Destination given bearing on a great circle

Average Error: 0.1 → 0.2

Time: 30.6s

Precision: binary64

Cost: 72064

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\left(\cos delta + 0.5 \cdot \left(\cos delta \cdot \left(\cos \left(\phi_1 \cdot 2\right) + -1\right)\right)\right) - \cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)} \]

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
   (* (cos phi1) (* (sin theta) (sin delta)))
   (-
    (+ (cos delta) (* 0.5 (* (cos delta) (+ (cos (* phi1 2.0)) -1.0))))
    (* (cos phi1) (* (sin delta) (* (cos theta) (sin phi1))))))))

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((cos(phi1) * (sin(theta) * sin(delta))), ((cos(delta) + (0.5 * (cos(delta) * (cos((phi1 * 2.0)) + -1.0)))) - (cos(phi1) * (sin(delta) * (cos(theta) * sin(phi1))))));
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))))
end function

↓

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), ((cos(delta) + (0.5d0 * (cos(delta) * (cos((phi1 * 2.0d0)) + (-1.0d0))))) - (cos(phi1) * (sin(delta) * (cos(theta) * sin(phi1))))))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.sin(phi1) * Math.cos(delta)) + ((Math.cos(phi1) * Math.sin(delta)) * Math.cos(theta))))))));
}

↓

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), ((Math.cos(delta) + (0.5 * (Math.cos(delta) * (Math.cos((phi1 * 2.0)) + -1.0)))) - (Math.cos(phi1) * (Math.sin(delta) * (Math.cos(theta) * Math.sin(phi1))))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.sin(phi1) * math.cos(delta)) + ((math.cos(phi1) * math.sin(delta)) * math.cos(theta))))))))

↓

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), ((math.cos(delta) + (0.5 * (math.cos(delta) * (math.cos((phi1 * 2.0)) + -1.0)))) - (math.cos(phi1) * (math.sin(delta) * (math.cos(theta) * math.sin(phi1))))))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)))))))))
end

↓

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), Float64(Float64(cos(delta) + Float64(0.5 * Float64(cos(delta) * Float64(cos(Float64(phi1 * 2.0)) + -1.0)))) - Float64(cos(phi1) * Float64(sin(delta) * Float64(cos(theta) * sin(phi1)))))))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
end

↓

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), ((cos(delta) + (0.5 * (cos(delta) * (cos((phi1 * 2.0)) + -1.0)))) - (cos(phi1) * (sin(delta) * (cos(theta) * sin(phi1))))));
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[delta], $MachinePrecision] + N[(0.5 * N[(N[Cos[delta], $MachinePrecision] * N[(N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[theta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

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. Applied egg-rr 0.1

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

3. Applied egg-rr 0.2

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

4. Simplified 0.2

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

   [Start] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \frac{\cos delta \cdot \left(\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)\right)}{2}\right)} \]
   associate-/l* [=>] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \color{blue}{\frac{\cos delta}{\frac{2}{\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)}}}\right)} \]
   associate-/r/ [=>] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \color{blue}{\frac{\cos delta}{2} \cdot \left(\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)\right)}\right)} \]
   +-inverses [=>] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \frac{\cos delta}{2} \cdot \left(\cos \color{blue}{0} - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]
   cos-0 [=>] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \frac{\cos delta}{2} \cdot \left(\color{blue}{1} - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]

5. Applied egg-rr 0.2

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

6. Taylor expanded in delta around inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos delta - 0.5 \cdot \left(\cos delta \cdot \left(1 - \cos \left(2 \cdot \phi_1\right)\right)\right)\right)} + \left(-\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)} \]

7. Final simplification 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\left(\cos delta + 0.5 \cdot \left(\cos delta \cdot \left(\cos \left(\phi_1 \cdot 2\right) + -1\right)\right)\right) - \cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)} \]

Alternatives

Alternative 1
Error	0.2
Cost	72064

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

Alternative 2
Error	0.1
Cost	71680

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

Alternative 3
Error	3.3
Cost	65536

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

Alternative 4
Error	3.3
Cost	65536

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

Alternative 5
Error	3.3
Cost	65152

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

Alternative 6
Error	4.0
Cost	59017

\[\begin{array}{l} \mathbf{if}\;theta \leq -1.05 \cdot 10^{+34} \lor \neg \left(theta \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta + \left(\frac{\cos \left(\phi_1 + \phi_1\right)}{2} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \cos \phi_1\right)}\\ \end{array} \]

Alternative 7
Error	4.5
Cost	39424

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta + \left(\frac{\cos \left(\phi_1 + \phi_1\right)}{2} + -0.5\right)} \]

Alternative 8
Error	4.8
Cost	33417

\[\begin{array}{l} \mathbf{if}\;delta \leq -2 \lor \neg \left(delta \leq 0.0048\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \cos \phi_1\right)}{\mathsf{fma}\left(-0.5, delta \cdot delta, 0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)\right)}\\ \end{array} \]

Alternative 9
Error	4.7
Cost	32905

\[\begin{array}{l} \mathbf{if}\;delta \leq -0.00015 \lor \neg \left(delta \leq 0.005\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \cos \phi_1\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \]

Alternative 10
Error	6.9
Cost	32512

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

Alternative 11
Error	8.3
Cost	25984

\[\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta} \]

Alternative 12
Error	15.1
Cost	19849

\[\begin{array}{l} \mathbf{if}\;delta \leq -1.9 \cdot 10^{-177} \lor \neg \left(delta \leq 7.5 \cdot 10^{-46}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 13
Error	12.6
Cost	19849

\[\begin{array}{l} \mathbf{if}\;theta \leq -6.2 \cdot 10^{-76} \lor \neg \left(theta \leq 85000000000\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 14
Error	18.7
Cost	13316

\[\begin{array}{l} \mathbf{if}\;delta \leq 8 \cdot 10^{-46}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \end{array} \]

Alternative 15
Error	19.0
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023076 
(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))))))))))