Destination given bearing on a great circle

Average Error: 0.1 → 0.2

Time: 32.7s

Precision: binary64

Cost: 78208

Math

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

↓

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

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)
    (+
     (* (sin phi1) (* (cos phi1) (* (sin delta) (cos theta))))
     (* (sin phi1) (* (cos delta) (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) - ((sin(phi1) * (cos(phi1) * (sin(delta) * cos(theta)))) + (sin(phi1) * (cos(delta) * 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) - ((sin(phi1) * (cos(phi1) * (sin(delta) * cos(theta)))) + (sin(phi1) * (cos(delta) * 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) - ((Math.sin(phi1) * (Math.cos(phi1) * (Math.sin(delta) * Math.cos(theta)))) + (Math.sin(phi1) * (Math.cos(delta) * 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) - ((math.sin(phi1) * (math.cos(phi1) * (math.sin(delta) * math.cos(theta)))) + (math.sin(phi1) * (math.cos(delta) * 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(cos(delta) - Float64(Float64(sin(phi1) * Float64(cos(phi1) * Float64(sin(delta) * cos(theta)))) + Float64(sin(phi1) * Float64(cos(delta) * 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) - ((sin(phi1) * (cos(phi1) * (sin(delta) * cos(theta)))) + (sin(phi1) * (cos(delta) * 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[Cos[delta], $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[delta], $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 0.2

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

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.1
Cost	78144

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

Alternative 2
Error	0.2
Cost	72064

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

Alternative 3
Error	0.1
Cost	71680

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

Alternative 4
Error	3.3
Cost	65152

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

Alternative 5
Error	4.9
Cost	45576

\[\begin{array}{l} t_1 := \sin theta \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -628.3065843612982:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.1442322950092507:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1 \cdot \left(delta + {delta}^{3} \cdot -0.16666666666666666\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot t_1}{\log \left(e^{\cos delta}\right)}\\ \end{array} \]

Alternative 6
Error	4.8
Cost	45504

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

Alternative 7
Error	4.9
Cost	39624

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -628.3065843612982:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 0.1442322950092507:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \cos \phi_1\right) \cdot \left(delta + {delta}^{3} \cdot -0.16666666666666666\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	4.9
Cost	39304

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -628.3065843612982:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 0.1442322950092507:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	4.9
Cost	39240

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -628.3065843612982:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 0.1442322950092507:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	4.9
Cost	32904

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -628.3065843612982:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 0.1442322950092507:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	7.0
Cost	32512

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

Alternative 12
Error	8.5
Cost	25984

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

Alternative 13
Error	11.9
Cost	19848

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{if}\;theta \leq -3.0233410817471927 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 9.008592846041128 \cdot 10^{-9}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	18.7
Cost	19720

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -8.591334748073943 \cdot 10^{-275}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\lambda_1 \leq 1.4896687908702544 \cdot 10^{-210}:\\ \;\;\;\;\tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 15
Error	14.9
Cost	19584

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

Alternative 16
Error	18.5
Cost	13448

\[\begin{array}{l} \mathbf{if}\;delta \leq 7.057143486762459 \cdot 10^{-77}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;delta \leq 8.247438702273748 \cdot 10^{+272}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 17
Error	18.4
Cost	13448

\[\begin{array}{l} \mathbf{if}\;delta \leq 7.057143486762459 \cdot 10^{-77}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;delta \leq 8.247438702273748 \cdot 10^{+272}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{1}\\ \end{array} \]

Alternative 18
Error	18.8
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2022300 
(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))))))))))