Distance on a great circle

Percentage Accurate: 61.9% → 78.9%

Time: 45.7s

Alternatives: 24

Speedup: 1.2×

• Could not converge on a ground truth

  1. R = 8.469010508863919e-107
  2. lambda1 = -2.9615738330261346e+100
  3. lambda2 = -4.802165281475614e+99
  4. phi1 = 2.0650505322744054e-103
  5. phi2 = 3.9557708808148694e+39

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    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
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0)
    code = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt(t_1), Math.sqrt((1.0 - t_1))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0)
	return R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    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
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0)
    code = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt(t_1), Math.sqrt((1.0 - t_1))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0)
	return R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ \left(\tan^{-1}_* \frac{\sqrt{\left(t\_1 \cdot t\_0\right) \cdot t\_0 + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + t\_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* (cos phi2) (cos phi1))))
   (*
    (*
     (atan2
      (sqrt
       (+
        (* (* t_1 t_0) t_0)
        (pow
         (-
          (* (cos (* phi2 0.5)) (sin (* 0.5 phi1)))
          (* (sin (* phi2 0.5)) (cos (* 0.5 phi1))))
         2.0)))
      (sqrt
       (fma
        (fma
         -0.5
         (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))
         0.5)
        (* (- (cos phi2)) (cos phi1))
        (fma (+ (* (sin phi2) (sin phi1)) t_1) 0.5 0.5))))
     2.0)
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = cos(phi2) * cos(phi1);
	return (atan2(sqrt((((t_1 * t_0) * t_0) + pow(((cos((phi2 * 0.5)) * sin((0.5 * phi1))) - (sin((phi2 * 0.5)) * cos((0.5 * phi1)))), 2.0))), sqrt(fma(fma(-0.5, fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))), 0.5), (-cos(phi2) * cos(phi1)), fma(((sin(phi2) * sin(phi1)) + t_1), 0.5, 0.5)))) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(cos(phi2) * cos(phi1))
	return Float64(Float64(atan(sqrt(Float64(Float64(Float64(t_1 * t_0) * t_0) + (Float64(Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1)))) ^ 2.0))), sqrt(fma(fma(-0.5, fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))), 0.5), Float64(Float64(-cos(phi2)) * cos(phi1)), fma(Float64(Float64(sin(phi2) * sin(phi1)) + t_1), 0.5, 0.5)))) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(-0.5 * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Initial program 66.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. div-inv N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. lower-*.f64 67.6

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 67.6%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Applied rewrites 67.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   2. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \color{blue}{\left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   3. cos-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   4. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   5. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   6. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   8. +-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   9. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   10. lower-+.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   11. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   13. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   14. lower-sin.f64 79.4

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

8. Applied rewrites 79.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, 0.5, 0.5\right)\right)}}\right) \]
9. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   2. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}, \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   3. cos-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2}, \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   4. +-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2}, \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   5. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2, \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}, \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   7. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\color{blue}{\sin \lambda_2}, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\sin \lambda_2, \color{blue}{\sin \lambda_1}, \cos \lambda_1 \cdot \cos \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   9. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   10. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 79.9

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2} \cdot \cos \lambda_1\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

10. Applied rewrites 79.9%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)}, 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

11. Final simplification 79.9%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
12. Add Preprocessing

Alternative 2: 61.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_2 - \phi_1\right)\\ t_1 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t\_2\right) \cdot t\_2\\ t_4 := \sqrt{t\_3}\\ \mathbf{if}\;\tan^{-1}_* \frac{t\_4}{\sqrt{1 - t\_3}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{t\_4}{\sqrt{1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(\cos \left(\phi_2 + \phi_1\right) + t\_0\right) \cdot t\_1, 0.5, 0.5 - t\_0 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_1, \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- phi2 phi1)))
        (t_1 (fma -0.5 (cos (- lambda1 lambda2)) 0.5))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi2) (cos phi1)) t_2) t_2)))
        (t_4 (sqrt t_3)))
   (if (<= (atan2 t_4 (sqrt (- 1.0 t_3))) 2e-16)
     (*
      (*
       (atan2 t_4 (sqrt (- 1.0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))))
       2.0)
      R)
     (*
      (*
       (atan2
        (sqrt
         (fma (* (+ (cos (+ phi2 phi1)) t_0) t_1) 0.5 (- 0.5 (* t_0 0.5))))
        (sqrt
         (fma
          t_1
          (* (- (cos phi2)) (cos phi1))
          (fma (cos (- phi1 phi2)) 0.5 0.5))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi2 - phi1));
	double t_1 = fma(-0.5, cos((lambda1 - lambda2)), 0.5);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi2) * cos(phi1)) * t_2) * t_2);
	double t_4 = sqrt(t_3);
	double tmp;
	if (atan2(t_4, sqrt((1.0 - t_3))) <= 2e-16) {
		tmp = (atan2(t_4, sqrt((1.0 - pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)))) * 2.0) * R;
	} else {
		tmp = (atan2(sqrt(fma(((cos((phi2 + phi1)) + t_0) * t_1), 0.5, (0.5 - (t_0 * 0.5)))), sqrt(fma(t_1, (-cos(phi2) * cos(phi1)), fma(cos((phi1 - phi2)), 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi2 - phi1))
	t_1 = fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5)
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi2) * cos(phi1)) * t_2) * t_2))
	t_4 = sqrt(t_3)
	tmp = 0.0
	if (atan(t_4, sqrt(Float64(1.0 - t_3))) <= 2e-16)
		tmp = Float64(Float64(atan(t_4, sqrt(Float64(1.0 - (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0)))) * 2.0) * R);
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(Float64(cos(Float64(phi2 + phi1)) + t_0) * t_1), 0.5, Float64(0.5 - Float64(t_0 * 0.5)))), sqrt(fma(t_1, Float64(Float64(-cos(phi2)) * cos(phi1)), fma(cos(Float64(phi1 - phi2)), 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$3], $MachinePrecision]}, If[LessEqual[N[ArcTan[t$95$4 / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2e-16], N[(N[(N[ArcTan[t$95$4 / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * 0.5 + N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 * N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))) < 2e-16

   1. Initial program 93.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left(1 + \phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right) + 1\right)} - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. associate--l+ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right) + \left(1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)\right)}}}\right) \]

      3. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\phi_2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} + \left(1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)\right)}}\right) \]

      4. associate-*r* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left(\phi_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)} + \left(1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)\right)}}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\phi_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), 1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)\right)}}}\right) \]

   5. Applied rewrites 93.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \phi_2, \cos \left(\phi_1 \cdot -0.5\right), {\cos \left(\phi_1 \cdot -0.5\right)}^{2} - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} \cdot \cos \phi_1\right)}}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 93.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \color{blue}{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}}\right) \]

   if 2e-16 < (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))

   1. Initial program 65.2%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 66.2

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 66.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 66.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]

   8. Applied rewrites 65.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \mathsf{fma}\left(-0.5, t\_1, 0.5\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(t\_0 \cdot t\_3\right) \cdot t\_3\\ t_5 := \cos \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}, t\_0, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - t\_1 \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(\cos \left(\phi_2 + \phi_1\right) + t\_5\right) \cdot t\_2, 0.5, 0.5 - t\_5 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_2, \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (fma -0.5 t_1 0.5))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* t_0 t_3) t_3)))
        (t_5 (cos (- phi2 phi1))))
   (if (<= (atan2 (sqrt t_4) (sqrt (- 1.0 t_4))) 2e-16)
     (*
      (* 2.0 R)
      (atan2
       (sqrt
        (fma
         (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0)
         t_0
         (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
       (sqrt (+ (* (- 0.5 (- 0.5 (* t_1 0.5))) (cos phi1)) 0.5))))
     (*
      (*
       (atan2
        (sqrt
         (fma (* (+ (cos (+ phi2 phi1)) t_5) t_2) 0.5 (- 0.5 (* t_5 0.5))))
        (sqrt
         (fma
          t_2
          (* (- (cos phi2)) (cos phi1))
          (fma (cos (- phi1 phi2)) 0.5 0.5))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = fma(-0.5, t_1, 0.5);
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((t_0 * t_3) * t_3);
	double t_5 = cos((phi2 - phi1));
	double tmp;
	if (atan2(sqrt(t_4), sqrt((1.0 - t_4))) <= 2e-16) {
		tmp = (2.0 * R) * atan2(sqrt(fma(pow(sin(((lambda1 - lambda2) * 0.5)), 2.0), t_0, (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt((((0.5 - (0.5 - (t_1 * 0.5))) * cos(phi1)) + 0.5)));
	} else {
		tmp = (atan2(sqrt(fma(((cos((phi2 + phi1)) + t_5) * t_2), 0.5, (0.5 - (t_5 * 0.5)))), sqrt(fma(t_2, (-cos(phi2) * cos(phi1)), fma(cos((phi1 - phi2)), 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = fma(-0.5, t_1, 0.5)
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(t_0 * t_3) * t_3))
	t_5 = cos(Float64(phi2 - phi1))
	tmp = 0.0
	if (atan(sqrt(t_4), sqrt(Float64(1.0 - t_4))) <= 2e-16)
		tmp = Float64(Float64(2.0 * R) * atan(sqrt(fma((sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0), t_0, Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(t_1 * 0.5))) * cos(phi1)) + 0.5))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(Float64(cos(Float64(phi2 + phi1)) + t_5) * t_2), 0.5, Float64(0.5 - Float64(t_5 * 0.5)))), sqrt(fma(t_2, Float64(Float64(-cos(phi2)) * cos(phi1)), fma(cos(Float64(phi1 - phi2)), 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * t$95$1 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2e-16], N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision] * t$95$2), $MachinePrecision] * 0.5 + N[(0.5 - N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$2 * N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))) < 2e-16

   1. Initial program 93.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 10.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 10.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 10.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      9. sqr-sin-a N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      18. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      19. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   9. Applied rewrites 55.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   if 2e-16 < (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))

   1. Initial program 65.2%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 66.2

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 66.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 66.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]

   8. Applied rewrites 65.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(t\_0 \cdot t\_2\right) \cdot t\_2\\ t_4 := \mathsf{fma}\left(-0.5, t\_1, 0.5\right)\\ \mathbf{if}\;\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}, t\_0, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - t\_1 \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_4 \cdot \cos \phi_1, \cos \phi_2, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_4, \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* t_0 t_2) t_2)))
        (t_4 (fma -0.5 t_1 0.5)))
   (if (<= (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) 2e-16)
     (*
      (* 2.0 R)
      (atan2
       (sqrt
        (fma
         (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0)
         t_0
         (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
       (sqrt (+ (* (- 0.5 (- 0.5 (* t_1 0.5))) (cos phi1)) 0.5))))
     (*
      (*
       (atan2
        (sqrt
         (fma
          (* t_4 (cos phi1))
          (cos phi2)
          (- 0.5 (* (cos (- phi2 phi1)) 0.5))))
        (sqrt
         (fma
          t_4
          (* (- (cos phi2)) (cos phi1))
          (fma (cos (- phi1 phi2)) 0.5 0.5))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((t_0 * t_2) * t_2);
	double t_4 = fma(-0.5, t_1, 0.5);
	double tmp;
	if (atan2(sqrt(t_3), sqrt((1.0 - t_3))) <= 2e-16) {
		tmp = (2.0 * R) * atan2(sqrt(fma(pow(sin(((lambda1 - lambda2) * 0.5)), 2.0), t_0, (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt((((0.5 - (0.5 - (t_1 * 0.5))) * cos(phi1)) + 0.5)));
	} else {
		tmp = (atan2(sqrt(fma((t_4 * cos(phi1)), cos(phi2), (0.5 - (cos((phi2 - phi1)) * 0.5)))), sqrt(fma(t_4, (-cos(phi2) * cos(phi1)), fma(cos((phi1 - phi2)), 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(t_0 * t_2) * t_2))
	t_4 = fma(-0.5, t_1, 0.5)
	tmp = 0.0
	if (atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) <= 2e-16)
		tmp = Float64(Float64(2.0 * R) * atan(sqrt(fma((sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0), t_0, Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(t_1 * 0.5))) * cos(phi1)) + 0.5))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(t_4 * cos(phi1)), cos(phi2), Float64(0.5 - Float64(cos(Float64(phi2 - phi1)) * 0.5)))), sqrt(fma(t_4, Float64(Float64(-cos(phi2)) * cos(phi1)), fma(cos(Float64(phi1 - phi2)), 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * t$95$1 + 0.5), $MachinePrecision]}, If[LessEqual[N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2e-16], N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(t$95$4 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$4 * N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))) < 2e-16

   1. Initial program 93.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 10.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 10.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 10.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      9. sqr-sin-a N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      18. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      19. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   9. Applied rewrites 55.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   if 2e-16 < (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))

   1. Initial program 65.2%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 66.2

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 66.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 66.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]

   8. Applied rewrites 65.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right) \cdot \cos \phi_1, \cos \phi_2, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right) \cdot \cos \phi_1, \cos \phi_2, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, t\_1\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + t\_0, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\lambda_2 \leq -0.38:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 8.2 \cdot 10^{-18}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\left(t\_0 \cdot t\_3\right) \cdot t\_3 + t\_1}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_0\right), 0.5, \mathsf{fma}\left(-\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \cos \phi_2, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1
         (pow
          (-
           (* (cos (* phi2 0.5)) (sin (* 0.5 phi1)))
           (* (sin (* phi2 0.5)) (cos (* 0.5 phi1))))
          2.0))
        (t_2
         (*
          (*
           (atan2
            (sqrt
             (fma
              (* (pow (sin (* -0.5 lambda2)) 2.0) (cos phi2))
              (cos phi1)
              t_1))
            (sqrt
             (fma
              (fma -0.5 (cos (- lambda1 lambda2)) 0.5)
              (* (- (cos phi2)) (cos phi1))
              (fma (+ (* (sin phi2) (sin phi1)) t_0) 0.5 0.5))))
           2.0)
          R))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (<= lambda2 -0.38)
     t_2
     (if (<= lambda2 8.2e-18)
       (*
        (*
         (atan2
          (sqrt (+ (* (* t_0 t_3) t_3) t_1))
          (sqrt
           (fma
            (fma (sin phi2) (sin phi1) t_0)
            0.5
            (fma
             (- (cos phi1))
             (* (fma (cos lambda1) -0.5 0.5) (cos phi2))
             0.5))))
         2.0)
        R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = pow(((cos((phi2 * 0.5)) * sin((0.5 * phi1))) - (sin((phi2 * 0.5)) * cos((0.5 * phi1)))), 2.0);
	double t_2 = (atan2(sqrt(fma((pow(sin((-0.5 * lambda2)), 2.0) * cos(phi2)), cos(phi1), t_1)), sqrt(fma(fma(-0.5, cos((lambda1 - lambda2)), 0.5), (-cos(phi2) * cos(phi1)), fma(((sin(phi2) * sin(phi1)) + t_0), 0.5, 0.5)))) * 2.0) * R;
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if (lambda2 <= -0.38) {
		tmp = t_2;
	} else if (lambda2 <= 8.2e-18) {
		tmp = (atan2(sqrt((((t_0 * t_3) * t_3) + t_1)), sqrt(fma(fma(sin(phi2), sin(phi1), t_0), 0.5, fma(-cos(phi1), (fma(cos(lambda1), -0.5, 0.5) * cos(phi2)), 0.5)))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1)))) ^ 2.0
	t_2 = Float64(Float64(atan(sqrt(fma(Float64((sin(Float64(-0.5 * lambda2)) ^ 2.0) * cos(phi2)), cos(phi1), t_1)), sqrt(fma(fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5), Float64(Float64(-cos(phi2)) * cos(phi1)), fma(Float64(Float64(sin(phi2) * sin(phi1)) + t_0), 0.5, 0.5)))) * 2.0) * R)
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if (lambda2 <= -0.38)
		tmp = t_2;
	elseif (lambda2 <= 8.2e-18)
		tmp = Float64(Float64(atan(sqrt(Float64(Float64(Float64(t_0 * t_3) * t_3) + t_1)), sqrt(fma(fma(sin(phi2), sin(phi1), t_0), 0.5, fma(Float64(-cos(phi1)), Float64(fma(cos(lambda1), -0.5, 0.5) * cos(phi2)), 0.5)))) * 2.0) * R);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -0.38], t$95$2, If[LessEqual[lambda2, 8.2e-18], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(t$95$0 * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + N[((-N[Cos[phi1], $MachinePrecision]) * N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -0.38 or 8.1999999999999995e-18 < lambda2

   1. Initial program 53.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 54.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 54.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 54.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \color{blue}{\left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 62.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

   8. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, 0.5, 0.5\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) \cdot \cos \phi_1} + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \color{blue}{\left(\lambda_2 \cdot \frac{-1}{2}\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \color{blue}{\left(\lambda_2 \cdot \frac{-1}{2}\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       9. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_2 \cdot \frac{-1}{2}\right)}^{2} \cdot \color{blue}{\cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       10. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_2 \cdot \frac{-1}{2}\right)}^{2} \cdot \cos \phi_2, \color{blue}{\cos \phi_1}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_2 \cdot \frac{-1}{2}\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   11. Applied rewrites 62.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left({\sin \left(\lambda_2 \cdot -0.5\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

   if -0.38 < lambda2 < 8.1999999999999995e-18

   1. Initial program 80.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 81.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 81.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 81.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \color{blue}{\left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 97.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

   8. Applied rewrites 97.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, 0.5, 0.5\right)\right)}}\right) \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(-1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_1\right)\right)\right) + \frac{1}{2} \cdot \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)\right)}}}\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left(-1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_1\right)\right)\right) + \frac{1}{2} \cdot \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)\right) + \frac{1}{2}}}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right) + -1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_1\right)\right)\right)\right)} + \frac{1}{2}}}\right) \]

       3. associate-+l+ N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right) + \left(-1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_1\right)\right)\right) + \frac{1}{2}\right)}}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right) \cdot \frac{1}{2}} + \left(-1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_1\right)\right)\right) + \frac{1}{2}\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, -1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_1\right)\right)\right) + \frac{1}{2}\right)}}}\right) \]

   11. Applied rewrites 97.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \cos \phi_1\right), 0.5, \mathsf{fma}\left(-\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \cos \phi_2, 0.5\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.38:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 8.2 \cdot 10^{-18}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \cos \phi_1\right), 0.5, \mathsf{fma}\left(-\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \cos \phi_2, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t\_0\right) \cdot t\_0 + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    (*
     (atan2
      (sqrt
       (+
        (* (* (* (cos phi2) (cos phi1)) t_0) t_0)
        (pow
         (-
          (* (cos (* phi2 0.5)) (sin (* 0.5 phi1)))
          (* (sin (* phi2 0.5)) (cos (* 0.5 phi1))))
         2.0)))
      (sqrt
       (fma
        (fma -0.5 (cos (- lambda1 lambda2)) 0.5)
        (* (- (cos phi2)) (cos phi1))
        (fma (fma (cos phi2) (cos phi1) (* (sin phi2) (sin phi1))) 0.5 0.5))))
     2.0)
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return (atan2(sqrt(((((cos(phi2) * cos(phi1)) * t_0) * t_0) + pow(((cos((phi2 * 0.5)) * sin((0.5 * phi1))) - (sin((phi2 * 0.5)) * cos((0.5 * phi1)))), 2.0))), sqrt(fma(fma(-0.5, cos((lambda1 - lambda2)), 0.5), (-cos(phi2) * cos(phi1)), fma(fma(cos(phi2), cos(phi1), (sin(phi2) * sin(phi1))), 0.5, 0.5)))) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(Float64(atan(sqrt(Float64(Float64(Float64(Float64(cos(phi2) * cos(phi1)) * t_0) * t_0) + (Float64(Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1)))) ^ 2.0))), sqrt(fma(fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5), Float64(Float64(-cos(phi2)) * cos(phi1)), fma(fma(cos(phi2), cos(phi1), Float64(sin(phi2) * sin(phi1))), 0.5, 0.5)))) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 66.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. div-inv N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. lower-*.f64 67.6

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 67.6%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Applied rewrites 67.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   2. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \color{blue}{\left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   3. cos-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   4. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   5. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   6. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   7. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   9. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right), \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right), \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_1} \cdot \sin \phi_2\right), \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   12. lower-sin.f64 79.4

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \color{blue}{\sin \phi_2}\right), 0.5, 0.5\right)\right)}}\right) \]

8. Applied rewrites 79.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)}, 0.5, 0.5\right)\right)}}\right) \]

9. Final simplification 79.4%

   \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
10. Add Preprocessing

Alternative 7: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}\\ t_1 := {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot 0.5\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, t\_1\right)}}{t\_0} \cdot 2\right) \cdot R\\ \mathbf{if}\;\lambda_1 \leq -0.03:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_1 \leq 7 \cdot 10^{-24}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, t\_1\right)}}{t\_0} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (sqrt
          (fma
           (fma -0.5 (cos (- lambda1 lambda2)) 0.5)
           (* (- (cos phi2)) (cos phi1))
           (fma
            (+ (* (sin phi2) (sin phi1)) (* (cos phi2) (cos phi1)))
            0.5
            0.5))))
        (t_1
         (pow
          (-
           (* (cos (* phi2 0.5)) (sin (* 0.5 phi1)))
           (* (sin (* phi2 0.5)) (cos (* 0.5 phi1))))
          2.0))
        (t_2
         (*
          (*
           (atan2
            (sqrt
             (fma
              (* (pow (sin (* lambda1 0.5)) 2.0) (cos phi2))
              (cos phi1)
              t_1))
            t_0)
           2.0)
          R)))
   (if (<= lambda1 -0.03)
     t_2
     (if (<= lambda1 7e-24)
       (*
        (*
         (atan2
          (sqrt
           (fma
            (* (pow (sin (* -0.5 lambda2)) 2.0) (cos phi2))
            (cos phi1)
            t_1))
          t_0)
         2.0)
        R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sqrt(fma(fma(-0.5, cos((lambda1 - lambda2)), 0.5), (-cos(phi2) * cos(phi1)), fma(((sin(phi2) * sin(phi1)) + (cos(phi2) * cos(phi1))), 0.5, 0.5)));
	double t_1 = pow(((cos((phi2 * 0.5)) * sin((0.5 * phi1))) - (sin((phi2 * 0.5)) * cos((0.5 * phi1)))), 2.0);
	double t_2 = (atan2(sqrt(fma((pow(sin((lambda1 * 0.5)), 2.0) * cos(phi2)), cos(phi1), t_1)), t_0) * 2.0) * R;
	double tmp;
	if (lambda1 <= -0.03) {
		tmp = t_2;
	} else if (lambda1 <= 7e-24) {
		tmp = (atan2(sqrt(fma((pow(sin((-0.5 * lambda2)), 2.0) * cos(phi2)), cos(phi1), t_1)), t_0) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sqrt(fma(fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5), Float64(Float64(-cos(phi2)) * cos(phi1)), fma(Float64(Float64(sin(phi2) * sin(phi1)) + Float64(cos(phi2) * cos(phi1))), 0.5, 0.5)))
	t_1 = Float64(Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1)))) ^ 2.0
	t_2 = Float64(Float64(atan(sqrt(fma(Float64((sin(Float64(lambda1 * 0.5)) ^ 2.0) * cos(phi2)), cos(phi1), t_1)), t_0) * 2.0) * R)
	tmp = 0.0
	if (lambda1 <= -0.03)
		tmp = t_2;
	elseif (lambda1 <= 7e-24)
		tmp = Float64(Float64(atan(sqrt(fma(Float64((sin(Float64(-0.5 * lambda2)) ^ 2.0) * cos(phi2)), cos(phi1), t_1)), t_0) * 2.0) * R);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sqrt[N[(N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Power[N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$0], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[lambda1, -0.03], t$95$2, If[LessEqual[lambda1, 7e-24], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$0], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -0.029999999999999999 or 6.9999999999999993e-24 < lambda1

   1. Initial program 48.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 49.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \color{blue}{\left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 59.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

   8. Applied rewrites 59.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, 0.5, 0.5\right)\right)}}\right) \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}\right) \cdot \cos \phi_1} + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       9. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}^{2} \cdot \color{blue}{\cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       10. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}^{2} \cdot \cos \phi_2, \color{blue}{\cos \phi_1}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   11. Applied rewrites 59.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot 0.5\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

   if -0.029999999999999999 < lambda1 < 6.9999999999999993e-24

   1. Initial program 82.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 83.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 83.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 83.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \color{blue}{\left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 97.2

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

   8. Applied rewrites 97.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, 0.5, 0.5\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) \cdot \cos \phi_1} + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \color{blue}{\left(\lambda_2 \cdot \frac{-1}{2}\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \color{blue}{\left(\lambda_2 \cdot \frac{-1}{2}\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       9. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_2 \cdot \frac{-1}{2}\right)}^{2} \cdot \color{blue}{\cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       10. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_2 \cdot \frac{-1}{2}\right)}^{2} \cdot \cos \phi_2, \color{blue}{\cos \phi_1}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_2 \cdot \frac{-1}{2}\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   11. Applied rewrites 95.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left({\sin \left(\lambda_2 \cdot -0.5\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.03:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot 0.5\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq 7 \cdot 10^{-24}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot 0.5\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_4 := \sqrt{\left(t\_0 \cdot t\_2\right) \cdot t\_2 + t\_3}\\ t_5 := \cos \left(\phi_2 - \phi_1\right)\\ t_6 := \left(\cos \left(\phi_2 + \phi_1\right) + t\_5\right) \cdot t\_1\\ \mathbf{if}\;\lambda_2 \leq -3.55 \cdot 10^{-31}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{t\_4}{\sqrt{{\left(\frac{1}{\left(t\_5 + 1\right) - t\_6}\right)}^{-1} \cdot 0.5}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot 0.5\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, t\_3\right)}}{\sqrt{\mathsf{fma}\left(t\_1, \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + t\_0, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{t\_4}{\sqrt{\frac{\left(\cos \left(\phi_1 - \phi_2\right) + 1\right) - t\_6}{2}}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (fma -0.5 (cos (- lambda1 lambda2)) 0.5))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (pow
          (-
           (* (cos (* phi2 0.5)) (sin (* 0.5 phi1)))
           (* (sin (* phi2 0.5)) (cos (* 0.5 phi1))))
          2.0))
        (t_4 (sqrt (+ (* (* t_0 t_2) t_2) t_3)))
        (t_5 (cos (- phi2 phi1)))
        (t_6 (* (+ (cos (+ phi2 phi1)) t_5) t_1)))
   (if (<= lambda2 -3.55e-31)
     (*
      (* (atan2 t_4 (sqrt (* (pow (/ 1.0 (- (+ t_5 1.0) t_6)) -1.0) 0.5))) 2.0)
      R)
     (if (<= lambda2 2.8e-40)
       (*
        (*
         (atan2
          (sqrt
           (fma (* (pow (sin (* lambda1 0.5)) 2.0) (cos phi2)) (cos phi1) t_3))
          (sqrt
           (fma
            t_1
            (* (- (cos phi2)) (cos phi1))
            (fma (+ (* (sin phi2) (sin phi1)) t_0) 0.5 0.5))))
         2.0)
        R)
       (*
        (* (atan2 t_4 (sqrt (/ (- (+ (cos (- phi1 phi2)) 1.0) t_6) 2.0))) 2.0)
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = fma(-0.5, cos((lambda1 - lambda2)), 0.5);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(((cos((phi2 * 0.5)) * sin((0.5 * phi1))) - (sin((phi2 * 0.5)) * cos((0.5 * phi1)))), 2.0);
	double t_4 = sqrt((((t_0 * t_2) * t_2) + t_3));
	double t_5 = cos((phi2 - phi1));
	double t_6 = (cos((phi2 + phi1)) + t_5) * t_1;
	double tmp;
	if (lambda2 <= -3.55e-31) {
		tmp = (atan2(t_4, sqrt((pow((1.0 / ((t_5 + 1.0) - t_6)), -1.0) * 0.5))) * 2.0) * R;
	} else if (lambda2 <= 2.8e-40) {
		tmp = (atan2(sqrt(fma((pow(sin((lambda1 * 0.5)), 2.0) * cos(phi2)), cos(phi1), t_3)), sqrt(fma(t_1, (-cos(phi2) * cos(phi1)), fma(((sin(phi2) * sin(phi1)) + t_0), 0.5, 0.5)))) * 2.0) * R;
	} else {
		tmp = (atan2(t_4, sqrt((((cos((phi1 - phi2)) + 1.0) - t_6) / 2.0))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5)
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1)))) ^ 2.0
	t_4 = sqrt(Float64(Float64(Float64(t_0 * t_2) * t_2) + t_3))
	t_5 = cos(Float64(phi2 - phi1))
	t_6 = Float64(Float64(cos(Float64(phi2 + phi1)) + t_5) * t_1)
	tmp = 0.0
	if (lambda2 <= -3.55e-31)
		tmp = Float64(Float64(atan(t_4, sqrt(Float64((Float64(1.0 / Float64(Float64(t_5 + 1.0) - t_6)) ^ -1.0) * 0.5))) * 2.0) * R);
	elseif (lambda2 <= 2.8e-40)
		tmp = Float64(Float64(atan(sqrt(fma(Float64((sin(Float64(lambda1 * 0.5)) ^ 2.0) * cos(phi2)), cos(phi1), t_3)), sqrt(fma(t_1, Float64(Float64(-cos(phi2)) * cos(phi1)), fma(Float64(Float64(sin(phi2) * sin(phi1)) + t_0), 0.5, 0.5)))) * 2.0) * R);
	else
		tmp = Float64(Float64(atan(t_4, sqrt(Float64(Float64(Float64(cos(Float64(phi1 - phi2)) + 1.0) - t_6) / 2.0))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[lambda2, -3.55e-31], N[(N[(N[ArcTan[t$95$4 / N[Sqrt[N[(N[Power[N[(1.0 / N[(N[(t$95$5 + 1.0), $MachinePrecision] - t$95$6), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 2.8e-40], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Power[N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 * N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], N[(N[(N[ArcTan[t$95$4 / N[Sqrt[N[(N[(N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] - t$95$6), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -3.5499999999999999e-31

   1. Initial program 54.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 55.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 55.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 55.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{0.5 \cdot {\left(\frac{1}{\left(\cos \left(\phi_2 - \phi_1\right) + 1\right) - \left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)}\right)}^{-1}}}}\right) \]

   if -3.5499999999999999e-31 < lambda2 < 2.8e-40

   1. Initial program 80.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 81.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 81.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 81.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \color{blue}{\left(\phi_1 - \phi_2\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 98.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

   8. Applied rewrites 98.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1}, 0.5, 0.5\right)\right)}}\right) \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}\right) \cdot \cos \phi_1} + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       9. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}^{2} \cdot \color{blue}{\cos \phi_2}, \cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       10. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}^{2} \cdot \cos \phi_2, \color{blue}{\cos \phi_1}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_1 - \lambda_2\right), \frac{1}{2}\right), \mathsf{neg}\left(\cos \phi_2 \cdot \cos \phi_1\right), \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   11. Applied rewrites 95.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot 0.5\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}}\right) \]

   if 2.8e-40 < lambda2

   1. Initial program 58.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. div-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. div-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 59.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 59.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 59.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\frac{\left(\cos \left(\phi_1 - \phi_2\right) + 1\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right) \cdot \left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right)}{2}}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.55 \cdot 10^{-31}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{{\left(\frac{1}{\left(\cos \left(\phi_2 - \phi_1\right) + 1\right) - \left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)}\right)}^{-1} \cdot 0.5}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\lambda_1 \cdot 0.5\right)}^{2} \cdot \cos \phi_2, \cos \phi_1, {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\frac{\left(\cos \left(\phi_1 - \phi_2\right) + 1\right) - \left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)}{2}}} \cdot 2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t\_0\right) \cdot t\_0 + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\frac{\left(\cos \left(\phi_1 - \phi_2\right) + 1\right) - \left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)}{2}}} \cdot 2\right) \cdot R \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    (*
     (atan2
      (sqrt
       (+
        (* (* (* (cos phi2) (cos phi1)) t_0) t_0)
        (pow
         (-
          (* (cos (* phi2 0.5)) (sin (* 0.5 phi1)))
          (* (sin (* phi2 0.5)) (cos (* 0.5 phi1))))
         2.0)))
      (sqrt
       (/
        (-
         (+ (cos (- phi1 phi2)) 1.0)
         (*
          (+ (cos (+ phi2 phi1)) (cos (- phi2 phi1)))
          (fma -0.5 (cos (- lambda1 lambda2)) 0.5)))
        2.0)))
     2.0)
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return (atan2(sqrt(((((cos(phi2) * cos(phi1)) * t_0) * t_0) + pow(((cos((phi2 * 0.5)) * sin((0.5 * phi1))) - (sin((phi2 * 0.5)) * cos((0.5 * phi1)))), 2.0))), sqrt((((cos((phi1 - phi2)) + 1.0) - ((cos((phi2 + phi1)) + cos((phi2 - phi1))) * fma(-0.5, cos((lambda1 - lambda2)), 0.5))) / 2.0))) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(Float64(atan(sqrt(Float64(Float64(Float64(Float64(cos(phi2) * cos(phi1)) * t_0) * t_0) + (Float64(Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1)))) ^ 2.0))), sqrt(Float64(Float64(Float64(cos(Float64(phi1 - phi2)) + 1.0) - Float64(Float64(cos(Float64(phi2 + phi1)) + cos(Float64(phi2 - phi1))) * fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5))) / 2.0))) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 66.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. div-inv N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. lower-*.f64 67.6

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 67.6%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Applied rewrites 68.1%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\frac{\left(\cos \left(\phi_1 - \phi_2\right) + 1\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right) \cdot \left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right)}{2}}}}\right) \]

7. Final simplification 68.1%

   \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\frac{\left(\cos \left(\phi_1 - \phi_2\right) + 1\right) - \left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)}{2}}} \cdot 2\right) \cdot R \]
8. Add Preprocessing

Alternative 10: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t\_0\right) \cdot t\_0 + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    (*
     (atan2
      (sqrt
       (+
        (* (* (* (cos phi2) (cos phi1)) t_0) t_0)
        (pow
         (-
          (* (cos (* phi2 0.5)) (sin (* 0.5 phi1)))
          (* (sin (* phi2 0.5)) (cos (* 0.5 phi1))))
         2.0)))
      (sqrt
       (fma
        (fma -0.5 (cos (- lambda1 lambda2)) 0.5)
        (* (- (cos phi2)) (cos phi1))
        (fma (cos (- phi1 phi2)) 0.5 0.5))))
     2.0)
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return (atan2(sqrt(((((cos(phi2) * cos(phi1)) * t_0) * t_0) + pow(((cos((phi2 * 0.5)) * sin((0.5 * phi1))) - (sin((phi2 * 0.5)) * cos((0.5 * phi1)))), 2.0))), sqrt(fma(fma(-0.5, cos((lambda1 - lambda2)), 0.5), (-cos(phi2) * cos(phi1)), fma(cos((phi1 - phi2)), 0.5, 0.5)))) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(Float64(atan(sqrt(Float64(Float64(Float64(Float64(cos(phi2) * cos(phi1)) * t_0) * t_0) + (Float64(Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1)))) ^ 2.0))), sqrt(fma(fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5), Float64(Float64(-cos(phi2)) * cos(phi1)), fma(cos(Float64(phi1 - phi2)), 0.5, 0.5)))) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 66.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. div-inv N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. lower-*.f64 67.6

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 67.6%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Applied rewrites 67.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), -\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}}}\right) \]

7. Final simplification 67.8%

   \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_2\right) \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
8. Add Preprocessing

Alternative 11: 62.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_2 - \phi_1\right)\\ \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{0.5} \cdot \sqrt{\left(t\_0 + 1\right) - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right) \cdot \left(\cos \left(\phi_2 + \phi_1\right) + t\_0\right)}} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- phi2 phi1))))
   (*
    (* 2.0 R)
    (atan2
     (sqrt
      (fma
       (* (cos phi2) (cos phi1))
       (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0)
       (pow (sin (* (- phi1 phi2) 0.5)) 2.0)))
     (*
      (sqrt 0.5)
      (sqrt
       (-
        (+ t_0 1.0)
        (*
         (fma (cos (- lambda2 lambda1)) -0.5 0.5)
         (+ (cos (+ phi2 phi1)) t_0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi2 - phi1));
	return (2.0 * R) * atan2(sqrt(fma((cos(phi2) * cos(phi1)), pow(sin(((lambda2 - lambda1) * -0.5)), 2.0), pow(sin(((phi1 - phi2) * 0.5)), 2.0))), (sqrt(0.5) * sqrt(((t_0 + 1.0) - (fma(cos((lambda2 - lambda1)), -0.5, 0.5) * (cos((phi2 + phi1)) + t_0))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi2 - phi1))
	return Float64(Float64(2.0 * R) * atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), (sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0), (sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0))), Float64(sqrt(0.5) * sqrt(Float64(Float64(t_0 + 1.0) - Float64(fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5) * Float64(cos(Float64(phi2 + phi1)) + t_0)))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(N[(t$95$0 + 1.0), $MachinePrecision] - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-*.f64 43.2

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\phi_1 \cdot 0.5\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 43.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\phi_1 \cdot 0.5\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Applied rewrites 43.6%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\frac{\left(\cos \left(\phi_1 - \phi_2\right) + 1\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right) \cdot \left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right)}{2}}}}\right) \]

8. Taylor expanded in lambda1 around -inf

   \[\leadsto \color{blue}{2 \cdot \left(R \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{\frac{1}{2}} \cdot \sqrt{\left(1 + \cos \left(\phi_1 - \phi_2\right)\right) - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right) \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)}}\right)} \]

10. Applied rewrites 67.1%

    \[\leadsto \color{blue}{\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\left(\cos \left(\phi_2 - \phi_1\right) + 1\right) - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right)} \cdot \sqrt{0.5}}} \]

11. Final simplification 67.1%

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{0.5} \cdot \sqrt{\left(\cos \left(\phi_2 - \phi_1\right) + 1\right) - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right) \cdot \left(\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)\right)}} \]
12. Add Preprocessing

Alternative 12: 60.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot 0.5\\ t_1 := 0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_4 := \sqrt{\mathsf{fma}\left(0.5 - \cos \left(t\_0 \cdot 2\right) \cdot 0.5, t\_2, t\_3\right)}\\ \mathbf{if}\;\phi_2 \leq -950000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_4}{\sqrt{t\_1 \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin t\_0}^{2}, t\_2, t\_3\right)}}{\sqrt{t\_1 \cdot \cos \phi_1 + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_4}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5, 0.5\right) - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right) \cdot \cos \phi_2}} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) 0.5))
        (t_1 (- 0.5 (- 0.5 (* (cos (- lambda1 lambda2)) 0.5))))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5)))
        (t_4 (sqrt (fma (- 0.5 (* (cos (* t_0 2.0)) 0.5)) t_2 t_3))))
   (if (<= phi2 -950000.0)
     (* (atan2 t_4 (sqrt (+ (* t_1 (cos phi2)) 0.5))) (* 2.0 R))
     (if (<= phi2 3.8e-6)
       (*
        (* 2.0 R)
        (atan2
         (sqrt (fma (pow (sin t_0) 2.0) t_2 t_3))
         (sqrt (+ (* t_1 (cos phi1)) 0.5))))
       (*
        (atan2
         t_4
         (sqrt
          (-
           (fma (cos phi2) 0.5 0.5)
           (* (fma (cos (- lambda2 lambda1)) -0.5 0.5) (cos phi2)))))
        (* 2.0 R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * 0.5;
	double t_1 = 0.5 - (0.5 - (cos((lambda1 - lambda2)) * 0.5));
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = 0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5);
	double t_4 = sqrt(fma((0.5 - (cos((t_0 * 2.0)) * 0.5)), t_2, t_3));
	double tmp;
	if (phi2 <= -950000.0) {
		tmp = atan2(t_4, sqrt(((t_1 * cos(phi2)) + 0.5))) * (2.0 * R);
	} else if (phi2 <= 3.8e-6) {
		tmp = (2.0 * R) * atan2(sqrt(fma(pow(sin(t_0), 2.0), t_2, t_3)), sqrt(((t_1 * cos(phi1)) + 0.5)));
	} else {
		tmp = atan2(t_4, sqrt((fma(cos(phi2), 0.5, 0.5) - (fma(cos((lambda2 - lambda1)), -0.5, 0.5) * cos(phi2))))) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * 0.5)
	t_1 = Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5)))
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5))
	t_4 = sqrt(fma(Float64(0.5 - Float64(cos(Float64(t_0 * 2.0)) * 0.5)), t_2, t_3))
	tmp = 0.0
	if (phi2 <= -950000.0)
		tmp = Float64(atan(t_4, sqrt(Float64(Float64(t_1 * cos(phi2)) + 0.5))) * Float64(2.0 * R));
	elseif (phi2 <= 3.8e-6)
		tmp = Float64(Float64(2.0 * R) * atan(sqrt(fma((sin(t_0) ^ 2.0), t_2, t_3)), sqrt(Float64(Float64(t_1 * cos(phi1)) + 0.5))));
	else
		tmp = Float64(atan(t_4, sqrt(Float64(fma(cos(phi2), 0.5, 0.5) - Float64(fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5) * cos(phi2))))) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -950000.0], N[(N[ArcTan[t$95$4 / N[Sqrt[N[(N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.8e-6], N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[t$95$4 / N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.5e5

   1. Initial program 55.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\frac{1}{2} \cdot \color{blue}{\cos \phi_2} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_2 \cdot \frac{1}{2}} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      9. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. lower--.f64 56.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 56.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   if -9.5e5 < phi2 < 3.8e-6

   1. Initial program 76.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 67.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 67.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      9. sqr-sin-a N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      18. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      19. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   9. Applied rewrites 72.5%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   if 3.8e-6 < phi2

   1. Initial program 59.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 22.1

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 22.1%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{1}{2}\right)} - \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} \cdot \color{blue}{\cos \phi_2} + \frac{1}{2}\right) - \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\color{blue}{\cos \phi_2 \cdot \frac{1}{2}} + \frac{1}{2}\right) - \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2}, \frac{1}{2}\right)} - \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \frac{1}{2}, \frac{1}{2}\right) - \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2}, \frac{1}{2}\right) - \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_2}}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2}, \frac{1}{2}\right) - \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_2}}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 59.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5, 0.5\right) - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right) \cdot \cos \phi_2}}} \cdot \left(R \cdot 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -950000:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5, 0.5\right) - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right) \cdot \cos \phi_2}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot 0.5\\ t_1 := 0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_4 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(t\_0 \cdot 2\right) \cdot 0.5, t\_2, t\_3\right)}}{\sqrt{t\_1 \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_2 \leq -950000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin t\_0}^{2}, t\_2, t\_3\right)}}{\sqrt{t\_1 \cdot \cos \phi_1 + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) 0.5))
        (t_1 (- 0.5 (- 0.5 (* (cos (- lambda1 lambda2)) 0.5))))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5)))
        (t_4
         (*
          (atan2
           (sqrt (fma (- 0.5 (* (cos (* t_0 2.0)) 0.5)) t_2 t_3))
           (sqrt (+ (* t_1 (cos phi2)) 0.5)))
          (* 2.0 R))))
   (if (<= phi2 -950000.0)
     t_4
     (if (<= phi2 3.8e-6)
       (*
        (* 2.0 R)
        (atan2
         (sqrt (fma (pow (sin t_0) 2.0) t_2 t_3))
         (sqrt (+ (* t_1 (cos phi1)) 0.5))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * 0.5;
	double t_1 = 0.5 - (0.5 - (cos((lambda1 - lambda2)) * 0.5));
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = 0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5);
	double t_4 = atan2(sqrt(fma((0.5 - (cos((t_0 * 2.0)) * 0.5)), t_2, t_3)), sqrt(((t_1 * cos(phi2)) + 0.5))) * (2.0 * R);
	double tmp;
	if (phi2 <= -950000.0) {
		tmp = t_4;
	} else if (phi2 <= 3.8e-6) {
		tmp = (2.0 * R) * atan2(sqrt(fma(pow(sin(t_0), 2.0), t_2, t_3)), sqrt(((t_1 * cos(phi1)) + 0.5)));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * 0.5)
	t_1 = Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5)))
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5))
	t_4 = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(t_0 * 2.0)) * 0.5)), t_2, t_3)), sqrt(Float64(Float64(t_1 * cos(phi2)) + 0.5))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi2 <= -950000.0)
		tmp = t_4;
	elseif (phi2 <= 3.8e-6)
		tmp = Float64(Float64(2.0 * R) * atan(sqrt(fma((sin(t_0) ^ 2.0), t_2, t_3)), sqrt(Float64(Float64(t_1 * cos(phi1)) + 0.5))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -950000.0], t$95$4, If[LessEqual[phi2, 3.8e-6], N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -9.5e5 or 3.8e-6 < phi2

   1. Initial program 57.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\frac{1}{2} \cdot \color{blue}{\cos \phi_2} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_2 \cdot \frac{1}{2}} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      9. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. lower--.f64 58.2

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 58.2%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   if -9.5e5 < phi2 < 3.8e-6

   1. Initial program 76.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 67.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 67.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      9. sqr-sin-a N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      18. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      19. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   9. Applied rewrites 72.5%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -950000:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := 0.5 - \left(0.5 - t\_2 \cdot 0.5\right)\\ t_4 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, -0.5, 0.5\right), t\_0, t\_1\right)}}{\sqrt{t\_3 \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -820000000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_1 \leq 5.4 \cdot 10^{+39}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, t\_0, t\_1\right)}}{\sqrt{t\_3 \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5)))
        (t_2 (cos (- lambda1 lambda2)))
        (t_3 (- 0.5 (- 0.5 (* t_2 0.5))))
        (t_4
         (*
          (atan2
           (sqrt (fma (fma t_2 -0.5 0.5) t_0 t_1))
           (sqrt (+ (* t_3 (cos phi1)) 0.5)))
          (* 2.0 R))))
   (if (<= phi1 -820000000.0)
     t_4
     (if (<= phi1 5.4e+39)
       (*
        (atan2
         (sqrt
          (fma
           (- 0.5 (* (cos (* (* (- lambda1 lambda2) 0.5) 2.0)) 0.5))
           t_0
           t_1))
         (sqrt (+ (* t_3 (cos phi2)) 0.5)))
        (* 2.0 R))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = 0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5);
	double t_2 = cos((lambda1 - lambda2));
	double t_3 = 0.5 - (0.5 - (t_2 * 0.5));
	double t_4 = atan2(sqrt(fma(fma(t_2, -0.5, 0.5), t_0, t_1)), sqrt(((t_3 * cos(phi1)) + 0.5))) * (2.0 * R);
	double tmp;
	if (phi1 <= -820000000.0) {
		tmp = t_4;
	} else if (phi1 <= 5.4e+39) {
		tmp = atan2(sqrt(fma((0.5 - (cos((((lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, t_1)), sqrt(((t_3 * cos(phi2)) + 0.5))) * (2.0 * R);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5))
	t_2 = cos(Float64(lambda1 - lambda2))
	t_3 = Float64(0.5 - Float64(0.5 - Float64(t_2 * 0.5)))
	t_4 = Float64(atan(sqrt(fma(fma(t_2, -0.5, 0.5), t_0, t_1)), sqrt(Float64(Float64(t_3 * cos(phi1)) + 0.5))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -820000000.0)
		tmp = t_4;
	elseif (phi1 <= 5.4e+39)
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(Float64(Float64(lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, t_1)), sqrt(Float64(Float64(t_3 * cos(phi2)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[ArcTan[N[Sqrt[N[(N[(t$95$2 * -0.5 + 0.5), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$3 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -820000000.0], t$95$4, If[LessEqual[phi1, 5.4e+39], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$3 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.2e8 or 5.40000000000000007e39 < phi1

   1. Initial program 54.2%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 55.0

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 55.0%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{1}{2}}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      9. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \color{blue}{\left(\left(2 \cdot \frac{1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(\color{blue}{1} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. lower-fma.f64 55.0

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   9. Applied rewrites 55.0%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   if -8.2e8 < phi1 < 5.40000000000000007e39

   1. Initial program 76.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\frac{1}{2} \cdot \color{blue}{\cos \phi_2} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_2 \cdot \frac{1}{2}} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      9. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. lower--.f64 68.3

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 68.3%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -820000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 5.4 \cdot 10^{+39}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_2 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), t\_0, t\_1\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -950000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 0.00165:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, -0.5, 0.5\right), t\_0, t\_1\right)}}{\sqrt{\left(0.5 - \left(0.5 - t\_3 \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5)))
        (t_2
         (*
          (atan2
           (sqrt (fma (fma (cos lambda1) -0.5 0.5) t_0 t_1))
           (sqrt
            (fma
             (cos phi2)
             (- 0.5 (fma (cos (- lambda2 lambda1)) -0.5 0.5))
             0.5)))
          (* 2.0 R)))
        (t_3 (cos (- lambda1 lambda2))))
   (if (<= phi2 -950000.0)
     t_2
     (if (<= phi2 0.00165)
       (*
        (atan2
         (sqrt (fma (fma t_3 -0.5 0.5) t_0 t_1))
         (sqrt (+ (* (- 0.5 (- 0.5 (* t_3 0.5))) (cos phi1)) 0.5)))
        (* 2.0 R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = 0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5);
	double t_2 = atan2(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), t_0, t_1)), sqrt(fma(cos(phi2), (0.5 - fma(cos((lambda2 - lambda1)), -0.5, 0.5)), 0.5))) * (2.0 * R);
	double t_3 = cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -950000.0) {
		tmp = t_2;
	} else if (phi2 <= 0.00165) {
		tmp = atan2(sqrt(fma(fma(t_3, -0.5, 0.5), t_0, t_1)), sqrt((((0.5 - (0.5 - (t_3 * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5))
	t_2 = Float64(atan(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), t_0, t_1)), sqrt(fma(cos(phi2), Float64(0.5 - fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5)), 0.5))) * Float64(2.0 * R))
	t_3 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= -950000.0)
		tmp = t_2;
	elseif (phi2 <= 0.00165)
		tmp = Float64(atan(sqrt(fma(fma(t_3, -0.5, 0.5), t_0, t_1)), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(t_3 * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -950000.0], t$95$2, If[LessEqual[phi2, 0.00165], N[(N[ArcTan[N[Sqrt[N[(N[(t$95$3 * -0.5 + 0.5), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -9.5e5 or 0.00165 < phi2

   1. Initial program 57.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 20.8

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 20.8%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 20.8

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 20.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       2. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \phi_2}\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

       5. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \color{blue}{-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       6. associate-+r+ N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_2 + -1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       7. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\left(-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2} \cdot \cos \phi_2\right)}}} \cdot \left(R \cdot 2\right) \]

       8. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2} \cdot \cos \phi_2\right) + \frac{1}{2}}}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 43.0%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}}} \cdot \left(R \cdot 2\right) \]

   if -9.5e5 < phi2 < 0.00165

   1. Initial program 76.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 67.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 67.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{1}{2}}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      9. associate-*r* N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \color{blue}{\left(\left(2 \cdot \frac{1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(\color{blue}{1} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. lower-fma.f64 67.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   9. Applied rewrites 67.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -950000:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00165:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_2 := \sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}\\ t_3 := \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), t\_0, t\_1\right)}\\ \mathbf{if}\;\phi_1 \leq -4.35 \cdot 10^{-35}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, -0.5, 0.5\right), t\_0, t\_1\right)}}{t\_2} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{t\_2} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5)))
        (t_2
         (sqrt
          (+
           (* (- 0.5 (- 0.5 (* (cos (- lambda1 lambda2)) 0.5))) (cos phi1))
           0.5)))
        (t_3 (sqrt (fma (fma (cos lambda1) -0.5 0.5) t_0 t_1))))
   (if (<= phi1 -4.35e-35)
     (*
      (atan2 (sqrt (fma (fma (cos lambda2) -0.5 0.5) t_0 t_1)) t_2)
      (* 2.0 R))
     (if (<= phi1 4.8e-8)
       (*
        (atan2
         t_3
         (sqrt
          (fma
           (cos phi2)
           (- 0.5 (fma (cos (- lambda2 lambda1)) -0.5 0.5))
           0.5)))
        (* 2.0 R))
       (* (atan2 t_3 t_2) (* 2.0 R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = 0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5);
	double t_2 = sqrt((((0.5 - (0.5 - (cos((lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5));
	double t_3 = sqrt(fma(fma(cos(lambda1), -0.5, 0.5), t_0, t_1));
	double tmp;
	if (phi1 <= -4.35e-35) {
		tmp = atan2(sqrt(fma(fma(cos(lambda2), -0.5, 0.5), t_0, t_1)), t_2) * (2.0 * R);
	} else if (phi1 <= 4.8e-8) {
		tmp = atan2(t_3, sqrt(fma(cos(phi2), (0.5 - fma(cos((lambda2 - lambda1)), -0.5, 0.5)), 0.5))) * (2.0 * R);
	} else {
		tmp = atan2(t_3, t_2) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5))
	t_2 = sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5))
	t_3 = sqrt(fma(fma(cos(lambda1), -0.5, 0.5), t_0, t_1))
	tmp = 0.0
	if (phi1 <= -4.35e-35)
		tmp = Float64(atan(sqrt(fma(fma(cos(lambda2), -0.5, 0.5), t_0, t_1)), t_2) * Float64(2.0 * R));
	elseif (phi1 <= 4.8e-8)
		tmp = Float64(atan(t_3, sqrt(fma(cos(phi2), Float64(0.5 - fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5)), 0.5))) * Float64(2.0 * R));
	else
		tmp = Float64(atan(t_3, t_2) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -4.35e-35], N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[lambda2], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4.8e-8], N[(N[ArcTan[t$95$3 / N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[t$95$3 / t$95$2], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.34999999999999976e-35

   1. Initial program 55.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 52.6

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 52.6%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower-cos.f64 45.9

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 45.9%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   if -4.34999999999999976e-35 < phi1 < 4.79999999999999997e-8

   1. Initial program 79.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 35.3

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 35.3%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 27.1

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 27.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       2. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \phi_2}\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

       5. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \color{blue}{-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       6. associate-+r+ N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_2 + -1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       7. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\left(-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2} \cdot \cos \phi_2\right)}}} \cdot \left(R \cdot 2\right) \]

       8. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2} \cdot \cos \phi_2\right) + \frac{1}{2}}}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 48.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}}} \cdot \left(R \cdot 2\right) \]

   if 4.79999999999999997e-8 < phi1

   1. Initial program 50.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 50.8

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 50.8%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 42.1

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 42.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.35 \cdot 10^{-35}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}\\ t_2 := \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)\\ t_3 := \sqrt{\mathsf{fma}\left(t\_2, t\_0, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}\\ \mathbf{if}\;\phi_1 \leq -850000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, t\_0, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{t\_1} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{t\_1} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1
         (sqrt
          (+
           (* (- 0.5 (- 0.5 (* (cos (- lambda1 lambda2)) 0.5))) (cos phi1))
           0.5)))
        (t_2 (fma (cos lambda1) -0.5 0.5))
        (t_3
         (sqrt
          (fma t_2 t_0 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))))
   (if (<= phi1 -850000000.0)
     (* (atan2 (sqrt (fma t_2 t_0 (- 0.5 (* (cos phi1) 0.5)))) t_1) (* 2.0 R))
     (if (<= phi1 4.8e-8)
       (*
        (atan2
         t_3
         (sqrt
          (fma
           (cos phi2)
           (- 0.5 (fma (cos (- lambda2 lambda1)) -0.5 0.5))
           0.5)))
        (* 2.0 R))
       (* (atan2 t_3 t_1) (* 2.0 R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = sqrt((((0.5 - (0.5 - (cos((lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5));
	double t_2 = fma(cos(lambda1), -0.5, 0.5);
	double t_3 = sqrt(fma(t_2, t_0, (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5))));
	double tmp;
	if (phi1 <= -850000000.0) {
		tmp = atan2(sqrt(fma(t_2, t_0, (0.5 - (cos(phi1) * 0.5)))), t_1) * (2.0 * R);
	} else if (phi1 <= 4.8e-8) {
		tmp = atan2(t_3, sqrt(fma(cos(phi2), (0.5 - fma(cos((lambda2 - lambda1)), -0.5, 0.5)), 0.5))) * (2.0 * R);
	} else {
		tmp = atan2(t_3, t_1) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5))
	t_2 = fma(cos(lambda1), -0.5, 0.5)
	t_3 = sqrt(fma(t_2, t_0, Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5))))
	tmp = 0.0
	if (phi1 <= -850000000.0)
		tmp = Float64(atan(sqrt(fma(t_2, t_0, Float64(0.5 - Float64(cos(phi1) * 0.5)))), t_1) * Float64(2.0 * R));
	elseif (phi1 <= 4.8e-8)
		tmp = Float64(atan(t_3, sqrt(fma(cos(phi2), Float64(0.5 - fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5)), 0.5))) * Float64(2.0 * R));
	else
		tmp = Float64(atan(t_3, t_1) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * t$95$0 + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -850000000.0], N[(N[ArcTan[N[Sqrt[N[(t$95$2 * t$95$0 + N[(0.5 - N[(N[Cos[phi1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4.8e-8], N[(N[ArcTan[t$95$3 / N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[t$95$3 / t$95$1], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -8.5e8

   1. Initial program 53.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 54.9

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 54.9%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 42.5

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 42.5%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       2. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_1 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_1 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       4. lower-cos.f64 42.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \color{blue}{\cos \phi_1} \cdot 0.5\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 42.7%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{0.5 - \cos \phi_1 \cdot 0.5}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   if -8.5e8 < phi1 < 4.79999999999999997e-8

   1. Initial program 78.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 35.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 35.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 26.3

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 26.3%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       2. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \phi_2}\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

       5. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \color{blue}{-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       6. associate-+r+ N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_2 + -1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       7. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\left(-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2} \cdot \cos \phi_2\right)}}} \cdot \left(R \cdot 2\right) \]

       8. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2} \cdot \cos \phi_2\right) + \frac{1}{2}}}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 47.3%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}}} \cdot \left(R \cdot 2\right) \]

   if 4.79999999999999997e-8 < phi1

   1. Initial program 50.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 50.8

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 50.8%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 42.1

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 42.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -850000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)\\ t_2 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_2 \leq -950000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (fma (cos lambda1) -0.5 0.5))
        (t_2
         (*
          (atan2
           (sqrt
            (fma t_1 t_0 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
           (sqrt
            (fma
             (cos phi2)
             (- 0.5 (fma (cos (- lambda2 lambda1)) -0.5 0.5))
             0.5)))
          (* 2.0 R))))
   (if (<= phi2 -950000.0)
     t_2
     (if (<= phi2 8e-7)
       (*
        (atan2
         (sqrt (fma t_1 t_0 (- 0.5 (* (cos phi1) 0.5))))
         (sqrt
          (+
           (* (- 0.5 (- 0.5 (* (cos (- lambda1 lambda2)) 0.5))) (cos phi1))
           0.5)))
        (* 2.0 R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = fma(cos(lambda1), -0.5, 0.5);
	double t_2 = atan2(sqrt(fma(t_1, t_0, (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(cos(phi2), (0.5 - fma(cos((lambda2 - lambda1)), -0.5, 0.5)), 0.5))) * (2.0 * R);
	double tmp;
	if (phi2 <= -950000.0) {
		tmp = t_2;
	} else if (phi2 <= 8e-7) {
		tmp = atan2(sqrt(fma(t_1, t_0, (0.5 - (cos(phi1) * 0.5)))), sqrt((((0.5 - (0.5 - (cos((lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = fma(cos(lambda1), -0.5, 0.5)
	t_2 = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(cos(phi2), Float64(0.5 - fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5)), 0.5))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi2 <= -950000.0)
		tmp = t_2;
	elseif (phi2 <= 8e-7)
		tmp = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - Float64(cos(phi1) * 0.5)))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -950000.0], t$95$2, If[LessEqual[phi2, 8e-7], N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - N[(N[Cos[phi1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -9.5e5 or 7.9999999999999996e-7 < phi2

   1. Initial program 57.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 20.8

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 20.8%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 20.8

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 20.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       2. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \phi_2}\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \left(\mathsf{neg}\left(\cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

       5. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_2\right) + \color{blue}{-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       6. associate-+r+ N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_2 + -1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       7. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\left(-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2} \cdot \cos \phi_2\right)}}} \cdot \left(R \cdot 2\right) \]

       8. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(-1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2} \cdot \cos \phi_2\right) + \frac{1}{2}}}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 43.0%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}}} \cdot \left(R \cdot 2\right) \]

   if -9.5e5 < phi2 < 7.9999999999999996e-7

   1. Initial program 76.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 67.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 67.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 47.2

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 47.2%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       2. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_1 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_1 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       4. lower-cos.f64 47.2

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \color{blue}{\cos \phi_1} \cdot 0.5\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 47.2%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{0.5 - \cos \phi_1 \cdot 0.5}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -950000:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 33.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)\\ t_2 := \sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}\\ \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 0.0009:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{t\_2} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - \cos \phi_2 \cdot 0.5\right)}}{t\_2} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (fma (cos lambda1) -0.5 0.5))
        (t_2
         (sqrt
          (+
           (* (- 0.5 (- 0.5 (* (cos (- lambda1 lambda2)) 0.5))) (cos phi1))
           0.5))))
   (if (<= phi2 -1.45e-5)
     (*
      (atan2
       (sqrt (fma t_1 t_0 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
       (sqrt (fma (cos (- lambda2 lambda1)) 0.5 0.5)))
      (* 2.0 R))
     (if (<= phi2 0.0009)
       (*
        (atan2 (sqrt (fma t_1 t_0 (- 0.5 (* (cos phi1) 0.5)))) t_2)
        (* 2.0 R))
       (*
        (atan2 (sqrt (fma t_1 t_0 (- 0.5 (* (cos phi2) 0.5)))) t_2)
        (* 2.0 R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = fma(cos(lambda1), -0.5, 0.5);
	double t_2 = sqrt((((0.5 - (0.5 - (cos((lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5));
	double tmp;
	if (phi2 <= -1.45e-5) {
		tmp = atan2(sqrt(fma(t_1, t_0, (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(cos((lambda2 - lambda1)), 0.5, 0.5))) * (2.0 * R);
	} else if (phi2 <= 0.0009) {
		tmp = atan2(sqrt(fma(t_1, t_0, (0.5 - (cos(phi1) * 0.5)))), t_2) * (2.0 * R);
	} else {
		tmp = atan2(sqrt(fma(t_1, t_0, (0.5 - (cos(phi2) * 0.5)))), t_2) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = fma(cos(lambda1), -0.5, 0.5)
	t_2 = sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5))
	tmp = 0.0
	if (phi2 <= -1.45e-5)
		tmp = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(cos(Float64(lambda2 - lambda1)), 0.5, 0.5))) * Float64(2.0 * R));
	elseif (phi2 <= 0.0009)
		tmp = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - Float64(cos(phi1) * 0.5)))), t_2) * Float64(2.0 * R));
	else
		tmp = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - Float64(cos(phi2) * 0.5)))), t_2) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -1.45e-5], N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.0009], N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - N[(N[Cos[phi1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.45e-5

   1. Initial program 54.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 19.3

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 19.3%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 19.9

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 20.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{0.5}, 0.5\right)}} \cdot \left(R \cdot 2\right) \]

   if -1.45e-5 < phi2 < 8.9999999999999998e-4

   1. Initial program 77.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 67.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 67.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 47.4

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 47.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       2. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_1 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_1 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       4. lower-cos.f64 47.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \color{blue}{\cos \phi_1} \cdot 0.5\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 47.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{0.5 - \cos \phi_1 \cdot 0.5}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   if 8.9999999999999998e-4 < phi2

   1. Initial program 59.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 22.1

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 22.1%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 21.7

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 21.7%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \phi_2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       2. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       3. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_2 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_2 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       5. lower-cos.f64 21.9

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \color{blue}{\cos \phi_2} \cdot 0.5\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{0.5 - \cos \phi_2 \cdot 0.5}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 0.0009:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_2 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 33.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)\\ t_2 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (fma (cos lambda1) -0.5 0.5))
        (t_2
         (*
          (atan2
           (sqrt
            (fma t_1 t_0 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
           (sqrt (fma (cos (- lambda2 lambda1)) 0.5 0.5)))
          (* 2.0 R))))
   (if (<= phi2 -1.45e-5)
     t_2
     (if (<= phi2 1.1e-6)
       (*
        (atan2
         (sqrt (fma t_1 t_0 (- 0.5 (* (cos phi1) 0.5))))
         (sqrt
          (+
           (* (- 0.5 (- 0.5 (* (cos (- lambda1 lambda2)) 0.5))) (cos phi1))
           0.5)))
        (* 2.0 R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = fma(cos(lambda1), -0.5, 0.5);
	double t_2 = atan2(sqrt(fma(t_1, t_0, (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(cos((lambda2 - lambda1)), 0.5, 0.5))) * (2.0 * R);
	double tmp;
	if (phi2 <= -1.45e-5) {
		tmp = t_2;
	} else if (phi2 <= 1.1e-6) {
		tmp = atan2(sqrt(fma(t_1, t_0, (0.5 - (cos(phi1) * 0.5)))), sqrt((((0.5 - (0.5 - (cos((lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = fma(cos(lambda1), -0.5, 0.5)
	t_2 = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(cos(Float64(lambda2 - lambda1)), 0.5, 0.5))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi2 <= -1.45e-5)
		tmp = t_2;
	elseif (phi2 <= 1.1e-6)
		tmp = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - Float64(cos(phi1) * 0.5)))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.45e-5], t$95$2, If[LessEqual[phi2, 1.1e-6], N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - N[(N[Cos[phi1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.45e-5 or 1.1000000000000001e-6 < phi2

   1. Initial program 57.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 20.8

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 20.8%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 20.8

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 20.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 20.9%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{0.5}, 0.5\right)}} \cdot \left(R \cdot 2\right) \]

   if -1.45e-5 < phi2 < 1.1000000000000001e-6

   1. Initial program 77.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 67.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 67.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 47.4

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 47.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       2. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_1 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \color{blue}{\cos \phi_1 \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       4. lower-cos.f64 47.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \color{blue}{\cos \phi_1} \cdot 0.5\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 47.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \color{blue}{0.5 - \cos \phi_1 \cdot 0.5}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 33.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(\cos \lambda_1 \cdot 0.5\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (atan2
   (sqrt
    (fma
     (fma (cos lambda1) -0.5 0.5)
     (* (cos phi2) (cos phi1))
     (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
   (sqrt (+ (* (* (cos lambda1) 0.5) (cos phi1)) 0.5)))
  (* 2.0 R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), (cos(phi2) * cos(phi1)), (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt((((cos(lambda1) * 0.5) * cos(phi1)) + 0.5))) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(atan(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(Float64(Float64(Float64(cos(lambda1) * 0.5) * cos(phi1)) + 0.5))) * Float64(2.0 * R))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

4. Applied rewrites 62.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   4. distribute-lft-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

   9. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   12. lower--.f64 43.3

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

7. Applied rewrites 43.3%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
9. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   2. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   3. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   6. lower-cos.f64 33.6

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

10. Applied rewrites 33.6%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

11. Taylor expanded in lambda2 around 0

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} \cdot \color{blue}{\cos \lambda_1}\right)}} \cdot \left(R \cdot 2\right) \]
12. Step-by-step derivation

13. Applied rewrites 33.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \color{blue}{0.5}\right)}} \cdot \left(R \cdot 2\right) \]

14. Final simplification 33.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(\cos \lambda_1 \cdot 0.5\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right) \]
15. Add Preprocessing

Alternative 22: 33.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \lambda_1 \cdot 0.5, \cos \phi_1, 0.5\right)}} \cdot \left(2 \cdot R\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (atan2
   (sqrt
    (fma
     (fma (cos lambda1) -0.5 0.5)
     (* (cos phi2) (cos phi1))
     (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
   (sqrt (fma (* (cos lambda1) 0.5) (cos phi1) 0.5)))
  (* 2.0 R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), (cos(phi2) * cos(phi1)), (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma((cos(lambda1) * 0.5), cos(phi1), 0.5))) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(atan(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(Float64(cos(lambda1) * 0.5), cos(phi1), 0.5))) * Float64(2.0 * R))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

4. Applied rewrites 62.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   4. distribute-lft-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

   9. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   12. lower--.f64 43.3

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

7. Applied rewrites 43.3%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
9. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   2. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   3. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   6. lower-cos.f64 33.6

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

10. Applied rewrites 33.6%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

11. Taylor expanded in lambda2 around 0

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)}}} \cdot \left(R \cdot 2\right) \]
12. Step-by-step derivation

13. Applied rewrites 33.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \lambda_1 \cdot 0.5, \color{blue}{\cos \phi_1}, 0.5\right)}} \cdot \left(R \cdot 2\right) \]

14. Final simplification 33.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \lambda_1 \cdot 0.5, \cos \phi_1, 0.5\right)}} \cdot \left(2 \cdot R\right) \]
15. Add Preprocessing

Alternative 23: 29.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_2 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), t\_0, t\_1\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\lambda_1 \leq -7.8 \cdot 10^{-16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_1 \leq 1.15:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.25 \cdot \left(\lambda_1 \cdot \lambda_1\right), t\_0, t\_1\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5)))
        (t_2
         (*
          (atan2
           (sqrt (fma (fma (cos lambda1) -0.5 0.5) t_0 t_1))
           (sqrt (fma (cos (- lambda2 lambda1)) 0.5 0.5)))
          (* 2.0 R))))
   (if (<= lambda1 -7.8e-16)
     t_2
     (if (<= lambda1 1.15)
       (*
        (atan2
         (sqrt (fma (* 0.25 (* lambda1 lambda1)) t_0 t_1))
         (sqrt
          (+
           (* (- 0.5 (- 0.5 (* (cos (- lambda1 lambda2)) 0.5))) (cos phi1))
           0.5)))
        (* 2.0 R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = 0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5);
	double t_2 = atan2(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), t_0, t_1)), sqrt(fma(cos((lambda2 - lambda1)), 0.5, 0.5))) * (2.0 * R);
	double tmp;
	if (lambda1 <= -7.8e-16) {
		tmp = t_2;
	} else if (lambda1 <= 1.15) {
		tmp = atan2(sqrt(fma((0.25 * (lambda1 * lambda1)), t_0, t_1)), sqrt((((0.5 - (0.5 - (cos((lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5))
	t_2 = Float64(atan(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), t_0, t_1)), sqrt(fma(cos(Float64(lambda2 - lambda1)), 0.5, 0.5))) * Float64(2.0 * R))
	tmp = 0.0
	if (lambda1 <= -7.8e-16)
		tmp = t_2;
	elseif (lambda1 <= 1.15)
		tmp = Float64(atan(sqrt(fma(Float64(0.25 * Float64(lambda1 * lambda1)), t_0, t_1)), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -7.8e-16], t$95$2, If[LessEqual[lambda1, 1.15], N[(N[ArcTan[N[Sqrt[N[(N[(0.25 * N[(lambda1 * lambda1), $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -7.79999999999999954e-16 or 1.1499999999999999 < lambda1

   1. Initial program 48.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 38.9

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 38.9%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 38.5

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 38.5%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 30.0%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{0.5}, 0.5\right)}} \cdot \left(R \cdot 2\right) \]

   if -7.79999999999999954e-16 < lambda1 < 1.1499999999999999

   1. Initial program 82.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. lower--.f64 47.3

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 47.3%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      3. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 29.2

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 29.2%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in lambda1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{4} \cdot \color{blue}{{\lambda_1}^{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 31.7%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(\lambda_1 \cdot \lambda_1\right) \cdot \color{blue}{0.25}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -7.8 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.15:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.25 \cdot \left(\lambda_1 \cdot \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 24.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (atan2
   (sqrt
    (fma
     (fma (cos lambda1) -0.5 0.5)
     (* (cos phi2) (cos phi1))
     (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
   (sqrt (fma (cos (- lambda2 lambda1)) 0.5 0.5)))
  (* 2.0 R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), (cos(phi2) * cos(phi1)), (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(cos((lambda2 - lambda1)), 0.5, 0.5))) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(atan(sqrt(fma(fma(cos(lambda1), -0.5, 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(cos(Float64(lambda2 - lambda1)), 0.5, 0.5))) * Float64(2.0 * R))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

4. Applied rewrites 62.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   4. distribute-lft-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

   9. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   12. lower--.f64 43.3

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

7. Applied rewrites 43.3%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
9. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   2. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   3. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \cos \lambda_1 + \frac{1}{2}}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \lambda_1 \cdot \frac{-1}{2}} + \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right)}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   6. lower-cos.f64 33.6

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1}, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

10. Applied rewrites 33.6%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right)}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}} \cdot \left(R \cdot 2\right) \]
12. Step-by-step derivation

13. Applied rewrites 23.9%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{0.5}, 0.5\right)}} \cdot \left(R \cdot 2\right) \]

14. Final simplification 23.9%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, 0.5\right)}} \cdot \left(2 \cdot R\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))