Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 26.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sin \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\\ t_2 := \sin \phi_1 \cdot t_1\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\cos delta}^{3} - t_2 \cdot \left(t_2 \cdot t_2\right)}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(t_1 \cdot \mathsf{fma}\left(\sin \phi_1, t_1, \cos delta\right)\right)\right)}} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :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))))))))))

↓

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (sin
          (asin
           (fma
            (sin phi1)
            (cos delta)
            (* (sin delta) (* (cos phi1) (cos theta)))))))
        (t_2 (* (sin phi1) t_1)))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (/
      (- (pow (cos delta) 3.0) (* t_2 (* t_2 t_2)))
      (fma
       (cos delta)
       (cos delta)
       (* (sin phi1) (* t_1 (fma (sin phi1) t_1 (cos delta))))))))))

C

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) {
	double t_1 = sin(asin(fma(sin(phi1), cos(delta), (sin(delta) * (cos(phi1) * cos(theta))))));
	double t_2 = sin(phi1) * t_1;
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), ((pow(cos(delta), 3.0) - (t_2 * (t_2 * t_2))) / fma(cos(delta), cos(delta), (sin(phi1) * (t_1 * fma(sin(phi1), t_1, cos(delta)))))));
}

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

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

2. Simplified 0.2

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

3. Applied flip3--_binary64 0.2

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

4. Simplified 0.2

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

5. Simplified 0.2

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

6. Applied cube-mult_binary64 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022097 
(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))))))))))