Distance on a great circle

Percentage Accurate: 62.7% → 64.5%

Time: 31.6s

Alternatives: 25

Speedup: 1.1×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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: 62.7% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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: 64.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

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

   3. lift-/.f64 N/A

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

   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. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 63.6

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

3. Applied rewrites 63.6%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lift--.f64 64.1

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

5. Applied rewrites 64.1%

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

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 64.5

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

7. Applied rewrites 64.5%

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

Alternative 2: 62.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\lambda_1 - \lambda_2}{2}\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\\ t_2 := \sin t\_0\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := t\_3 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}} \leq 0.1:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + \left(\cos \phi_2 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot t\_2}}{\sqrt{1 - \left(t\_3 + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}, t\_1, t\_3\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, t\_1, t\_3\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (/ (- lambda1 lambda2) 2.0))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 t_0)))))
        (t_2 (sin t_0))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4 (+ t_3 (* (* (* (cos phi1) (cos phi2)) t_2) t_2))))
   (if (<= (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))) 0.1)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_3 (* (* (cos phi2) (sin (* 0.5 (- lambda1 lambda2)))) t_2)))
        (sqrt
         (-
          1.0
          (+
           t_3
           (*
            (*
             (*
              (fma
               (-
                (*
                 (fma -0.001388888888888889 (* phi1 phi1) 0.041666666666666664)
                 (* phi1 phi1))
                0.5)
               (* phi1 phi1)
               1.0)
              (cos phi2))
             t_2)
            t_2)))))))
     (*
      (*
       (atan2
        (sqrt
         (fma (/ (+ (cos (+ phi2 phi1)) (cos (- phi2 phi1))) 2.0) t_1 t_3))
        (sqrt (- 1.0 (fma (* (cos phi2) (cos phi1)) t_1 t_3))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) / 2.0;
	double t_1 = 0.5 - (0.5 * cos((2.0 * t_0)));
	double t_2 = sin(t_0);
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = t_3 + (((cos(phi1) * cos(phi2)) * t_2) * t_2);
	double tmp;
	if ((2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4)))) <= 0.1) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + ((cos(phi2) * sin((0.5 * (lambda1 - lambda2)))) * t_2))), sqrt((1.0 - (t_3 + (((fma(((fma(-0.001388888888888889, (phi1 * phi1), 0.041666666666666664) * (phi1 * phi1)) - 0.5), (phi1 * phi1), 1.0) * cos(phi2)) * t_2) * t_2))))));
	} else {
		tmp = (atan2(sqrt(fma(((cos((phi2 + phi1)) + cos((phi2 - phi1))) / 2.0), t_1, t_3)), sqrt((1.0 - fma((cos(phi2) * cos(phi1)), t_1, t_3)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0))))
	t_2 = sin(t_0)
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = Float64(t_3 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))) <= 0.1)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + Float64(Float64(cos(phi2) * sin(Float64(0.5 * Float64(lambda1 - lambda2)))) * t_2))), sqrt(Float64(1.0 - Float64(t_3 + Float64(Float64(Float64(fma(Float64(Float64(fma(-0.001388888888888889, Float64(phi1 * phi1), 0.041666666666666664) * Float64(phi1 * phi1)) - 0.5), Float64(phi1 * phi1), 1.0) * cos(phi2)) * t_2) * t_2)))))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(Float64(cos(Float64(phi2 + phi1)) + cos(Float64(phi2 - phi1))) / 2.0), t_1, t_3)), sqrt(Float64(1.0 - fma(Float64(cos(phi2) * cos(phi1)), t_1, t_3)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.1], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$3 + N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(phi1 * phi1), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (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))))))))) < 0.10000000000000001

   1. Initial program 91.4%

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

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\left(1 + {\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\left({\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\left(\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right) \cdot {\phi_1}^{2} + 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, \color{blue}{{\phi_1}^{2}}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, {\color{blue}{\phi_1}}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {\phi_1}^{2} + \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {\phi_1}^{2}, \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \color{blue}{\phi_1}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-*.f64 88.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \color{blue}{\phi_1}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 88.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right)} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift--.f64 88.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 88.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   if 0.10000000000000001 < (*.f64 #s(literal 2 binary64) (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 60.4%

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

   3. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-mult N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 60.9

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

   5. Applied rewrites 60.9%

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

Alternative 3: 63.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\lambda_1 - \lambda_2}{2}\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\\ t_2 := \sin t\_0\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := t\_3 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + \left(\left(\cos \phi_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right) \cdot t\_2\right) \cdot t\_2}}{\sqrt{1 - \left(t\_3 + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot 1\right) \cdot t\_2\right) \cdot t\_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}, t\_1, t\_3\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, t\_1, t\_3\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (/ (- lambda1 lambda2) 2.0))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 t_0)))))
        (t_2 (sin t_0))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4 (+ t_3 (* (* (* (cos phi1) (cos phi2)) t_2) t_2))))
   (if (<= (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))) 4e-5)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+ t_3 (* (* (* (cos phi1) (+ 1.0 (* -0.5 (* phi2 phi2)))) t_2) t_2)))
        (sqrt
         (-
          1.0
          (+
           t_3
           (*
            (*
             (*
              (fma
               (-
                (*
                 (fma -0.001388888888888889 (* phi1 phi1) 0.041666666666666664)
                 (* phi1 phi1))
                0.5)
               (* phi1 phi1)
               1.0)
              1.0)
             t_2)
            t_2)))))))
     (*
      (*
       (atan2
        (sqrt
         (fma (/ (+ (cos (+ phi2 phi1)) (cos (- phi2 phi1))) 2.0) t_1 t_3))
        (sqrt (- 1.0 (fma (* (cos phi2) (cos phi1)) t_1 t_3))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) / 2.0;
	double t_1 = 0.5 - (0.5 * cos((2.0 * t_0)));
	double t_2 = sin(t_0);
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = t_3 + (((cos(phi1) * cos(phi2)) * t_2) * t_2);
	double tmp;
	if ((2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4)))) <= 4e-5) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + (((cos(phi1) * (1.0 + (-0.5 * (phi2 * phi2)))) * t_2) * t_2))), sqrt((1.0 - (t_3 + (((fma(((fma(-0.001388888888888889, (phi1 * phi1), 0.041666666666666664) * (phi1 * phi1)) - 0.5), (phi1 * phi1), 1.0) * 1.0) * t_2) * t_2))))));
	} else {
		tmp = (atan2(sqrt(fma(((cos((phi2 + phi1)) + cos((phi2 - phi1))) / 2.0), t_1, t_3)), sqrt((1.0 - fma((cos(phi2) * cos(phi1)), t_1, t_3)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0))))
	t_2 = sin(t_0)
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = Float64(t_3 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))) <= 4e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + Float64(Float64(Float64(cos(phi1) * Float64(1.0 + Float64(-0.5 * Float64(phi2 * phi2)))) * t_2) * t_2))), sqrt(Float64(1.0 - Float64(t_3 + Float64(Float64(Float64(fma(Float64(Float64(fma(-0.001388888888888889, Float64(phi1 * phi1), 0.041666666666666664) * Float64(phi1 * phi1)) - 0.5), Float64(phi1 * phi1), 1.0) * 1.0) * t_2) * t_2)))))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(Float64(cos(Float64(phi2 + phi1)) + cos(Float64(phi2 - phi1))) / 2.0), t_1, t_3)), sqrt(Float64(1.0 - fma(Float64(cos(phi2) * cos(phi1)), t_1, t_3)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$3 + N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(phi1 * phi1), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (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))))))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\left(1 + {\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\left({\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\left(\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right) \cdot {\phi_1}^{2} + 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, \color{blue}{{\phi_1}^{2}}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, {\color{blue}{\phi_1}}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {\phi_1}^{2} + \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {\phi_1}^{2}, \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \color{blue}{\phi_1}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-*.f64 95.2

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \color{blue}{\phi_1}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 95.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right)} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + \frac{-1}{2} \cdot \left(\phi_2 \cdot \color{blue}{\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 95.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \color{blue}{\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 95.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + \frac{-1}{2} \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   if 4.00000000000000033e-5 < (*.f64 #s(literal 2 binary64) (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 60.7%

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

   3. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-mult N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 61.0

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

   5. Applied rewrites 61.0%

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

Alternative 4: 64.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

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

   3. lift-/.f64 N/A

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

   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. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 63.6

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

3. Applied rewrites 63.6%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lift--.f64 64.1

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

5. Applied rewrites 64.1%

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

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 64.5

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

7. Applied rewrites 64.5%

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

8. Taylor expanded in lambda1 around 0

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

10. Applied rewrites 64.5%

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

Alternative 5: 62.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\\ \mathbf{if}\;t\_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1 + \left(\left(\cos \phi_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{1 - \left(t\_1 + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot 1\right) \cdot t\_0\right) \cdot t\_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2
         (fma
          (cos phi1)
          (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))
   (if (<= (+ t_1 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)) 2e-10)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+ t_1 (* (* (* (cos phi1) (+ 1.0 (* -0.5 (* phi2 phi2)))) t_0) t_0)))
        (sqrt
         (-
          1.0
          (+
           t_1
           (*
            (*
             (*
              (fma
               (-
                (*
                 (fma -0.001388888888888889 (* phi1 phi1) 0.041666666666666664)
                 (* phi1 phi1))
                0.5)
               (* phi1 phi1)
               1.0)
              1.0)
             t_0)
            t_0)))))))
     (* (* (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))) 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 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(sin((0.5 * (phi1 - phi2))), 2.0));
	double tmp;
	if ((t_1 + (((cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 2e-10) {
		tmp = R * (2.0 * atan2(sqrt((t_1 + (((cos(phi1) * (1.0 + (-0.5 * (phi2 * phi2)))) * t_0) * t_0))), sqrt((1.0 - (t_1 + (((fma(((fma(-0.001388888888888889, (phi1 * phi1), 0.041666666666666664) * (phi1 * phi1)) - 0.5), (phi1 * phi1), 1.0) * 1.0) * t_0) * t_0))))));
	} else {
		tmp = (atan2(sqrt(t_2), sqrt((1.0 - t_2))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 2e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_1 + Float64(Float64(Float64(cos(phi1) * Float64(1.0 + Float64(-0.5 * Float64(phi2 * phi2)))) * t_0) * t_0))), sqrt(Float64(1.0 - Float64(t_1 + Float64(Float64(Float64(fma(Float64(Float64(fma(-0.001388888888888889, Float64(phi1 * phi1), 0.041666666666666664) * Float64(phi1 * phi1)) - 0.5), Float64(phi1 * phi1), 1.0) * 1.0) * t_0) * t_0)))))));
	else
		tmp = Float64(Float64(atan(sqrt(t_2), sqrt(Float64(1.0 - t_2))) * 2.0) * R);
	end
	return tmp
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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2e-10], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 + N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(phi1 * phi1), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))))) < 2.00000000000000007e-10

   1. Initial program 70.6%

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

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\left(1 + {\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\left({\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\left(\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right) \cdot {\phi_1}^{2} + 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, \color{blue}{{\phi_1}^{2}}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, {\color{blue}{\phi_1}}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {\phi_1}^{2} + \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {\phi_1}^{2}, \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot {\phi_1}^{2} - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, {\phi_1}^{2}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \color{blue}{\phi_1}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-*.f64 70.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \color{blue}{\phi_1}, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 70.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right)} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_2}^{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + \frac{-1}{2} \cdot \left(\phi_2 \cdot \color{blue}{\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 72.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \color{blue}{\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 72.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + \frac{-1}{2} \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \phi_1 \cdot \phi_1, \frac{1}{24}\right) \cdot \left(\phi_1 \cdot \phi_1\right) - \frac{1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \phi_1 \cdot \phi_1, 0.041666666666666664\right) \cdot \left(\phi_1 \cdot \phi_1\right) - 0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   if 2.00000000000000007e-10 < (+.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 62.0%

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

   6. Applied rewrites 62.0%

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

Alternative 6: 61.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_3
         (fma
          (cos phi1)
          (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))
   (if (<= (+ t_1 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)) 2e-10)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma (cos phi1) t_2 (pow (sin (* 0.5 phi1)) 2.0)))
        (sqrt (- 1.0 (+ t_1 (* (cos phi1) t_2)))))))
     (* (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) 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 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_3 = fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(sin((0.5 * (phi1 - phi2))), 2.0));
	double tmp;
	if ((t_1 + (((cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 2e-10) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_2, pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - (t_1 + (cos(phi1) * t_2))))));
	} else {
		tmp = (atan2(sqrt(t_3), sqrt((1.0 - t_3))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_3 = fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 2e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_2, (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - Float64(t_1 + Float64(cos(phi1) * t_2)))))));
	else
		tmp = Float64(Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) * 2.0) * R);
	end
	return tmp
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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2e-10], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$2 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))))) < 2.00000000000000007e-10

   1. Initial program 70.6%

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

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

      3. lift-/.f64 N/A

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

      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. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-/.f64 75.9

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

   3. Applied rewrites 75.9%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 75.9

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

   6. Applied rewrites 75.9%

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

   7. Taylor expanded in phi2 around 0

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

      1. sin-diff-rev N/A

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

      2. div-sub N/A

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

      3. lower-fma.f64 N/A

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

   9. Applied rewrites 60.3%

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

   if 2.00000000000000007e-10 < (+.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 62.0%

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

   6. Applied rewrites 62.0%

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

Alternative 7: 64.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    code = r * (2.0d0 * atan2(sqrt(((((sin((phi1 / 2.0d0)) * cos((phi2 / 2.0d0))) - (cos((phi1 / 2.0d0)) * sin((phi2 / 2.0d0)))) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt((1.0d0 - ((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + ((((cos((phi1 + phi2)) + cos((phi1 - phi2))) / 2.0d0) * t_0) * t_0))))))
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));
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(((Math.sin((phi1 / 2.0)) * Math.cos((phi2 / 2.0))) - (Math.cos((phi1 / 2.0)) * Math.sin((phi2 / 2.0)))), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0))), Math.sqrt((1.0 - (Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + ((((Math.cos((phi1 + phi2)) + Math.cos((phi1 - phi2))) / 2.0) * t_0) * t_0))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

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

   3. lift-/.f64 N/A

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

   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. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 63.6

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

3. Applied rewrites 63.6%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lift--.f64 64.1

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

5. Applied rewrites 64.1%

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

Alternative 8: 64.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

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

   3. lift-/.f64 N/A

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

   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. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 63.6

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

3. Applied rewrites 63.6%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lift--.f64 64.1

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

5. Applied rewrites 64.1%

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

6. Taylor expanded in lambda1 around 0

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

8. Applied rewrites 64.1%

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

Alternative 9: 63.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    code = r * (2.0d0 * atan2(sqrt(((((sin((phi1 / 2.0d0)) * cos((phi2 / 2.0d0))) - sin((0.5d0 * phi2))) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt((1.0d0 - ((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + ((((cos((phi1 + phi2)) + cos((phi1 - phi2))) / 2.0d0) * t_0) * t_0))))))
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));
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(((Math.sin((phi1 / 2.0)) * Math.cos((phi2 / 2.0))) - Math.sin((0.5 * phi2))), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0))), Math.sqrt((1.0 - (Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + ((((Math.cos((phi1 + phi2)) + Math.cos((phi1 - phi2))) / 2.0) * t_0) * t_0))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	return R * (2.0 * math.atan2(math.sqrt((math.pow(((math.sin((phi1 / 2.0)) * math.cos((phi2 / 2.0))) - math.sin((0.5 * phi2))), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0))), math.sqrt((1.0 - (math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + ((((math.cos((phi1 + phi2)) + math.cos((phi1 - phi2))) / 2.0) * t_0) * t_0))))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	tmp = R * (2.0 * atan2(sqrt(((((sin((phi1 / 2.0)) * cos((phi2 / 2.0))) - sin((0.5 * phi2))) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt((1.0 - ((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + ((((cos((phi1 + phi2)) + cos((phi1 - phi2))) / 2.0) * t_0) * t_0))))));
end

Wolfram

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

Derivation

1. Initial program 62.7%

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

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

   3. lift-/.f64 N/A

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

   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. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 63.6

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

3. Applied rewrites 63.6%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lift--.f64 64.1

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

5. Applied rewrites 64.1%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower-*.f64 63.5

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

8. Applied rewrites 63.5%

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

Alternative 10: 63.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow
         (- (sin (* 0.5 phi1)) (* (cos (/ phi1 2.0)) (sin (/ phi2 2.0))))
         2.0)
        t_1))
      (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) 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 = ((cos(phi1) * cos(phi2)) * t_0) * t_0;
	return R * (2.0 * atan2(sqrt((pow((sin((0.5 * phi1)) - (cos((phi1 / 2.0)) * sin((phi2 / 2.0)))), 2.0) + t_1)), sqrt((1.0 - (pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 = ((cos(phi1) * cos(phi2)) * t_0) * t_0
    code = r * (2.0d0 * atan2(sqrt((((sin((0.5d0 * phi1)) - (cos((phi1 / 2.0d0)) * sin((phi2 / 2.0d0)))) ** 2.0d0) + t_1)), sqrt((1.0d0 - ((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + 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.cos(phi1) * Math.cos(phi2)) * t_0) * t_0;
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow((Math.sin((0.5 * phi1)) - (Math.cos((phi1 / 2.0)) * Math.sin((phi2 / 2.0)))), 2.0) + t_1)), Math.sqrt((1.0 - (Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = ((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0
	return R * (2.0 * math.atan2(math.sqrt((math.pow((math.sin((0.5 * phi1)) - (math.cos((phi1 / 2.0)) * math.sin((phi2 / 2.0)))), 2.0) + t_1)), math.sqrt((1.0 - (math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((Float64(sin(Float64(0.5 * phi1)) - Float64(cos(Float64(phi1 / 2.0)) * sin(Float64(phi2 / 2.0)))) ^ 2.0) + t_1)), sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_1))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = ((cos(phi1) * cos(phi2)) * t_0) * t_0;
	tmp = R * (2.0 * atan2(sqrt((((sin((0.5 * phi1)) - (cos((phi1 / 2.0)) * sin((phi2 / 2.0)))) ^ 2.0) + t_1)), sqrt((1.0 - ((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + 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[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] - N[(N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.7%

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

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

   3. lift-/.f64 N/A

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

   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. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 63.6

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

3. Applied rewrites 63.6%

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

4. Taylor expanded in phi2 around 0

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

   1. lower-sin.f64 N/A

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

   2. lower-*.f64 63.0

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

6. Applied rewrites 63.0%

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

Alternative 11: 63.2% accurate, 1.0× speedup

Math

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

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)))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        t_1
        (* (* (/ (+ (cos (+ phi2 phi1)) (cos (- phi2 phi1))) 2.0) t_0) t_0)))
      (sqrt (- 1.0 (+ t_1 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))))))))

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);
	return R * (2.0 * atan2(sqrt((t_1 + ((((cos((phi2 + phi1)) + cos((phi2 - phi1))) / 2.0) * t_0) * t_0))), sqrt((1.0 - (t_1 + (((cos(phi1) * cos(phi2)) * t_0) * t_0))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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
    code = r * (2.0d0 * atan2(sqrt((t_1 + ((((cos((phi2 + phi1)) + cos((phi2 - phi1))) / 2.0d0) * t_0) * t_0))), sqrt((1.0d0 - (t_1 + (((cos(phi1) * cos(phi2)) * t_0) * t_0))))))
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);
	return R * (2.0 * Math.atan2(Math.sqrt((t_1 + ((((Math.cos((phi2 + phi1)) + Math.cos((phi2 - phi1))) / 2.0) * t_0) * t_0))), Math.sqrt((1.0 - (t_1 + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0))))));
}

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)
	return R * (2.0 * math.atan2(math.sqrt((t_1 + ((((math.cos((phi2 + phi1)) + math.cos((phi2 - phi1))) / 2.0) * t_0) * t_0))), math.sqrt((1.0 - (t_1 + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(t_1 + Float64(Float64(Float64(Float64(cos(Float64(phi2 + phi1)) + cos(Float64(phi2 - phi1))) / 2.0) * t_0) * t_0))), sqrt(Float64(1.0 - Float64(t_1 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)))))))
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;
	tmp = R * (2.0 * atan2(sqrt((t_1 + ((((cos((phi2 + phi1)) + cos((phi2 - phi1))) / 2.0) * t_0) * t_0))), sqrt((1.0 - (t_1 + (((cos(phi1) * cos(phi2)) * t_0) * t_0))))));
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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + N[(N[(N[(N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.7%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. cos-mult N/A

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

   6. lower-/.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower--.f64 63.2

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

3. Applied rewrites 63.2%

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

Alternative 12: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ t_2 := \frac{\phi_1 - \phi_2}{2}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin t\_2}^{2} + t\_1}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_2\right)\right) + t\_1\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* (* (* (cos phi1) (cos phi2)) t_0) t_0))
        (t_2 (/ (- phi1 phi2) 2.0)))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (+ (pow (sin t_2) 2.0) t_1))
      (sqrt (- 1.0 (+ (- 0.5 (* 0.5 (cos (* 2.0 t_2)))) 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 = ((cos(phi1) * cos(phi2)) * t_0) * t_0;
	double t_2 = (phi1 - phi2) / 2.0;
	return R * (2.0 * atan2(sqrt((pow(sin(t_2), 2.0) + t_1)), sqrt((1.0 - ((0.5 - (0.5 * cos((2.0 * t_2)))) + t_1)))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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
    real(8) :: t_2
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = ((cos(phi1) * cos(phi2)) * t_0) * t_0
    t_2 = (phi1 - phi2) / 2.0d0
    code = r * (2.0d0 * atan2(sqrt(((sin(t_2) ** 2.0d0) + t_1)), sqrt((1.0d0 - ((0.5d0 - (0.5d0 * cos((2.0d0 * t_2)))) + 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.cos(phi1) * Math.cos(phi2)) * t_0) * t_0;
	double t_2 = (phi1 - phi2) / 2.0;
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(t_2), 2.0) + t_1)), Math.sqrt((1.0 - ((0.5 - (0.5 * Math.cos((2.0 * t_2)))) + t_1)))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = ((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0
	t_2 = (phi1 - phi2) / 2.0
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(t_2), 2.0) + t_1)), math.sqrt((1.0 - ((0.5 - (0.5 * math.cos((2.0 * t_2)))) + t_1)))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)
	t_2 = Float64(Float64(phi1 - phi2) / 2.0)
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(t_2) ^ 2.0) + t_1)), sqrt(Float64(1.0 - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_2)))) + t_1))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = ((cos(phi1) * cos(phi2)) * t_0) * t_0;
	t_2 = (phi1 - phi2) / 2.0;
	tmp = R * (2.0 * atan2(sqrt(((sin(t_2) ^ 2.0) + t_1)), sqrt((1.0 - ((0.5 - (0.5 * cos((2.0 * t_2)))) + 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[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 62.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-sin-a N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 62.8

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

3. Applied rewrites 62.8%

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

Alternative 13: 62.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Applied rewrites 62.8%

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

4. Taylor expanded in lambda1 around 0

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

6. Applied rewrites 62.8%

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

Alternative 14: 61.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), t\_1\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -12000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.9 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left(t\_1 + \cos \phi_1 \cdot t\_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2
         (*
          (*
           (atan2
            (sqrt
             (fma
              (cos phi2)
              (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
              (pow (sin (* -0.5 phi2)) 2.0)))
            (sqrt
             (-
              1.0
              (fma
               (* (cos phi2) (cos phi1))
               (- 0.5 (* 0.5 (cos (* 2.0 (/ (- lambda1 lambda2) 2.0)))))
               t_1))))
           2.0)
          R)))
   (if (<= phi2 -12000000.0)
     t_2
     (if (<= phi2 1.9e-21)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt (- 1.0 (+ t_1 (* (cos phi1) t_0)))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = (atan2(sqrt(fma(cos(phi2), (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - fma((cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) / 2.0))))), t_1)))) * 2.0) * R;
	double tmp;
	if (phi2 <= -12000000.0) {
		tmp = t_2;
	} else if (phi2 <= 1.9e-21) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - (t_1 + (cos(phi1) * t_0))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = Float64(Float64(atan(sqrt(fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - fma(Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) / 2.0))))), t_1)))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -12000000.0)
		tmp = t_2;
	elseif (phi2 <= 1.9e-21)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - Float64(t_1 + Float64(cos(phi1) * t_0)))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -12000000.0], t$95$2, If[LessEqual[phi2, 1.9e-21], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.2e7 or 1.8999999999999999e-21 < phi2

   1. Initial program 48.5%

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

   3. Applied rewrites 48.5%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 48.7

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

   6. Applied rewrites 48.7%

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

   if -1.2e7 < phi2 < 1.8999999999999999e-21

   1. Initial program 77.6%

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

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

      3. lift-/.f64 N/A

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

      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. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-/.f64 77.6

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

   3. Applied rewrites 77.6%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 77.2

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

   6. Applied rewrites 77.2%

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

   7. Taylor expanded in phi2 around 0

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

      1. sin-diff-rev N/A

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

      2. div-sub N/A

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

      3. lower-fma.f64 N/A

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

   9. Applied rewrites 74.7%

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

Alternative 15: 60.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_1 \leq -0.205:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, t\_3, t\_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(\phi_1 \cdot \phi_1\right)\right) - 0.5\right)\right), t\_3, t\_1\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2 (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R))
        (t_3 (- 0.5 (* 0.5 (cos (* 2.0 (/ (- lambda1 lambda2) 2.0)))))))
   (if (<= phi1 -0.205)
     t_2
     (if (<= phi1 5e-24)
       (*
        (*
         (atan2
          (sqrt (fma (* (cos phi2) (cos phi1)) t_3 t_1))
          (sqrt
           (-
            1.0
            (fma
             (*
              (cos phi2)
              (+
               1.0
               (*
                (* phi1 phi1)
                (-
                 (*
                  (* phi1 phi1)
                  (+
                   0.041666666666666664
                   (* -0.001388888888888889 (* phi1 phi1))))
                 0.5))))
             t_3
             t_1))))
         2.0)
        R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
	double t_3 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) / 2.0))));
	double tmp;
	if (phi1 <= -0.205) {
		tmp = t_2;
	} else if (phi1 <= 5e-24) {
		tmp = (atan2(sqrt(fma((cos(phi2) * cos(phi1)), t_3, t_1)), sqrt((1.0 - fma((cos(phi2) * (1.0 + ((phi1 * phi1) * (((phi1 * phi1) * (0.041666666666666664 + (-0.001388888888888889 * (phi1 * phi1)))) - 0.5)))), t_3, t_1)))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) / 2.0)))))
	tmp = 0.0
	if (phi1 <= -0.205)
		tmp = t_2;
	elseif (phi1 <= 5e-24)
		tmp = Float64(Float64(atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), t_3, t_1)), sqrt(Float64(1.0 - fma(Float64(cos(phi2) * Float64(1.0 + Float64(Float64(phi1 * phi1) * Float64(Float64(Float64(phi1 * phi1) * Float64(0.041666666666666664 + Float64(-0.001388888888888889 * Float64(phi1 * phi1)))) - 0.5)))), t_3, t_1)))) * 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[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.205], t$95$2, If[LessEqual[phi1, 5e-24], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$3 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[(1.0 + N[(N[(phi1 * phi1), $MachinePrecision] * N[(N[(N[(phi1 * phi1), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.001388888888888889 * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.204999999999999988 or 4.9999999999999998e-24 < phi1

   1. Initial program 47.7%

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

   3. Applied rewrites 47.7%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 47.6

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

   6. Applied rewrites 47.6%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-cos.f64 48.5

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

   9. Applied rewrites 48.5%

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

   if -0.204999999999999988 < phi1 < 4.9999999999999998e-24

   1. Initial program 78.1%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\left(1 + {\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]
   5. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(\color{blue}{1} + {\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \color{blue}{{\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + {\phi_1}^{2} \cdot \color{blue}{\left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      4. pow2 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\color{blue}{{\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right)} - \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\color{blue}{{\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right)} - \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \color{blue}{\frac{1}{2}}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      8. pow2 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      12. pow2 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot \left(\phi_1 \cdot \phi_1\right)\right) - \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

      13. lift-*.f64 73.2

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(\phi_1 \cdot \phi_1\right)\right) - 0.5\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   6. Applied rewrites 73.2%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(\phi_1 \cdot \phi_1\right)\right) - 0.5\right)\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 60.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_1 \leq -0.078:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, t\_3, t\_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, t\_3, t\_1\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2 (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R))
        (t_3 (- 0.5 (* 0.5 (cos (* 2.0 (/ (- lambda1 lambda2) 2.0)))))))
   (if (<= phi1 -0.078)
     t_2
     (if (<= phi1 5e-24)
       (*
        (*
         (atan2
          (sqrt (fma (* (cos phi2) (cos phi1)) t_3 t_1))
          (sqrt (- 1.0 (fma (cos phi2) t_3 t_1))))
         2.0)
        R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
	double t_3 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) / 2.0))));
	double tmp;
	if (phi1 <= -0.078) {
		tmp = t_2;
	} else if (phi1 <= 5e-24) {
		tmp = (atan2(sqrt(fma((cos(phi2) * cos(phi1)), t_3, t_1)), sqrt((1.0 - fma(cos(phi2), t_3, t_1)))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) / 2.0)))))
	tmp = 0.0
	if (phi1 <= -0.078)
		tmp = t_2;
	elseif (phi1 <= 5e-24)
		tmp = Float64(Float64(atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), t_3, t_1)), sqrt(Float64(1.0 - fma(cos(phi2), t_3, t_1)))) * 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[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.078], t$95$2, If[LessEqual[phi1, 5e-24], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$3 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * t$95$3 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0779999999999999999 or 4.9999999999999998e-24 < phi1

   1. Initial program 47.7%

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

   3. Applied rewrites 47.7%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 47.6

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

   6. Applied rewrites 47.6%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-cos.f64 48.5

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

   9. Applied rewrites 48.5%

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

   if -0.0779999999999999999 < phi1 < 4.9999999999999998e-24

   1. Initial program 78.1%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 73.1

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

   6. Applied rewrites 73.1%

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

Alternative 17: 41.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := \frac{\lambda_1 - \lambda_2}{2}\\ t_4 := 0.5 - 0.5 \cdot \cos \left(2 \cdot t\_3\right)\\ t_5 := \sin t\_3\\ t_6 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;t\_5 \leq -0.34:\\ \;\;\;\;\left(\tan^{-1}_* \frac{t\_1}{\sqrt{1 - \left(t\_6 \cdot t\_4 + t\_2\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;t\_5 \leq 0.33:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_6, t\_4, t\_2\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{t\_1}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
        (t_1 (sqrt t_0))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3 (/ (- lambda1 lambda2) 2.0))
        (t_4 (- 0.5 (* 0.5 (cos (* 2.0 t_3)))))
        (t_5 (sin t_3))
        (t_6 (* (cos phi2) (cos phi1))))
   (if (<= t_5 -0.34)
     (* (* (atan2 t_1 (sqrt (- 1.0 (+ (* t_6 t_4) t_2)))) 2.0) R)
     (if (<= t_5 0.33)
       (*
        (*
         (atan2
          (sqrt (fma t_6 t_4 t_2))
          (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))
         2.0)
        R)
       (*
        (*
         (atan2
          t_1
          (sqrt (- 1.0 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))))
         2.0)
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((lambda1 - lambda2)));
	double t_1 = sqrt(t_0);
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = (lambda1 - lambda2) / 2.0;
	double t_4 = 0.5 - (0.5 * cos((2.0 * t_3)));
	double t_5 = sin(t_3);
	double t_6 = cos(phi2) * cos(phi1);
	double tmp;
	if (t_5 <= -0.34) {
		tmp = (atan2(t_1, sqrt((1.0 - ((t_6 * t_4) + t_2)))) * 2.0) * R;
	} else if (t_5 <= 0.33) {
		tmp = (atan2(sqrt(fma(t_6, t_4, t_2)), sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))) * 2.0) * R;
	} else {
		tmp = (atan2(t_1, sqrt((1.0 - fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))
	t_1 = sqrt(t_0)
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_4 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_3))))
	t_5 = sin(t_3)
	t_6 = Float64(cos(phi2) * cos(phi1))
	tmp = 0.0
	if (t_5 <= -0.34)
		tmp = Float64(Float64(atan(t_1, sqrt(Float64(1.0 - Float64(Float64(t_6 * t_4) + t_2)))) * 2.0) * R);
	elseif (t_5 <= 0.33)
		tmp = Float64(Float64(atan(sqrt(fma(t_6, t_4, t_2)), sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0)))) * 2.0) * R);
	else
		tmp = Float64(Float64(atan(t_1, sqrt(Float64(1.0 - fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sin[t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -0.34], N[(N[(N[ArcTan[t$95$1 / N[Sqrt[N[(1.0 - N[(N[(t$95$6 * t$95$4), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[t$95$5, 0.33], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$6 * t$95$4 + t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], N[(N[(N[ArcTan[t$95$1 / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.340000000000000024

   1. Initial program 57.9%

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

   3. Applied rewrites 58.0%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 45.8

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

   6. Applied rewrites 45.8%

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

   7. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 34.3

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

   9. Applied rewrites 34.3%

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

   11. Applied rewrites 34.3%

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

   if -0.340000000000000024 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.330000000000000016

   1. Initial program 70.0%

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

   3. Applied rewrites 63.9%

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

   4. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 57.2

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

   6. Applied rewrites 57.2%

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

   7. Taylor expanded in lambda1 around 0

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 50.2

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

   9. Applied rewrites 50.2%

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

   if 0.330000000000000016 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 57.4%

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

   3. Applied rewrites 57.5%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 46.3

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

   6. Applied rewrites 46.3%

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

   7. Taylor expanded in phi1 around 0

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 34.6

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

   9. Applied rewrites 34.6%

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

   10. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift--.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift--.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 34.9

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

   12. Applied rewrites 34.9%

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

Alternative 18: 60.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -0.0031:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
        (t_1 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2 (* (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))) 2.0) R)))
   (if (<= phi1 -0.0031)
     t_2
     (if (<= phi1 5e-24)
       (*
        (*
         (atan2
          (sqrt
           (fma
            (* (cos phi2) (cos phi1))
            (- 0.5 (* 0.5 (cos (* 2.0 (/ (- lambda1 lambda2) 2.0)))))
            (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
          (sqrt (- 1.0 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))))
         2.0)
        R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((lambda1 - lambda2)));
	double t_1 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_2 = (atan2(sqrt(t_1), sqrt((1.0 - t_1))) * 2.0) * R;
	double tmp;
	if (phi1 <= -0.0031) {
		tmp = t_2;
	} else if (phi1 <= 5e-24) {
		tmp = (atan2(sqrt(fma((cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) / 2.0))))), pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0))))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))
	t_1 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = Float64(Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -0.0031)
		tmp = t_2;
	elseif (phi1 <= 5e-24)
		tmp = Float64(Float64(atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) / 2.0))))), (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))))) * 2.0) * R);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -0.0031], t$95$2, If[LessEqual[phi1, 5e-24], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.00309999999999999989 or 4.9999999999999998e-24 < phi1

   1. Initial program 47.8%

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

   3. Applied rewrites 47.7%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 47.6

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

   6. Applied rewrites 47.6%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-cos.f64 48.5

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

   9. Applied rewrites 48.5%

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

   if -0.00309999999999999989 < phi1 < 4.9999999999999998e-24

   1. Initial program 78.2%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 73.1

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

   6. Applied rewrites 73.1%

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

Alternative 19: 60.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t\_1, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -12000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (* (cos phi2) (cos phi1))
          (- 0.5 (* 0.5 (cos (* 2.0 (/ (- lambda1 lambda2) 2.0)))))
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (t_1 (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
        (t_2
         (*
          (*
           (atan2
            (sqrt (fma (cos phi2) t_1 (pow (sin (* -0.5 phi2)) 2.0)))
            (sqrt (- 1.0 t_0)))
           2.0)
          R)))
   (if (<= phi2 -12000000.0)
     t_2
     (if (<= phi2 1.4e-5)
       (*
        (*
         (atan2
          (sqrt t_0)
          (sqrt (- 1.0 (fma (cos phi1) t_1 (pow (sin (* 0.5 phi1)) 2.0)))))
         2.0)
        R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) / 2.0))))), pow(sin(((phi1 - phi2) / 2.0)), 2.0));
	double t_1 = 0.5 - (0.5 * cos((lambda1 - lambda2)));
	double t_2 = (atan2(sqrt(fma(cos(phi2), t_1, pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - t_0))) * 2.0) * R;
	double tmp;
	if (phi2 <= -12000000.0) {
		tmp = t_2;
	} else if (phi2 <= 1.4e-5) {
		tmp = (atan2(sqrt(t_0), sqrt((1.0 - fma(cos(phi1), t_1, pow(sin((0.5 * phi1)), 2.0))))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) / 2.0))))), (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))
	t_2 = Float64(Float64(atan(sqrt(fma(cos(phi2), t_1, (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - t_0))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -12000000.0)
		tmp = t_2;
	elseif (phi2 <= 1.4e-5)
		tmp = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - fma(cos(phi1), t_1, (sin(Float64(0.5 * phi1)) ^ 2.0))))) * 2.0) * R);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * t$95$1 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -12000000.0], t$95$2, If[LessEqual[phi2, 1.4e-5], N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.2e7 or 1.39999999999999998e-5 < phi2

   1. Initial program 47.8%

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

   3. Applied rewrites 47.8%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 48.5

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

   6. Applied rewrites 48.5%

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

   if -1.2e7 < phi2 < 1.39999999999999998e-5

   1. Initial program 77.6%

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

   3. Applied rewrites 72.7%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 72.2

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

   6. Applied rewrites 72.2%

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

Alternative 20: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_2 := 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot \cos \phi_1\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t\_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_3, t\_0, {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -12000000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_2 \leq 1.9 \cdot 10^{-21}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_2, t\_1\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_3, t\_0, t\_1\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (/ (- lambda1 lambda2) 2.0))))))
        (t_1 (pow (sin (* 0.5 phi1)) 2.0))
        (t_2 (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
        (t_3 (* (cos phi2) (cos phi1)))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma (cos phi2) t_2 (pow (sin (* -0.5 phi2)) 2.0)))
            (sqrt (- 1.0 (fma t_3 t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))))
           2.0)
          R)))
   (if (<= phi2 -12000000.0)
     t_4
     (if (<= phi2 1.9e-21)
       (*
        (*
         (atan2
          (sqrt (fma (cos phi1) t_2 t_1))
          (sqrt (- 1.0 (fma t_3 t_0 t_1))))
         2.0)
        R)
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) / 2.0))));
	double t_1 = pow(sin((0.5 * phi1)), 2.0);
	double t_2 = 0.5 - (0.5 * cos((lambda1 - lambda2)));
	double t_3 = cos(phi2) * cos(phi1);
	double t_4 = (atan2(sqrt(fma(cos(phi2), t_2, pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - fma(t_3, t_0, pow(sin(((phi1 - phi2) / 2.0)), 2.0))))) * 2.0) * R;
	double tmp;
	if (phi2 <= -12000000.0) {
		tmp = t_4;
	} else if (phi2 <= 1.9e-21) {
		tmp = (atan2(sqrt(fma(cos(phi1), t_2, t_1)), sqrt((1.0 - fma(t_3, t_0, t_1)))) * 2.0) * R;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) / 2.0)))))
	t_1 = sin(Float64(0.5 * phi1)) ^ 2.0
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))
	t_3 = Float64(cos(phi2) * cos(phi1))
	t_4 = Float64(Float64(atan(sqrt(fma(cos(phi2), t_2, (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_3, t_0, (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -12000000.0)
		tmp = t_4;
	elseif (phi2 <= 1.9e-21)
		tmp = Float64(Float64(atan(sqrt(fma(cos(phi1), t_2, t_1)), sqrt(Float64(1.0 - fma(t_3, t_0, t_1)))) * 2.0) * R);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * t$95$2 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$3 * t$95$0 + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -12000000.0], t$95$4, If[LessEqual[phi2, 1.9e-21], N[(N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$3 * t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.2e7 or 1.8999999999999999e-21 < phi2

   1. Initial program 48.5%

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

   3. Applied rewrites 48.5%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 48.7

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

   6. Applied rewrites 48.7%

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

   if -1.2e7 < phi2 < 1.8999999999999999e-21

   1. Initial program 77.6%

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 69.7

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in phi1 around inf

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

      1. lift-*.f64 69.7

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

   9. Applied rewrites 69.7%

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

Alternative 21: 54.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.00011:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1
         (*
          (*
           (atan2
            (sqrt
             (fma
              (* (cos phi2) (cos phi1))
              (- 0.5 (* 0.5 (cos (* 2.0 (/ (- lambda1 lambda2) 2.0)))))
              (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
            (sqrt
             (-
              1.0
              (fma
               (cos phi2)
               (- 0.5 (* 0.5 (cos lambda1)))
               (pow (sin (* -0.5 phi2)) 2.0)))))
           2.0)
          R)))
   (if (<= phi2 -0.00011)
     t_1
     (if (<= phi2 4e-16)
       (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R)
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = (atan2(sqrt(fma((cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) / 2.0))))), pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - fma(cos(phi2), (0.5 - (0.5 * cos(lambda1))), pow(sin((-0.5 * phi2)), 2.0))))) * 2.0) * R;
	double tmp;
	if (phi2 <= -0.00011) {
		tmp = t_1;
	} else if (phi2 <= 4e-16) {
		tmp = (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = Float64(Float64(atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) / 2.0))))), (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(lambda1))), (sin(Float64(-0.5 * phi2)) ^ 2.0))))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -0.00011)
		tmp = t_1;
	elseif (phi2 <= 4e-16)
		tmp = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -0.00011], t$95$1, If[LessEqual[phi2, 4e-16], N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.10000000000000004e-4 or 3.9999999999999999e-16 < phi2

   1. Initial program 48.2%

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

   3. Applied rewrites 48.2%

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

   4. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 39.3

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

   6. Applied rewrites 39.3%

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

   7. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lift-*.f64 39.6

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

   9. Applied rewrites 39.6%

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

   if -1.10000000000000004e-4 < phi2 < 3.9999999999999999e-16

   1. Initial program 78.3%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 70.7

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

   6. Applied rewrites 70.7%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-cos.f64 70.7

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

   9. Applied rewrites 70.7%

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

Alternative 22: 51.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ t_1 := 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := \frac{\lambda_1 - \lambda_2}{2}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{t\_3}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_1, t\_0\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot t\_4\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{t\_3}{\sqrt{1 - t\_2}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (t_1 (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
        (t_2 (fma (cos phi1) t_1 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (sqrt t_2))
        (t_4 (/ (- lambda1 lambda2) 2.0)))
   (if (<= t_4 -1e-5)
     (*
      (*
       (atan2 t_3 (sqrt (- 1.0 (fma (cos phi1) (* (cos phi2) t_1) t_0))))
       2.0)
      R)
     (if (<= t_4 2e-6)
       (*
        (*
         (atan2
          (sqrt
           (fma
            (* (cos phi2) (cos phi1))
            (- 0.5 (* 0.5 (cos (* 2.0 t_4))))
            (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
          (sqrt (- 1.0 t_0)))
         2.0)
        R)
       (* (* (atan2 t_3 (sqrt (- 1.0 t_2))) 2.0) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double t_1 = 0.5 - (0.5 * cos((lambda1 - lambda2)));
	double t_2 = fma(cos(phi1), t_1, pow(sin((0.5 * phi1)), 2.0));
	double t_3 = sqrt(t_2);
	double t_4 = (lambda1 - lambda2) / 2.0;
	double tmp;
	if (t_4 <= -1e-5) {
		tmp = (atan2(t_3, sqrt((1.0 - fma(cos(phi1), (cos(phi2) * t_1), t_0)))) * 2.0) * R;
	} else if (t_4 <= 2e-6) {
		tmp = (atan2(sqrt(fma((cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * t_4)))), pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - t_0))) * 2.0) * R;
	} else {
		tmp = (atan2(t_3, sqrt((1.0 - t_2))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))
	t_2 = fma(cos(phi1), t_1, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = sqrt(t_2)
	t_4 = Float64(Float64(lambda1 - lambda2) / 2.0)
	tmp = 0.0
	if (t_4 <= -1e-5)
		tmp = Float64(Float64(atan(t_3, sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * t_1), t_0)))) * 2.0) * R);
	elseif (t_4 <= 2e-6)
		tmp = Float64(Float64(atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_4)))), (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - t_0))) * 2.0) * R);
	else
		tmp = Float64(Float64(atan(t_3, sqrt(Float64(1.0 - t_2))) * 2.0) * R);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -1.00000000000000008e-5

   1. Initial program 58.0%

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

   3. Applied rewrites 57.9%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 45.2

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

   6. Applied rewrites 45.2%

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

   7. Taylor expanded in lambda1 around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-pow.f64 N/A

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

   9. Applied rewrites 45.2%

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

   if -1.00000000000000008e-5 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < 1.99999999999999991e-6

   1. Initial program 78.7%

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

   3. Applied rewrites 68.7%

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

   4. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 68.6

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

   6. Applied rewrites 68.6%

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

   7. Taylor expanded in lambda1 around 0

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 68.6

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

   9. Applied rewrites 68.6%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 57.5%

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

   3. Applied rewrites 57.4%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 45.9

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-cos.f64 46.2

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

   9. Applied rewrites 46.2%

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

Alternative 23: 51.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := \frac{\lambda_1 - \lambda_2}{2}\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot t\_1\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1 (/ (- lambda1 lambda2) 2.0))
        (t_2 (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R)))
   (if (<= t_1 -1e-5)
     t_2
     (if (<= t_1 2e-6)
       (*
        (*
         (atan2
          (sqrt
           (fma
            (* (cos phi2) (cos phi1))
            (- 0.5 (* 0.5 (cos (* 2.0 t_1))))
            (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
          (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))
         2.0)
        R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = (lambda1 - lambda2) / 2.0;
	double t_2 = (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
	double tmp;
	if (t_1 <= -1e-5) {
		tmp = t_2;
	} else if (t_1 <= 2e-6) {
		tmp = (atan2(sqrt(fma((cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * t_1)))), pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_2 = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R)
	tmp = 0.0
	if (t_1 <= -1e-5)
		tmp = t_2;
	elseif (t_1 <= 2e-6)
		tmp = Float64(Float64(atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_1)))), (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0)))) * 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[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-5], t$95$2, If[LessEqual[t$95$1, 2e-6], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -1.00000000000000008e-5 or 1.99999999999999991e-6 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 57.8%

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

   3. Applied rewrites 57.6%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 45.6

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

   6. Applied rewrites 45.6%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-cos.f64 45.8

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

   9. Applied rewrites 45.8%

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

   if -1.00000000000000008e-5 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < 1.99999999999999991e-6

   1. Initial program 78.7%

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

   3. Applied rewrites 68.7%

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

   4. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 68.6

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

   6. Applied rewrites 68.6%

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

   7. Taylor expanded in lambda1 around 0

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 68.6

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

   9. Applied rewrites 68.6%

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

Alternative 24: 27.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Applied rewrites 60.3%

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

4. Taylor expanded in phi2 around 0

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

   1. lower-fma.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-*.f64 44.0

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

6. Applied rewrites 44.0%

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

7. Taylor expanded in phi1 around 0

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 27.1

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

9. Applied rewrites 27.1%

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

10. Taylor expanded in phi1 around 0

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

    1. lower--.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lift-cos.f64 N/A

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

    4. lift-cos.f64 N/A

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

    5. lift--.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift--.f64 N/A

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

    8. lower-pow.f64 N/A

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

    9. lower-sin.f64 N/A

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

    10. lift-*.f64 27.3

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

12. Applied rewrites 27.3%

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

Alternative 25: 27.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Applied rewrites 60.3%

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

4. Taylor expanded in phi2 around 0

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

   1. lower-fma.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-*.f64 44.0

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

6. Applied rewrites 44.0%

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

7. Taylor expanded in phi1 around 0

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 27.1

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

9. Applied rewrites 27.1%

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

10. Taylor expanded in phi2 around 0

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

    1. lower--.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lift-cos.f64 N/A

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

    4. lift-cos.f64 N/A

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

    5. lift--.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift--.f64 N/A

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

    8. lower-pow.f64 N/A

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

    9. lower-sin.f64 N/A

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

    10. lower-*.f64 27.3

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

12. Applied rewrites 27.3%

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

Reproduce

herbie shell --seed 2025103 
(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))))))))))