Destination given bearing on a great circle

Average Error: 0.1 → 0.2

Time: 26.0s

Precision: binary64

Cost: 201600

Math

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

↓

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

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
         (-
          (cos delta)
          (*
           (sin phi1)
           (sin
            (asin
             (+
              (* (cos delta) (sin phi1))
              (* (cos phi1) (* (sin delta) (cos theta))))))))))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (* t_1 (* t_1 (/ 1.0 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 = cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(phi1) * (sin(delta) * cos(theta)))))));
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (t_1 * (t_1 * (1.0 / 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 = cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(phi1) * (sin(delta) * cos(theta)))))))
    code = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (t_1 * (t_1 * (1.0d0 / 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.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.cos(delta) * Math.sin(phi1)) + (Math.cos(phi1) * (Math.sin(delta) * Math.cos(theta)))))));
	return lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (t_1 * (t_1 * (1.0 / 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.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.cos(delta) * math.sin(phi1)) + (math.cos(phi1) * (math.sin(delta) * math.cos(theta)))))))
	return lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (t_1 * (t_1 * (1.0 / 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(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(phi1) * Float64(sin(delta) * cos(theta))))))))
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(t_1 * Float64(t_1 * Float64(1.0 / 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 = cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(phi1) * (sin(delta) * cos(theta)))))));
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (t_1 * (t_1 * (1.0 / 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[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$1 * N[(1.0 / t$95$1), $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

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

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.1
Cost	71680

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

Alternative 2
Error	0.1
Cost	71680

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

Alternative 3
Error	3.4
Cost	65152

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

Alternative 4
Error	4.2
Cost	59016

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{if}\;theta \leq -7.4 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 10^{-23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin delta \cdot theta\right) \cdot \cos \phi_1}{\cos delta - \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \cos \phi_1\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	5.0
Cost	45504

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

Alternative 6
Error	5.5
Cost	32968

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -0.000215:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{1 - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	7.2
Cost	32512

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

Alternative 8
Error	7.2
Cost	32512

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

Alternative 9
Error	7.9
Cost	26376

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;theta \leq -6 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	8.6
Cost	25984

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

Alternative 11
Error	15.2
Cost	19848

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{if}\;delta \leq -7.8 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 2.2 \cdot 10^{-192}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	12.2
Cost	19848

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}\\ \mathbf{if}\;theta \leq -7.8 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 31000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	12.4
Cost	19848

\[\begin{array}{l} \mathbf{if}\;theta \leq -7.4 \cdot 10^{+37}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}\\ \mathbf{elif}\;theta \leq 2.1 \cdot 10^{-17}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \]

Alternative 14
Error	18.8
Cost	13448

\[\begin{array}{l} \mathbf{if}\;theta \leq -6.8 \cdot 10^{-153}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;theta \leq 1560000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 15
Error	19.0
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023090 
(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))))))))))