Destination given bearing on a great circle

Average Accuracy: 99.8% → 99.7%

Time: 40.0s

Precision: binary64

Cost: 72064

Math

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

↓

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

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

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(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) + (((cos(delta) / 2.0d0) * (cos((phi1 + phi1)) + (-1.0d0))) - (sin(phi1) * (cos(phi1) * (sin(delta) * cos(theta)))))))
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.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (Math.cos(delta) + (((Math.cos(delta) / 2.0) * (Math.cos((phi1 + phi1)) + -1.0)) - (Math.sin(phi1) * (Math.cos(phi1) * (Math.sin(delta) * Math.cos(theta)))))));
}

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.sin(theta) * math.sin(delta)) * math.cos(phi1)), (math.cos(delta) + (((math.cos(delta) / 2.0) * (math.cos((phi1 + phi1)) + -1.0)) - (math.sin(phi1) * (math.cos(phi1) * (math.sin(delta) * math.cos(theta)))))))

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(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) + Float64(Float64(Float64(cos(delta) / 2.0) * Float64(cos(Float64(phi1 + phi1)) + -1.0)) - Float64(sin(phi1) * Float64(cos(phi1) * Float64(sin(delta) * cos(theta))))))))
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(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) + (((cos(delta) / 2.0) * (cos((phi1 + phi1)) + -1.0)) - (sin(phi1) * (cos(phi1) * (sin(delta) * cos(theta)))))));
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[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] + N[(N[(N[(N[Cos[delta], $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   sin-asin [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}} \]
   +-commutative [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \color{blue}{\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \sin \phi_1 \cdot \cos delta\right)}} \]
   distribute-rgt-in [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{\left(\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1 + \left(\sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1\right)}} \]
   associate-*l* [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\color{blue}{\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)} \cdot \sin \phi_1 + \left(\sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1\right)} \]
   *-commutative [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \color{blue}{\left(\cos delta \cdot \sin \phi_1\right)} \cdot \sin \phi_1\right)} \]

3. Applied egg-rr 99.7%

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

   [Start] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \left(\cos delta \cdot \sin \phi_1\right) \cdot \sin \phi_1\right)} \]
   associate-*l* [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \color{blue}{\cos delta \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)}\right)} \]
   sin-mult [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \cos delta \cdot \color{blue}{\frac{\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)}{2}}\right)} \]
   associate-*r/ [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \color{blue}{\frac{\cos delta \cdot \left(\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)\right)}{2}}\right)} \]

4. Simplified 99.7%

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

   [Start] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \frac{\cos delta \cdot \left(\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)\right)}{2}\right)} \]
   associate-/l* [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \color{blue}{\frac{\cos delta}{\frac{2}{\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)}}}\right)} \]
   associate-/r/ [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \color{blue}{\frac{\cos delta}{2} \cdot \left(\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)\right)}\right)} \]
   +-inverses [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \frac{\cos delta}{2} \cdot \left(\cos \color{blue}{0} - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]
   cos-0 [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \frac{\cos delta}{2} \cdot \left(\color{blue}{1} - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]

5. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	65536

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

Alternative 2
Accuracy	94.8%
Cost	65152

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

Alternative 3
Accuracy	94.8%
Cost	59008

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

Alternative 4
Accuracy	92.3%
Cost	39424

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

Alternative 5
Accuracy	89.2%
Cost	39172

\[\begin{array}{l} t_1 := \sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \]

Alternative 6
Accuracy	89.2%
Cost	39108

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq 2.2 \cdot 10^{+100}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \end{array} \]

Alternative 7
Accuracy	89.2%
Cost	33028

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq 1.5 \cdot 10^{+99}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{0.5 + \cos \left(\phi_1 + \phi_1\right) \cdot 0.5}\\ \end{array} \]

Alternative 8
Accuracy	91.8%
Cost	32904

\[\begin{array}{l} \mathbf{if}\;delta \leq -7600000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.42 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]

Alternative 9
Accuracy	88.5%
Cost	32512

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

Alternative 10
Accuracy	86.2%
Cost	25984

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

Alternative 11
Accuracy	80.1%
Cost	19849

\[\begin{array}{l} \mathbf{if}\;theta \leq -4000000000000 \lor \neg \left(theta \leq 2.02 \cdot 10^{+21}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 12
Accuracy	72.8%
Cost	19584

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

Alternative 13
Accuracy	66.8%
Cost	13184

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

Reproduce

herbie shell --seed 2023131 
(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))))))))))