Average Error: 0.1 → 0.2
Time: 16.9s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\frac{{\left({\left(\cos delta\right)}^{3}\right)}^{3} - {\left({\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)\right)}^{3}\right)}^{3}}{\mathsf{fma}\left({\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)\right)}^{3}, {\left(\cos delta\right)}^{3} + {\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)\right)}^{3}, {\left(\cos delta\right)}^{6}\right)}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right), \cos delta\right)\right)\right)}}\]
\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{\frac{{\left({\left(\cos delta\right)}^{3}\right)}^{3} - {\left({\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\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)\right)}^{3}\right)}^{3}}{\mathsf{fma}\left({\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\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)\right)}^{3}, {\left(\cos delta\right)}^{3} + {\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\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)\right)}^{3}, {\left(\cos delta\right)}^{6}\right)}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\mathsf{log1p}\left(\mathsf{expm1}\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) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\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), \cos delta\right)\right)\right)}}
double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return (lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))))));
}
↓
double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return (lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (((pow(pow(cos(delta), 3.0), 3.0) - pow(pow((sin(phi1) * sin(log1p(expm1(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))))), 3.0), 3.0)) / fma(pow((sin(phi1) * sin(log1p(expm1(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))))), 3.0), (pow(cos(delta), 3.0) + pow((sin(phi1) * sin(log1p(expm1(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))))), 3.0)), pow(cos(delta), 6.0))) / fma(cos(delta), cos(delta), (sin(phi1) * (sin(log1p(expm1(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))) * fma(sin(phi1), sin(log1p(expm1(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))), cos(delta))))))));
}

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 0.1

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

  3. Applied log1p-expm1-u0.2

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

  5. Applied flip3--0.2

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

  6. Simplified0.2

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

  8. Applied flip3--0.2

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

  9. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\frac{{\left({\left(\cos delta\right)}^{3}\right)}^{3} - {\left({\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)\right)}^{3}\right)}^{3}}{\color{blue}{\mathsf{fma}\left({\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)\right)}^{3}, {\left(\cos delta\right)}^{3} + {\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)\right)}^{3}, {\left(\cos delta\right)}^{6}\right)}}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right), \cos delta\right)\right)\right)}}\]

  10. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\frac{{\left({\left(\cos delta\right)}^{3}\right)}^{3} - {\left({\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)\right)}^{3}\right)}^{3}}{\mathsf{fma}\left({\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)\right)}^{3}, {\left(\cos delta\right)}^{3} + {\left(\sin \phi_1 \cdot \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)\right)}^{3}, {\left(\cos delta\right)}^{6}\right)}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right), \cos delta\right)\right)\right)}}\]

Reproduce

herbie shell --seed 2020078 +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))))))))))