Bearing on a great circle

Average Error: 13.5 → 13.0

Time: 32.2s

Precision: binary64

Cost: 65604

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

↓

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)\\ \mathbf{if}\;\lambda_2 \leq -5.6 \cdot 10^{+41}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\left(\left(-\frac{\sin \lambda_1}{\lambda_2}\right) + \cos \lambda_1\right) \cdot t_0\right) \cdot \cos \phi_2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0 \cdot \cos \phi_2}{t_1}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

↓

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1
         (-
          (* (cos phi1) (sin phi2))
          (* (sin phi1) (* (cos (- lambda1 lambda2)) (cos phi2))))))
   (if (<= lambda2 -5.6e+41)
     (atan2
      (* (* (+ (- (/ (sin lambda1) lambda2)) (cos lambda1)) t_0) (cos phi2))
      t_1)
     (atan2 (* t_0 (cos phi2)) t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}

↓

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = (cos(phi1) * sin(phi2)) - (sin(phi1) * (cos((lambda1 - lambda2)) * cos(phi2)));
	double tmp;
	if (lambda2 <= -5.6e+41) {
		tmp = atan2((((-(sin(lambda1) / lambda2) + cos(lambda1)) * t_0) * cos(phi2)), t_1);
	} else {
		tmp = atan2((t_0 * cos(phi2)), t_1);
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

↓

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = (cos(phi1) * sin(phi2)) - (sin(phi1) * (cos((lambda1 - lambda2)) * cos(phi2)))
    if (lambda2 <= (-5.6d+41)) then
        tmp = atan2((((-(sin(lambda1) / lambda2) + cos(lambda1)) * t_0) * cos(phi2)), t_1)
    else
        tmp = atan2((t_0 * cos(phi2)), t_1)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
}

↓

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = (Math.cos(phi1) * Math.sin(phi2)) - (Math.sin(phi1) * (Math.cos((lambda1 - lambda2)) * Math.cos(phi2)));
	double tmp;
	if (lambda2 <= -5.6e+41) {
		tmp = Math.atan2((((-(Math.sin(lambda1) / lambda2) + Math.cos(lambda1)) * t_0) * Math.cos(phi2)), t_1);
	} else {
		tmp = Math.atan2((t_0 * Math.cos(phi2)), t_1);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))

↓

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = (math.cos(phi1) * math.sin(phi2)) - (math.sin(phi1) * (math.cos((lambda1 - lambda2)) * math.cos(phi2)))
	tmp = 0
	if lambda2 <= -5.6e+41:
		tmp = math.atan2((((-(math.sin(lambda1) / lambda2) + math.cos(lambda1)) * t_0) * math.cos(phi2)), t_1)
	else:
		tmp = math.atan2((t_0 * math.cos(phi2)), t_1)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))))
end

↓

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * Float64(cos(Float64(lambda1 - lambda2)) * cos(phi2))))
	tmp = 0.0
	if (lambda2 <= -5.6e+41)
		tmp = atan(Float64(Float64(Float64(Float64(-Float64(sin(lambda1) / lambda2)) + cos(lambda1)) * t_0) * cos(phi2)), t_1);
	else
		tmp = atan(Float64(t_0 * cos(phi2)), t_1);
	end
	return tmp
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
end

↓

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = (cos(phi1) * sin(phi2)) - (sin(phi1) * (cos((lambda1 - lambda2)) * cos(phi2)));
	tmp = 0.0;
	if (lambda2 <= -5.6e+41)
		tmp = atan2((((-(sin(lambda1) / lambda2) + cos(lambda1)) * t_0) * cos(phi2)), t_1);
	else
		tmp = atan2((t_0 * cos(phi2)), t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -5.6e+41], N[ArcTan[N[(N[(N[((-N[(N[Sin[lambda1], $MachinePrecision] / lambda2), $MachinePrecision]) + N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision], N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -5.5999999999999999e41

   1. Initial program 27.2

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Simplified 27.2

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}} \]
      Proof

      [Start] 27.2	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.2	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.2	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}} \]

   3. Applied egg-rr 27.2

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\left(\sin \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{\sin \left(\lambda_1 - \lambda_2\right)}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \]

   4. Taylor expanded in lambda1 around 0 27.6

      \[\leadsto \tan^{-1}_* \frac{\left(\left(\sin \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{\color{blue}{\sin \left(-\lambda_2\right)}}\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \]

   5. Taylor expanded in lambda2 around 0 25.0

      \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\left(-1 \cdot \frac{\sin \lambda_1}{\lambda_2} + \cos \lambda_1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \]

   6. Simplified 25.0

      \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\left(\left(-\frac{\sin \lambda_1}{\lambda_2}\right) + \cos \lambda_1\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \]
      Proof

      [Start] 25.0	\[ \tan^{-1}_* \frac{\left(\left(-1 \cdot \frac{\sin \lambda_1}{\lambda_2} + \cos \lambda_1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 25.0	\[ \tan^{-1}_* \frac{\left(\left(\color{blue}{\frac{\sin \lambda_1}{\lambda_2} \cdot -1} + \cos \lambda_1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 25.0	\[ \tan^{-1}_* \frac{\left(\left(\color{blue}{\left(-\frac{\sin \lambda_1}{\lambda_2}\right)} + \cos \lambda_1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \]

   if -5.5999999999999999e41 < lambda2

   1. Initial program 9.6

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Simplified 9.6

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}} \]
      Proof

      [Start] 9.6	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 9.6	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 9.6	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5.6 \cdot 10^{+41}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\left(\left(-\frac{\sin \lambda_1}{\lambda_2}\right) + \cos \lambda_1\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.7
Cost	52424

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}\\ \mathbf{if}\;\lambda_2 \leq -0.042:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 0.0024:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	13.6
Cost	52424

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_2 \leq -8.8 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \left(-\lambda_2\right) \cdot \cos \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 0.0225:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}\\ \end{array} \]

Alternative 3
Error	16.8
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -0.0072:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 5.8 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	19.0
Cost	52232

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -2.3 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 540000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.5
Cost	52224

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \]

Alternative 6
Error	20.3
Cost	46216

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -4000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 \cdot \cos \phi_1 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	19.9
Cost	45960

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0 \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_2 \leq -0.00052:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 6300:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	22.2
Cost	45832

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \cos \lambda_1}\\ \mathbf{if}\;\phi_2 \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 6300:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	19.8
Cost	45832

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0 \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_2 \leq -0.00135:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 6300:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	23.1
Cost	39432

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.0014:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.16 \cdot 10^{+24}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	25.9
Cost	32968

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \lambda_1\right)}\\ \mathbf{if}\;\phi_1 \leq -0.108:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 1.65 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 + \cos \lambda_1 \cdot \left(-\phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	27.8
Cost	32840

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -2.85 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 2.55 \cdot 10^{+61}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	27.4
Cost	26376

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -3 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.16 \cdot 10^{+24}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{-\cos \left(-\lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	36.5
Cost	26184

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -3.1 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 3.4 \cdot 10^{-81}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	39.1
Cost	26120

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.0013:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 14500:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	33.3
Cost	25984

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

Alternative 17
Error	45.3
Cost	19656

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \mathbf{if}\;\lambda_1 \leq -2.9 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 7 \cdot 10^{-73}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	43.7
Cost	19456

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]

Alternative 19
Error	48.3
Cost	19328

\[\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2} \]

Reproduce

herbie shell --seed 2023090 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))