Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 32.2s

Precision: binary64

Cost: 130304

Math

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

↓

\[\begin{array}{l} t_1 := \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\frac{{\cos delta}^{2} - {t_1}^{2}}{\cos delta + t_1}} \end{array} \]

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 phi1)
          (+
           (* (cos delta) (sin phi1))
           (* (sin delta) (* (cos phi1) (cos theta)))))))
   (+
    lambda1
    (atan2
     (* (sin theta) (* (sin delta) (cos phi1)))
     (/ (- (pow (cos delta) 2.0) (pow t_1 2.0)) (+ (cos delta) t_1))))))

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(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * (cos(phi1) * cos(theta))));
	return lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), ((pow(cos(delta), 2.0) - pow(t_1, 2.0)) / (cos(delta) + t_1)));
}

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
    real(8) :: t_1
    t_1 = sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * (cos(phi1) * cos(theta))))
    code = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (((cos(delta) ** 2.0d0) - (t_1 ** 2.0d0)) / (cos(delta) + t_1)))
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) {
	double t_1 = Math.sin(phi1) * ((Math.cos(delta) * Math.sin(phi1)) + (Math.sin(delta) * (Math.cos(phi1) * Math.cos(theta))));
	return lambda1 + Math.atan2((Math.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), ((Math.pow(Math.cos(delta), 2.0) - Math.pow(t_1, 2.0)) / (Math.cos(delta) + t_1)));
}

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):
	t_1 = math.sin(phi1) * ((math.cos(delta) * math.sin(phi1)) + (math.sin(delta) * (math.cos(phi1) * math.cos(theta))))
	return lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), ((math.pow(math.cos(delta), 2.0) - math.pow(t_1, 2.0)) / (math.cos(delta) + t_1)))

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)
	t_1 = Float64(sin(phi1) * Float64(Float64(cos(delta) * sin(phi1)) + Float64(sin(delta) * Float64(cos(phi1) * cos(theta)))))
	return Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(Float64((cos(delta) ^ 2.0) - (t_1 ^ 2.0)) / Float64(cos(delta) + t_1))))
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)
	t_1 = sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * (cos(phi1) * cos(theta))));
	tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (((cos(delta) ^ 2.0) - (t_1 ^ 2.0)) / (cos(delta) + t_1)));
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_] := Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[Cos[delta], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

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

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

   [Start] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   associate-*l* [=>] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   *-commutative [=>] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1}} \]
   cancel-sign-sub-inv [=>] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta + \left(-\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \phi_1}} \]
   cancel-sign-sub [<=] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta - \left(-\left(-\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1}} \]
   *-commutative [=>] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\sin \phi_1 \cdot \left(-\left(-\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}} \]
   remove-double-neg [=>] 0.2	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \color{blue}{\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}} \]

3. Taylor expanded in delta around inf 0.2

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

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	84288

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

Alternative 2
Error	0.2
Cost	77952

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

Alternative 3
Error	0.2
Cost	71680

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

Alternative 4
Error	3.7
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 5
Error	4.0
Cost	64836

\[\begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \mathbf{if}\;theta \leq -0.029:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t_1}{\mathsf{fma}\left(\sin \sin^{-1} \sin \phi_1, -\sin \phi_1, \cos delta\right)}\\ \mathbf{elif}\;theta \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \left(t_1 + \cos delta \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta + \left(\frac{\cos \left(\phi_1 + \phi_1\right)}{2} + -0.5\right)}\\ \end{array} \]

Alternative 6
Error	4.0
Cost	59016

\[\begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;theta \leq -0.031:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{elif}\;theta \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \left(\sin delta \cdot \cos \phi_1 + \cos delta \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta + \left(\frac{\cos \left(\phi_1 + \phi_1\right)}{2} + -0.5\right)}\\ \end{array} \]

Alternative 7
Error	5.2
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	5.8
Cost	39304

\[\begin{array}{l} t_1 := \sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\\ \mathbf{if}\;delta \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \phi_1 \cdot \phi_1}\\ \end{array} \]

Alternative 9
Error	7.5
Cost	32512

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

Alternative 10
Error	9.2
Cost	26633

\[\begin{array}{l} \mathbf{if}\;theta \leq 3.1 \cdot 10^{-204} \lor \neg \left(theta \leq 7 \cdot 10^{-133}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta - \phi_1 \cdot \phi_1}\\ \end{array} \]

Alternative 11
Error	8.9
Cost	25984

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

Alternative 12
Error	15.3
Cost	19849

\[\begin{array}{l} \mathbf{if}\;delta \leq -0.00335 \lor \neg \left(delta \leq 4 \cdot 10^{-58}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 13
Error	12.6
Cost	19849

\[\begin{array}{l} \mathbf{if}\;delta \leq -0.23 \lor \neg \left(delta \leq 1.1 \cdot 10^{-17}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \]

Alternative 14
Error	12.2
Cost	19849

\[\begin{array}{l} \mathbf{if}\;theta \leq -0.105 \lor \neg \left(theta \leq 4.8 \cdot 10^{-17}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 15
Error	19.0
Cost	13316

\[\begin{array}{l} \mathbf{if}\;delta \leq 5.6 \cdot 10^{-56}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \end{array} \]

Alternative 16
Error	19.1
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023187 
(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))))))))))