Destination given bearing on a great circle

Average Error: 0.1 → 0.1

Time: 27.3s

Precision: binary64

Cost: 84480

Math

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

↓

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

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
 (+
  lambda1
  (atan2
   (* (sin theta) (* (sin delta) (cos phi1)))
   (-
    (cos delta)
    (*
     (sin phi1)
     (sin
      (asin
       (+
        (* (cos delta) (sin phi1))
        (* (* (sin delta) (cos phi1)) (cos theta))))))))))

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) {
	return lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (cos(delta) - (sin(phi1) * sin(asin((cos(delta) * sin(phi1)) + ((sin(delta) * cos(phi1)) * cos(theta)))))));
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Alternatives

Alternative 1
Error	0.2
Cost	220672

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

Alternative 2
Error	11.2
Cost	214656

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\sqrt[3]{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)} \cdot \sqrt[3]{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)}\right) \cdot \sqrt[3]{1 - \left(\sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right) + 0.5 \cdot \left(delta \cdot delta\right)\right)}}\]

Alternative 3
Error	0.2
Cost	181760

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

Alternative 4
Error	7.0
Cost	168704

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

Alternative 5
Error	41.5
Cost	162048

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

Alternative 6
Error	16.8
Cost	155776

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

Alternative 7
Error	31.5
Cost	142720

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

Alternative 8
Error	0.2
Cost	129600

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

Alternative 9
Error	28.9
Cost	116864

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

Alternative 10
Error	0.2
Cost	116736

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

Alternative 11
Error	12.9
Cost	110336

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

Alternative 12
Error	0.2
Cost	97344

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

Alternative 13
Error	16.8
Cost	97280

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

Alternative 14
Error	0.2
Cost	97280

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

Alternative 15
Error	8.8
Cost	97280

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

Alternative 16
Error	0.2
Cost	97280

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

Alternative 17
Error	12.9
Cost	90880

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

Alternative 18
Error	0.1
Cost	84480

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

Alternative 19
Error	0.1
Cost	84480

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

Alternative 20
Error	11.1
Cost	78464

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{1 - \left(\sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right) + 0.5 \cdot \left(delta \cdot delta\right)\right)}\]

Alternative 21
Error	14.2
Cost	78080

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

Alternative 22
Error	14.3
Cost	78080

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

Alternative 23
Error	11.7
Cost	78080

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

Alternative 24
Error	12.9
Cost	78080

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

Alternative 25
Error	7.2
Cost	45312

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

Alternative 26
Error	7.2
Cost	32512

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

Alternative 27
Error	60.9
Cost	64

\[1\]

Alternative 28
Error	62.0
Cost	64

\[0\]

Alternative 29
Error	60.9
Cost	64

\[-1\]

Derivation

1. Initial program 0.1

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

3. Applied associate-*l*_binary64_1724 0.1

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

4. Simplified 0.1

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

5. Simplified 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2021022 
(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))))))))))