96082;
        return r96083;
}

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. Taylor expanded around 0 0.2

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

  4. Applied flip3--0.2

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

  5. Simplified0.2

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

  7. Applied unpow-prod-down0.2

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

  8. Final simplification0.2

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

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))))))))))