Average Error: 0.1 → 0.1
Time: 37.5s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\mathsf{fma}\left(\cos delta, \cos delta, -{\left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)}{\mathsf{fma}\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right), \sin \phi_1, \cos delta\right)}}\]
\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{\mathsf{fma}\left(\cos delta, \cos delta, -{\left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)}{\mathsf{fma}\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right), \sin \phi_1, \cos delta\right)}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r101084 = lambda1;
        double r101085 = theta;
        double r101086 = sin(r101085);
        double r101087 = delta;
        double r101088 = sin(r101087);
        double r101089 = r101086 * r101088;
        double r101090 = phi1;
        double r101091 = cos(r101090);
        double r101092 = r101089 * r101091;
        double r101093 = cos(r101087);
        double r101094 = sin(r101090);
        double r101095 = r101094 * r101093;
        double r101096 = r101091 * r101088;
        double r101097 = cos(r101085);
        double r101098 = r101096 * r101097;
        double r101099 = r101095 + r101098;
        double r101100 = asin(r101099);
        double r101101 = sin(r101100);
        double r101102 = r101094 * r101101;
        double r101103 = r101093 - r101102;
        double r101104 = atan2(r101092, r101103);
        double r101105 = r101084 + r101104;
        return r101105;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r101106 = lambda1;
        double r101107 = theta;
        double r101108 = sin(r101107);
        double r101109 = delta;
        double r101110 = sin(r101109);
        double r101111 = r101108 * r101110;
        double r101112 = phi1;
        double r101113 = cos(r101112);
        double r101114 = r101111 * r101113;
        double r101115 = cos(r101109);
        double r101116 = cos(r101107);
        double r101117 = r101113 * r101116;
        double r101118 = sin(r101112);
        double r101119 = r101118 * r101115;
        double r101120 = fma(r101110, r101117, r101119);
        double r101121 = 2.0;
        double r101122 = pow(r101120, r101121);
        double r101123 = pow(r101118, r101121);
        double r101124 = r101122 * r101123;
        double r101125 = -r101124;
        double r101126 = fma(r101115, r101115, r101125);
        double r101127 = asin(r101120);
        double r101128 = sin(r101127);
        double r101129 = fma(r101128, r101118, r101115);
        double r101130 = r101126 / r101129;
        double r101131 = atan2(r101114, r101130);
        double r101132 = r101106 + r101131;
        return r101132;
}

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 0.1

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

  3. Applied expm1-log1p-u0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}}\]

  4. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1\right)}\right)}\]
  5. Using strategy rm

  6. Applied flip--0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{\cos delta \cdot \cos delta - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1\right)\right)}{\cos delta + \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1\right)\right)}}}\]

  7. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\mathsf{fma}\left(\cos delta, \cos delta, -{\left(\sqrt[3]{\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1}\right)}^{\left(2 \cdot 3\right)}\right)}}{\cos delta + \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1\right)\right)}}\]

  8. Simplified0.2

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

  9. Taylor expanded around inf 0.2

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019208 +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))))))))))