Equirectangular approximation to distance on a great circle

Percentage Accurate: 58.6% → 79.4%

Time: 7.1s

Alternatives: 14

Speedup: 0.5×

• 69.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 58.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \mathsf{min}\left(\phi_1, \phi_2\right)\\ t_1 := 0.5 \cdot \mathsf{max}\left(\phi_1, \phi_2\right)\\ t_2 := {\left(\cos t\_0 \cdot \cos t\_1 - \sin t\_0 \cdot \sin t\_1\right)}^{2}\\ t_3 := \mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\\ t_4 := \mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\\ t_5 := \sqrt{t\_2}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+130}:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot \left(-1 \cdot \frac{R \cdot \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot t\_2\right)}{\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot t\_5} + R \cdot t\_5\right)\right)\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_3 - \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{max}\left(\phi_1, \phi_2\right) + \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot -0.5\right)\right)\right) \cdot t\_4\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* 0.5 (fmin phi1 phi2)))
       (t_1 (* 0.5 (fmax phi1 phi2)))
       (t_2
        (pow (- (* (cos t_0) (cos t_1)) (* (sin t_0) (sin t_1))) 2.0))
       (t_3 (- (fmax phi1 phi2) (fmin phi1 phi2)))
       (t_4 (- (fmin lambda1 lambda2) (fmax lambda1 lambda2)))
       (t_5 (sqrt t_2)))
  (if (<= t_4 -1e+130)
    (*
     -1.0
     (*
      (fmin lambda1 lambda2)
      (+
       (*
        -1.0
        (/
         (* R (* (fmax lambda1 lambda2) t_2))
         (* (fmin lambda1 lambda2) t_5)))
       (* R t_5))))
    (if (<= t_4 -2e+63)
      (*
       (sqrt
        (-
         (* t_3 t_3)
         (*
          (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))
          (*
           (+
            0.5
            (*
             0.5
             (cos
              (*
               2.0
               (* (+ (fmax phi1 phi2) (fmin phi1 phi2)) -0.5)))))
           t_4))))
       R)
      (* t_3 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * fmin(phi1, phi2);
	double t_1 = 0.5 * fmax(phi1, phi2);
	double t_2 = pow(((cos(t_0) * cos(t_1)) - (sin(t_0) * sin(t_1))), 2.0);
	double t_3 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double t_4 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_5 = sqrt(t_2);
	double tmp;
	if (t_4 <= -1e+130) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_5))) + (R * t_5)));
	} else if (t_4 <= -2e+63) {
		tmp = sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_4)))) * R;
	} else {
		tmp = t_3 * R;
	}
	return tmp;
}

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
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = 0.5d0 * fmin(phi1, phi2)
    t_1 = 0.5d0 * fmax(phi1, phi2)
    t_2 = ((cos(t_0) * cos(t_1)) - (sin(t_0) * sin(t_1))) ** 2.0d0
    t_3 = fmax(phi1, phi2) - fmin(phi1, phi2)
    t_4 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
    t_5 = sqrt(t_2)
    if (t_4 <= (-1d+130)) then
        tmp = (-1.0d0) * (fmin(lambda1, lambda2) * (((-1.0d0) * ((r * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_5))) + (r * t_5)))
    else if (t_4 <= (-2d+63)) then
        tmp = sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5d0 + (0.5d0 * cos((2.0d0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * (-0.5d0)))))) * t_4)))) * r
    else
        tmp = t_3 * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * fmin(phi1, phi2);
	double t_1 = 0.5 * fmax(phi1, phi2);
	double t_2 = Math.pow(((Math.cos(t_0) * Math.cos(t_1)) - (Math.sin(t_0) * Math.sin(t_1))), 2.0);
	double t_3 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double t_4 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_5 = Math.sqrt(t_2);
	double tmp;
	if (t_4 <= -1e+130) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_5))) + (R * t_5)));
	} else if (t_4 <= -2e+63) {
		tmp = Math.sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * Math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_4)))) * R;
	} else {
		tmp = t_3 * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = 0.5 * fmin(phi1, phi2)
	t_1 = 0.5 * fmax(phi1, phi2)
	t_2 = math.pow(((math.cos(t_0) * math.cos(t_1)) - (math.sin(t_0) * math.sin(t_1))), 2.0)
	t_3 = fmax(phi1, phi2) - fmin(phi1, phi2)
	t_4 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
	t_5 = math.sqrt(t_2)
	tmp = 0
	if t_4 <= -1e+130:
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_5))) + (R * t_5)))
	elif t_4 <= -2e+63:
		tmp = math.sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_4)))) * R
	else:
		tmp = t_3 * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 * fmin(phi1, phi2))
	t_1 = Float64(0.5 * fmax(phi1, phi2))
	t_2 = Float64(Float64(cos(t_0) * cos(t_1)) - Float64(sin(t_0) * sin(t_1))) ^ 2.0
	t_3 = Float64(fmax(phi1, phi2) - fmin(phi1, phi2))
	t_4 = Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2))
	t_5 = sqrt(t_2)
	tmp = 0.0
	if (t_4 <= -1e+130)
		tmp = Float64(-1.0 * Float64(fmin(lambda1, lambda2) * Float64(Float64(-1.0 * Float64(Float64(R * Float64(fmax(lambda1, lambda2) * t_2)) / Float64(fmin(lambda1, lambda2) * t_5))) + Float64(R * t_5))));
	elseif (t_4 <= -2e+63)
		tmp = Float64(sqrt(Float64(Float64(t_3 * t_3) - Float64(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_4)))) * R);
	else
		tmp = Float64(t_3 * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = 0.5 * min(phi1, phi2);
	t_1 = 0.5 * max(phi1, phi2);
	t_2 = ((cos(t_0) * cos(t_1)) - (sin(t_0) * sin(t_1))) ^ 2.0;
	t_3 = max(phi1, phi2) - min(phi1, phi2);
	t_4 = min(lambda1, lambda2) - max(lambda1, lambda2);
	t_5 = sqrt(t_2);
	tmp = 0.0;
	if (t_4 <= -1e+130)
		tmp = -1.0 * (min(lambda1, lambda2) * ((-1.0 * ((R * (max(lambda1, lambda2) * t_2)) / (min(lambda1, lambda2) * t_5))) + (R * t_5)));
	elseif (t_4 <= -2e+63)
		tmp = sqrt(((t_3 * t_3) - ((max(lambda1, lambda2) - min(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((max(phi1, phi2) + min(phi1, phi2)) * -0.5))))) * t_4)))) * R;
	else
		tmp = t_3 * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$2], $MachinePrecision]}, If[LessEqual[t$95$4, -1e+130], N[(-1.0 * N[(N[Min[lambda1, lambda2], $MachinePrecision] * N[(N[(-1.0 * N[(N[(R * N[(N[Max[lambda1, lambda2], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[Min[lambda1, lambda2], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(R * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -2e+63], N[(N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[Max[phi1, phi2], $MachinePrecision] + N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(t$95$3 * R), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -1.0000000000000001e130

   1. Initial program 58.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. cos-sum N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

      23. mult-flip N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 56.0%

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

   3. Applied rewrites 56.0%

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. cos-sum N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

      23. mult-flip N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 59.2%

          \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   5. Applied rewrites 59.2%

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   6. Taylor expanded in lambda1 around -inf

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

   8. Applied rewrites 30.2%

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

   if -1.0000000000000001e130 < (-.f64 lambda1 lambda2) < -2.0000000000000001e63

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   if -2.0000000000000001e63 < (-.f64 lambda1 lambda2)

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := -\mathsf{max}\left(\phi_1, \phi_2\right)\\ t_1 := \mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\\ t_2 := 0.5 + 0.5 \cdot \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \cos t\_0 - \sin \left(-\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin t\_0\right)\\ t_3 := \mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\\ t_4 := \sqrt{--1 \cdot t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+130}:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot \left(-1 \cdot \frac{R \cdot \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot t\_2\right)}{\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot t\_4} + R \cdot t\_4\right)\right)\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{t\_1 \cdot t\_1 - \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{max}\left(\phi_1, \phi_2\right) + \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot -0.5\right)\right)\right) \cdot t\_3\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (- (fmax phi1 phi2)))
       (t_1 (- (fmax phi1 phi2) (fmin phi1 phi2)))
       (t_2
        (+
         0.5
         (*
          0.5
          (-
           (* (cos (fmin phi1 phi2)) (cos t_0))
           (* (sin (- (fmin phi1 phi2))) (sin t_0))))))
       (t_3 (- (fmin lambda1 lambda2) (fmax lambda1 lambda2)))
       (t_4 (sqrt (- (* -1.0 t_2)))))
  (if (<= t_3 -1e+130)
    (*
     -1.0
     (*
      (fmin lambda1 lambda2)
      (+
       (*
        -1.0
        (/
         (* R (* (fmax lambda1 lambda2) t_2))
         (* (fmin lambda1 lambda2) t_4)))
       (* R t_4))))
    (if (<= t_3 -2e+63)
      (*
       (sqrt
        (-
         (* t_1 t_1)
         (*
          (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))
          (*
           (+
            0.5
            (*
             0.5
             (cos
              (*
               2.0
               (* (+ (fmax phi1 phi2) (fmin phi1 phi2)) -0.5)))))
           t_3))))
       R)
      (* t_1 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = -fmax(phi1, phi2);
	double t_1 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double t_2 = 0.5 + (0.5 * ((cos(fmin(phi1, phi2)) * cos(t_0)) - (sin(-fmin(phi1, phi2)) * sin(t_0))));
	double t_3 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_4 = sqrt(-(-1.0 * t_2));
	double tmp;
	if (t_3 <= -1e+130) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_4))) + (R * t_4)));
	} else if (t_3 <= -2e+63) {
		tmp = sqrt(((t_1 * t_1) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_3)))) * R;
	} else {
		tmp = t_1 * R;
	}
	return tmp;
}

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
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = -fmax(phi1, phi2)
    t_1 = fmax(phi1, phi2) - fmin(phi1, phi2)
    t_2 = 0.5d0 + (0.5d0 * ((cos(fmin(phi1, phi2)) * cos(t_0)) - (sin(-fmin(phi1, phi2)) * sin(t_0))))
    t_3 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
    t_4 = sqrt(-((-1.0d0) * t_2))
    if (t_3 <= (-1d+130)) then
        tmp = (-1.0d0) * (fmin(lambda1, lambda2) * (((-1.0d0) * ((r * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_4))) + (r * t_4)))
    else if (t_3 <= (-2d+63)) then
        tmp = sqrt(((t_1 * t_1) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5d0 + (0.5d0 * cos((2.0d0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * (-0.5d0)))))) * t_3)))) * r
    else
        tmp = t_1 * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = -fmax(phi1, phi2);
	double t_1 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double t_2 = 0.5 + (0.5 * ((Math.cos(fmin(phi1, phi2)) * Math.cos(t_0)) - (Math.sin(-fmin(phi1, phi2)) * Math.sin(t_0))));
	double t_3 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_4 = Math.sqrt(-(-1.0 * t_2));
	double tmp;
	if (t_3 <= -1e+130) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_4))) + (R * t_4)));
	} else if (t_3 <= -2e+63) {
		tmp = Math.sqrt(((t_1 * t_1) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * Math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_3)))) * R;
	} else {
		tmp = t_1 * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = -fmax(phi1, phi2)
	t_1 = fmax(phi1, phi2) - fmin(phi1, phi2)
	t_2 = 0.5 + (0.5 * ((math.cos(fmin(phi1, phi2)) * math.cos(t_0)) - (math.sin(-fmin(phi1, phi2)) * math.sin(t_0))))
	t_3 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
	t_4 = math.sqrt(-(-1.0 * t_2))
	tmp = 0
	if t_3 <= -1e+130:
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_4))) + (R * t_4)))
	elif t_3 <= -2e+63:
		tmp = math.sqrt(((t_1 * t_1) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_3)))) * R
	else:
		tmp = t_1 * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(-fmax(phi1, phi2))
	t_1 = Float64(fmax(phi1, phi2) - fmin(phi1, phi2))
	t_2 = Float64(0.5 + Float64(0.5 * Float64(Float64(cos(fmin(phi1, phi2)) * cos(t_0)) - Float64(sin(Float64(-fmin(phi1, phi2))) * sin(t_0)))))
	t_3 = Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2))
	t_4 = sqrt(Float64(-Float64(-1.0 * t_2)))
	tmp = 0.0
	if (t_3 <= -1e+130)
		tmp = Float64(-1.0 * Float64(fmin(lambda1, lambda2) * Float64(Float64(-1.0 * Float64(Float64(R * Float64(fmax(lambda1, lambda2) * t_2)) / Float64(fmin(lambda1, lambda2) * t_4))) + Float64(R * t_4))));
	elseif (t_3 <= -2e+63)
		tmp = Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_3)))) * R);
	else
		tmp = Float64(t_1 * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = -max(phi1, phi2);
	t_1 = max(phi1, phi2) - min(phi1, phi2);
	t_2 = 0.5 + (0.5 * ((cos(min(phi1, phi2)) * cos(t_0)) - (sin(-min(phi1, phi2)) * sin(t_0))));
	t_3 = min(lambda1, lambda2) - max(lambda1, lambda2);
	t_4 = sqrt(-(-1.0 * t_2));
	tmp = 0.0;
	if (t_3 <= -1e+130)
		tmp = -1.0 * (min(lambda1, lambda2) * ((-1.0 * ((R * (max(lambda1, lambda2) * t_2)) / (min(lambda1, lambda2) * t_4))) + (R * t_4)));
	elseif (t_3 <= -2e+63)
		tmp = sqrt(((t_1 * t_1) - ((max(lambda1, lambda2) - min(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((max(phi1, phi2) + min(phi1, phi2)) * -0.5))))) * t_3)))) * R;
	else
		tmp = t_1 * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = (-N[Max[phi1, phi2], $MachinePrecision])}, Block[{t$95$1 = N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 + N[(0.5 * N[(N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[(-N[Min[phi1, phi2], $MachinePrecision])], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[(-N[(-1.0 * t$95$2), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[t$95$3, -1e+130], N[(-1.0 * N[(N[Min[lambda1, lambda2], $MachinePrecision] * N[(N[(-1.0 * N[(N[(R * N[(N[Max[lambda1, lambda2], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[Min[lambda1, lambda2], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(R * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e+63], N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[Max[phi1, phi2], $MachinePrecision] + N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(t$95$1 * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -1.0000000000000001e130

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   4. Taylor expanded in lambda1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 28.3%

      \[\leadsto \color{blue}{-1 \cdot \left(\lambda_1 \cdot \left(-1 \cdot \frac{R \cdot \left(\lambda_2 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)}{\lambda_1 \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}} + R \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. cos-sum N/A

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

      8. lower--.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

      17. mul-1-neg N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-sin.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. mul-1-neg N/A

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

      24. lower-neg.f64 N/A

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

      25. *-commutative N/A

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

      26. mul-1-neg N/A

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

   8. Applied rewrites 28.3%

      \[\leadsto -1 \cdot \left(\lambda_1 \cdot \left(-1 \cdot \frac{R \cdot \left(\lambda_2 \cdot \left(0.5 + 0.5 \cdot \left(\cos \phi_1 \cdot \cos \left(-\phi_2\right) - \sin \left(-\phi_1\right) \cdot \sin \left(-\phi_2\right)\right)\right)\right)}{\lambda_1 \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}} + R \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. cos-sum N/A

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

      8. lower--.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

      17. mul-1-neg N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-sin.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. mul-1-neg N/A

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

      24. lower-neg.f64 N/A

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

      25. *-commutative N/A

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

      26. mul-1-neg N/A

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

   10. Applied rewrites 29.0%

       \[\leadsto -1 \cdot \left(\lambda_1 \cdot \left(-1 \cdot \frac{R \cdot \left(\lambda_2 \cdot \left(0.5 + 0.5 \cdot \left(\cos \phi_1 \cdot \cos \left(-\phi_2\right) - \sin \left(-\phi_1\right) \cdot \sin \left(-\phi_2\right)\right)\right)\right)}{\lambda_1 \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \left(\cos \phi_1 \cdot \cos \left(-\phi_2\right) - \sin \left(-\phi_1\right) \cdot \sin \left(-\phi_2\right)\right)\right)}} + R \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right)\right) \]
   11. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. cos-sum N/A

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

       8. lower--.f64 N/A

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

       9. lift-cos.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-cos.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. mul-1-neg N/A

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

       14. cos-neg N/A

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

       15. lower-cos.f64 N/A

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

       16. *-commutative N/A

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

       17. mul-1-neg N/A

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

       18. lower-cos.f64 N/A

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

       19. lower-neg.f64 N/A

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

       20. lower-*.f64 N/A

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

       21. lower-sin.f64 N/A

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

       22. lift-*.f64 N/A

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

       23. mul-1-neg N/A

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

       24. lower-neg.f64 N/A

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

       25. *-commutative N/A

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

       26. mul-1-neg N/A

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

   12. Applied rewrites 30.1%

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

   if -1.0000000000000001e130 < (-.f64 lambda1 lambda2) < -2.0000000000000001e63

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   if -2.0000000000000001e63 < (-.f64 lambda1 lambda2)

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\phi_1, \phi_2\right) \cdot 0.5\\ t_1 := \mathsf{min}\left(\phi_1, \phi_2\right) - \mathsf{max}\left(\phi_1, \phi_2\right)\\ t_2 := {\cos \left(0.5 \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)}^{2}\\ t_3 := \sqrt{t\_2}\\ t_4 := \mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\\ t_5 := \mathsf{max}\left(\phi_1, \phi_2\right) \cdot 0.5\\ t_6 := t\_4 \cdot \left(\cos t\_5 \cdot \cos t\_0 - \sin t\_5 \cdot \sin t\_0\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot \left(-1 \cdot \frac{R \cdot \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot t\_2\right)}{\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot t\_3} + R \cdot t\_3\right)\right)\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \sqrt{t\_6 \cdot t\_6 + t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (fmin phi1 phi2) 0.5))
       (t_1 (- (fmin phi1 phi2) (fmax phi1 phi2)))
       (t_2
        (pow (cos (* 0.5 (+ (fmin phi1 phi2) (fmax phi1 phi2)))) 2.0))
       (t_3 (sqrt t_2))
       (t_4 (- (fmin lambda1 lambda2) (fmax lambda1 lambda2)))
       (t_5 (* (fmax phi1 phi2) 0.5))
       (t_6
        (* t_4 (- (* (cos t_5) (cos t_0)) (* (sin t_5) (sin t_0))))))
  (if (<= t_4 -5e+132)
    (*
     -1.0
     (*
      (fmin lambda1 lambda2)
      (+
       (*
        -1.0
        (/
         (* R (* (fmax lambda1 lambda2) t_2))
         (* (fmin lambda1 lambda2) t_3)))
       (* R t_3))))
    (if (<= t_4 -2e+63)
      (* R (sqrt (+ (* t_6 t_6) (* t_1 t_1))))
      (* (- (fmax phi1 phi2) (fmin phi1 phi2)) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fmin(phi1, phi2) * 0.5;
	double t_1 = fmin(phi1, phi2) - fmax(phi1, phi2);
	double t_2 = pow(cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0);
	double t_3 = sqrt(t_2);
	double t_4 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_5 = fmax(phi1, phi2) * 0.5;
	double t_6 = t_4 * ((cos(t_5) * cos(t_0)) - (sin(t_5) * sin(t_0)));
	double tmp;
	if (t_4 <= -5e+132) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_3))) + (R * t_3)));
	} else if (t_4 <= -2e+63) {
		tmp = R * sqrt(((t_6 * t_6) + (t_1 * t_1)));
	} else {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	}
	return tmp;
}

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
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = fmin(phi1, phi2) * 0.5d0
    t_1 = fmin(phi1, phi2) - fmax(phi1, phi2)
    t_2 = cos((0.5d0 * (fmin(phi1, phi2) + fmax(phi1, phi2)))) ** 2.0d0
    t_3 = sqrt(t_2)
    t_4 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
    t_5 = fmax(phi1, phi2) * 0.5d0
    t_6 = t_4 * ((cos(t_5) * cos(t_0)) - (sin(t_5) * sin(t_0)))
    if (t_4 <= (-5d+132)) then
        tmp = (-1.0d0) * (fmin(lambda1, lambda2) * (((-1.0d0) * ((r * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_3))) + (r * t_3)))
    else if (t_4 <= (-2d+63)) then
        tmp = r * sqrt(((t_6 * t_6) + (t_1 * t_1)))
    else
        tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fmin(phi1, phi2) * 0.5;
	double t_1 = fmin(phi1, phi2) - fmax(phi1, phi2);
	double t_2 = Math.pow(Math.cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0);
	double t_3 = Math.sqrt(t_2);
	double t_4 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_5 = fmax(phi1, phi2) * 0.5;
	double t_6 = t_4 * ((Math.cos(t_5) * Math.cos(t_0)) - (Math.sin(t_5) * Math.sin(t_0)));
	double tmp;
	if (t_4 <= -5e+132) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_3))) + (R * t_3)));
	} else if (t_4 <= -2e+63) {
		tmp = R * Math.sqrt(((t_6 * t_6) + (t_1 * t_1)));
	} else {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = fmin(phi1, phi2) * 0.5
	t_1 = fmin(phi1, phi2) - fmax(phi1, phi2)
	t_2 = math.pow(math.cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0)
	t_3 = math.sqrt(t_2)
	t_4 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
	t_5 = fmax(phi1, phi2) * 0.5
	t_6 = t_4 * ((math.cos(t_5) * math.cos(t_0)) - (math.sin(t_5) * math.sin(t_0)))
	tmp = 0
	if t_4 <= -5e+132:
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_3))) + (R * t_3)))
	elif t_4 <= -2e+63:
		tmp = R * math.sqrt(((t_6 * t_6) + (t_1 * t_1)))
	else:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(fmin(phi1, phi2) * 0.5)
	t_1 = Float64(fmin(phi1, phi2) - fmax(phi1, phi2))
	t_2 = cos(Float64(0.5 * Float64(fmin(phi1, phi2) + fmax(phi1, phi2)))) ^ 2.0
	t_3 = sqrt(t_2)
	t_4 = Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2))
	t_5 = Float64(fmax(phi1, phi2) * 0.5)
	t_6 = Float64(t_4 * Float64(Float64(cos(t_5) * cos(t_0)) - Float64(sin(t_5) * sin(t_0))))
	tmp = 0.0
	if (t_4 <= -5e+132)
		tmp = Float64(-1.0 * Float64(fmin(lambda1, lambda2) * Float64(Float64(-1.0 * Float64(Float64(R * Float64(fmax(lambda1, lambda2) * t_2)) / Float64(fmin(lambda1, lambda2) * t_3))) + Float64(R * t_3))));
	elseif (t_4 <= -2e+63)
		tmp = Float64(R * sqrt(Float64(Float64(t_6 * t_6) + Float64(t_1 * t_1))));
	else
		tmp = Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = min(phi1, phi2) * 0.5;
	t_1 = min(phi1, phi2) - max(phi1, phi2);
	t_2 = cos((0.5 * (min(phi1, phi2) + max(phi1, phi2)))) ^ 2.0;
	t_3 = sqrt(t_2);
	t_4 = min(lambda1, lambda2) - max(lambda1, lambda2);
	t_5 = max(phi1, phi2) * 0.5;
	t_6 = t_4 * ((cos(t_5) * cos(t_0)) - (sin(t_5) * sin(t_0)));
	tmp = 0.0;
	if (t_4 <= -5e+132)
		tmp = -1.0 * (min(lambda1, lambda2) * ((-1.0 * ((R * (max(lambda1, lambda2) * t_2)) / (min(lambda1, lambda2) * t_3))) + (R * t_3)));
	elseif (t_4 <= -2e+63)
		tmp = R * sqrt(((t_6 * t_6) + (t_1 * t_1)));
	else
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Min[phi1, phi2], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[phi1, phi2], $MachinePrecision] - N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Cos[N[(0.5 * N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Max[phi1, phi2], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * N[(N[(N[Cos[t$95$5], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$5], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+132], N[(-1.0 * N[(N[Min[lambda1, lambda2], $MachinePrecision] * N[(N[(-1.0 * N[(N[(R * N[(N[Max[lambda1, lambda2], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[Min[lambda1, lambda2], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(R * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -2e+63], N[(R * N[Sqrt[N[(N[(t$95$6 * t$95$6), $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -5.0000000000000001e132

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi1 around inf

      \[\leadsto \color{blue}{R \cdot \phi_1} \]
   3. Step-by-step derivation

      1. lower-*.f64 17.3%

         \[\leadsto R \cdot \color{blue}{\phi_1} \]

   4. Applied rewrites 17.3%

      \[\leadsto \color{blue}{R \cdot \phi_1} \]

   5. Taylor expanded in lambda1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

   7. Applied rewrites 28.3%

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

   if -5.0000000000000001e132 < (-.f64 lambda1 lambda2) < -2.0000000000000001e63

   1. Initial program 58.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. cos-sum N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

      23. mult-flip N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 56.0%

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

   3. Applied rewrites 56.0%

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. cos-sum N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

      23. mult-flip N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 59.2%

          \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   5. Applied rewrites 59.2%

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   if -2.0000000000000001e63 < (-.f64 lambda1 lambda2)

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := {\cos \left(0.5 \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)}^{2}\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\\ t_3 := \mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot \left(-1 \cdot \frac{R \cdot \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot t\_0\right)}{\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot t\_1} + R \cdot t\_1\right)\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_3 - \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{max}\left(\phi_1, \phi_2\right) + \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot -0.5\right)\right)\right) \cdot t\_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (pow (cos (* 0.5 (+ (fmin phi1 phi2) (fmax phi1 phi2)))) 2.0))
       (t_1 (sqrt t_0))
       (t_2 (- (fmin lambda1 lambda2) (fmax lambda1 lambda2)))
       (t_3 (- (fmax phi1 phi2) (fmin phi1 phi2))))
  (if (<= t_2 -5e+132)
    (*
     -1.0
     (*
      (fmin lambda1 lambda2)
      (+
       (*
        -1.0
        (/
         (* R (* (fmax lambda1 lambda2) t_0))
         (* (fmin lambda1 lambda2) t_1)))
       (* R t_1))))
    (if (<= t_2 -2e+63)
      (*
       (sqrt
        (-
         (* t_3 t_3)
         (*
          (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))
          (*
           (+
            0.5
            (*
             0.5
             (cos
              (*
               2.0
               (* (+ (fmax phi1 phi2) (fmin phi1 phi2)) -0.5)))))
           t_2))))
       R)
      (* t_3 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0);
	double t_1 = sqrt(t_0);
	double t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_3 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (t_2 <= -5e+132) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_0)) / (fmin(lambda1, lambda2) * t_1))) + (R * t_1)));
	} else if (t_2 <= -2e+63) {
		tmp = sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R;
	} else {
		tmp = t_3 * R;
	}
	return tmp;
}

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
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos((0.5d0 * (fmin(phi1, phi2) + fmax(phi1, phi2)))) ** 2.0d0
    t_1 = sqrt(t_0)
    t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
    t_3 = fmax(phi1, phi2) - fmin(phi1, phi2)
    if (t_2 <= (-5d+132)) then
        tmp = (-1.0d0) * (fmin(lambda1, lambda2) * (((-1.0d0) * ((r * (fmax(lambda1, lambda2) * t_0)) / (fmin(lambda1, lambda2) * t_1))) + (r * t_1)))
    else if (t_2 <= (-2d+63)) then
        tmp = sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5d0 + (0.5d0 * cos((2.0d0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * (-0.5d0)))))) * t_2)))) * r
    else
        tmp = t_3 * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0);
	double t_1 = Math.sqrt(t_0);
	double t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_3 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (t_2 <= -5e+132) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_0)) / (fmin(lambda1, lambda2) * t_1))) + (R * t_1)));
	} else if (t_2 <= -2e+63) {
		tmp = Math.sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * Math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R;
	} else {
		tmp = t_3 * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0)
	t_1 = math.sqrt(t_0)
	t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
	t_3 = fmax(phi1, phi2) - fmin(phi1, phi2)
	tmp = 0
	if t_2 <= -5e+132:
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_0)) / (fmin(lambda1, lambda2) * t_1))) + (R * t_1)))
	elif t_2 <= -2e+63:
		tmp = math.sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R
	else:
		tmp = t_3 * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * Float64(fmin(phi1, phi2) + fmax(phi1, phi2)))) ^ 2.0
	t_1 = sqrt(t_0)
	t_2 = Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2))
	t_3 = Float64(fmax(phi1, phi2) - fmin(phi1, phi2))
	tmp = 0.0
	if (t_2 <= -5e+132)
		tmp = Float64(-1.0 * Float64(fmin(lambda1, lambda2) * Float64(Float64(-1.0 * Float64(Float64(R * Float64(fmax(lambda1, lambda2) * t_0)) / Float64(fmin(lambda1, lambda2) * t_1))) + Float64(R * t_1))));
	elseif (t_2 <= -2e+63)
		tmp = Float64(sqrt(Float64(Float64(t_3 * t_3) - Float64(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R);
	else
		tmp = Float64(t_3 * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * (min(phi1, phi2) + max(phi1, phi2)))) ^ 2.0;
	t_1 = sqrt(t_0);
	t_2 = min(lambda1, lambda2) - max(lambda1, lambda2);
	t_3 = max(phi1, phi2) - min(phi1, phi2);
	tmp = 0.0;
	if (t_2 <= -5e+132)
		tmp = -1.0 * (min(lambda1, lambda2) * ((-1.0 * ((R * (max(lambda1, lambda2) * t_0)) / (min(lambda1, lambda2) * t_1))) + (R * t_1)));
	elseif (t_2 <= -2e+63)
		tmp = sqrt(((t_3 * t_3) - ((max(lambda1, lambda2) - min(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((max(phi1, phi2) + min(phi1, phi2)) * -0.5))))) * t_2)))) * R;
	else
		tmp = t_3 * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Cos[N[(0.5 * N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+132], N[(-1.0 * N[(N[Min[lambda1, lambda2], $MachinePrecision] * N[(N[(-1.0 * N[(N[(R * N[(N[Max[lambda1, lambda2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Min[lambda1, lambda2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(R * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+63], N[(N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[Max[phi1, phi2], $MachinePrecision] + N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(t$95$3 * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -5.0000000000000001e132

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi1 around inf

      \[\leadsto \color{blue}{R \cdot \phi_1} \]
   3. Step-by-step derivation

      1. lower-*.f64 17.3%

         \[\leadsto R \cdot \color{blue}{\phi_1} \]

   4. Applied rewrites 17.3%

      \[\leadsto \color{blue}{R \cdot \phi_1} \]

   5. Taylor expanded in lambda1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

   7. Applied rewrites 28.3%

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

   if -5.0000000000000001e132 < (-.f64 lambda1 lambda2) < -2.0000000000000001e63

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   if -2.0000000000000001e63 < (-.f64 lambda1 lambda2)

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\right) + 1\right) \cdot 0.5\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\\ t_3 := \mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\left(t\_1 \cdot R - \frac{\left(R \cdot \mathsf{max}\left(\lambda_1, \lambda_2\right)\right) \cdot t\_0}{t\_1 \cdot \mathsf{min}\left(\lambda_1, \lambda_2\right)}\right) \cdot \left(-\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_3 - \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{max}\left(\phi_1, \phi_2\right) + \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot -0.5\right)\right)\right) \cdot t\_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (* (+ (cos (+ (fmin phi1 phi2) (fmax phi1 phi2))) 1.0) 0.5))
       (t_1 (sqrt t_0))
       (t_2 (- (fmin lambda1 lambda2) (fmax lambda1 lambda2)))
       (t_3 (- (fmax phi1 phi2) (fmin phi1 phi2))))
  (if (<= t_2 -5e+132)
    (*
     (-
      (* t_1 R)
      (/
       (* (* R (fmax lambda1 lambda2)) t_0)
       (* t_1 (fmin lambda1 lambda2))))
     (- (fmin lambda1 lambda2)))
    (if (<= t_2 -2e+63)
      (*
       (sqrt
        (-
         (* t_3 t_3)
         (*
          (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))
          (*
           (+
            0.5
            (*
             0.5
             (cos
              (*
               2.0
               (* (+ (fmax phi1 phi2) (fmin phi1 phi2)) -0.5)))))
           t_2))))
       R)
      (* t_3 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos((fmin(phi1, phi2) + fmax(phi1, phi2))) + 1.0) * 0.5;
	double t_1 = sqrt(t_0);
	double t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_3 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (t_2 <= -5e+132) {
		tmp = ((t_1 * R) - (((R * fmax(lambda1, lambda2)) * t_0) / (t_1 * fmin(lambda1, lambda2)))) * -fmin(lambda1, lambda2);
	} else if (t_2 <= -2e+63) {
		tmp = sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R;
	} else {
		tmp = t_3 * R;
	}
	return tmp;
}

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
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (cos((fmin(phi1, phi2) + fmax(phi1, phi2))) + 1.0d0) * 0.5d0
    t_1 = sqrt(t_0)
    t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
    t_3 = fmax(phi1, phi2) - fmin(phi1, phi2)
    if (t_2 <= (-5d+132)) then
        tmp = ((t_1 * r) - (((r * fmax(lambda1, lambda2)) * t_0) / (t_1 * fmin(lambda1, lambda2)))) * -fmin(lambda1, lambda2)
    else if (t_2 <= (-2d+63)) then
        tmp = sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5d0 + (0.5d0 * cos((2.0d0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * (-0.5d0)))))) * t_2)))) * r
    else
        tmp = t_3 * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (Math.cos((fmin(phi1, phi2) + fmax(phi1, phi2))) + 1.0) * 0.5;
	double t_1 = Math.sqrt(t_0);
	double t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_3 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (t_2 <= -5e+132) {
		tmp = ((t_1 * R) - (((R * fmax(lambda1, lambda2)) * t_0) / (t_1 * fmin(lambda1, lambda2)))) * -fmin(lambda1, lambda2);
	} else if (t_2 <= -2e+63) {
		tmp = Math.sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * Math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R;
	} else {
		tmp = t_3 * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.cos((fmin(phi1, phi2) + fmax(phi1, phi2))) + 1.0) * 0.5
	t_1 = math.sqrt(t_0)
	t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
	t_3 = fmax(phi1, phi2) - fmin(phi1, phi2)
	tmp = 0
	if t_2 <= -5e+132:
		tmp = ((t_1 * R) - (((R * fmax(lambda1, lambda2)) * t_0) / (t_1 * fmin(lambda1, lambda2)))) * -fmin(lambda1, lambda2)
	elif t_2 <= -2e+63:
		tmp = math.sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R
	else:
		tmp = t_3 * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(Float64(fmin(phi1, phi2) + fmax(phi1, phi2))) + 1.0) * 0.5)
	t_1 = sqrt(t_0)
	t_2 = Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2))
	t_3 = Float64(fmax(phi1, phi2) - fmin(phi1, phi2))
	tmp = 0.0
	if (t_2 <= -5e+132)
		tmp = Float64(Float64(Float64(t_1 * R) - Float64(Float64(Float64(R * fmax(lambda1, lambda2)) * t_0) / Float64(t_1 * fmin(lambda1, lambda2)))) * Float64(-fmin(lambda1, lambda2)));
	elseif (t_2 <= -2e+63)
		tmp = Float64(sqrt(Float64(Float64(t_3 * t_3) - Float64(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R);
	else
		tmp = Float64(t_3 * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (cos((min(phi1, phi2) + max(phi1, phi2))) + 1.0) * 0.5;
	t_1 = sqrt(t_0);
	t_2 = min(lambda1, lambda2) - max(lambda1, lambda2);
	t_3 = max(phi1, phi2) - min(phi1, phi2);
	tmp = 0.0;
	if (t_2 <= -5e+132)
		tmp = ((t_1 * R) - (((R * max(lambda1, lambda2)) * t_0) / (t_1 * min(lambda1, lambda2)))) * -min(lambda1, lambda2);
	elseif (t_2 <= -2e+63)
		tmp = sqrt(((t_3 * t_3) - ((max(lambda1, lambda2) - min(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((max(phi1, phi2) + min(phi1, phi2)) * -0.5))))) * t_2)))) * R;
	else
		tmp = t_3 * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+132], N[(N[(N[(t$95$1 * R), $MachinePrecision] - N[(N[(N[(R * N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(t$95$1 * N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Min[lambda1, lambda2], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, -2e+63], N[(N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[Max[phi1, phi2], $MachinePrecision] + N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(t$95$3 * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -5.0000000000000001e132

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   4. Taylor expanded in lambda1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 28.3%

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

   8. Applied rewrites 28.3%

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

   if -5.0000000000000001e132 < (-.f64 lambda1 lambda2) < -2.0000000000000001e63

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   if -2.0000000000000001e63 < (-.f64 lambda1 lambda2)

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \left|\cos t\_0\right|\\ t_2 := \mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\\ t_3 := \mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+167}:\\ \;\;\;\;\left(\left(t\_1 - \frac{0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)}{t\_1 \cdot \mathsf{max}\left(\lambda_1, \lambda_2\right)} \cdot \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \mathsf{max}\left(\lambda_1, \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_3 - \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{max}\left(\phi_1, \phi_2\right) + \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot -0.5\right)\right)\right) \cdot t\_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* -0.5 (+ (fmin phi1 phi2) (fmax phi1 phi2))))
       (t_1 (fabs (cos t_0)))
       (t_2 (- (fmin lambda1 lambda2) (fmax lambda1 lambda2)))
       (t_3 (- (fmax phi1 phi2) (fmin phi1 phi2))))
  (if (<= t_2 -1e+167)
    (*
     (*
      (-
       t_1
       (*
        (/
         (+ 0.5 (* 0.5 (cos (* 2.0 t_0))))
         (* t_1 (fmax lambda1 lambda2)))
        (fmin lambda1 lambda2)))
      (fmax lambda1 lambda2))
     R)
    (if (<= t_2 -2e+63)
      (*
       (sqrt
        (-
         (* t_3 t_3)
         (*
          (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))
          (*
           (+
            0.5
            (*
             0.5
             (cos
              (*
               2.0
               (* (+ (fmax phi1 phi2) (fmin phi1 phi2)) -0.5)))))
           t_2))))
       R)
      (* t_3 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = -0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2));
	double t_1 = fabs(cos(t_0));
	double t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_3 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (t_2 <= -1e+167) {
		tmp = ((t_1 - (((0.5 + (0.5 * cos((2.0 * t_0)))) / (t_1 * fmax(lambda1, lambda2))) * fmin(lambda1, lambda2))) * fmax(lambda1, lambda2)) * R;
	} else if (t_2 <= -2e+63) {
		tmp = sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R;
	} else {
		tmp = t_3 * R;
	}
	return tmp;
}

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
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (-0.5d0) * (fmin(phi1, phi2) + fmax(phi1, phi2))
    t_1 = abs(cos(t_0))
    t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
    t_3 = fmax(phi1, phi2) - fmin(phi1, phi2)
    if (t_2 <= (-1d+167)) then
        tmp = ((t_1 - (((0.5d0 + (0.5d0 * cos((2.0d0 * t_0)))) / (t_1 * fmax(lambda1, lambda2))) * fmin(lambda1, lambda2))) * fmax(lambda1, lambda2)) * r
    else if (t_2 <= (-2d+63)) then
        tmp = sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5d0 + (0.5d0 * cos((2.0d0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * (-0.5d0)))))) * t_2)))) * r
    else
        tmp = t_3 * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = -0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2));
	double t_1 = Math.abs(Math.cos(t_0));
	double t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2);
	double t_3 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (t_2 <= -1e+167) {
		tmp = ((t_1 - (((0.5 + (0.5 * Math.cos((2.0 * t_0)))) / (t_1 * fmax(lambda1, lambda2))) * fmin(lambda1, lambda2))) * fmax(lambda1, lambda2)) * R;
	} else if (t_2 <= -2e+63) {
		tmp = Math.sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * Math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R;
	} else {
		tmp = t_3 * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = -0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2))
	t_1 = math.fabs(math.cos(t_0))
	t_2 = fmin(lambda1, lambda2) - fmax(lambda1, lambda2)
	t_3 = fmax(phi1, phi2) - fmin(phi1, phi2)
	tmp = 0
	if t_2 <= -1e+167:
		tmp = ((t_1 - (((0.5 + (0.5 * math.cos((2.0 * t_0)))) / (t_1 * fmax(lambda1, lambda2))) * fmin(lambda1, lambda2))) * fmax(lambda1, lambda2)) * R
	elif t_2 <= -2e+63:
		tmp = math.sqrt(((t_3 * t_3) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R
	else:
		tmp = t_3 * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(-0.5 * Float64(fmin(phi1, phi2) + fmax(phi1, phi2)))
	t_1 = abs(cos(t_0))
	t_2 = Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2))
	t_3 = Float64(fmax(phi1, phi2) - fmin(phi1, phi2))
	tmp = 0.0
	if (t_2 <= -1e+167)
		tmp = Float64(Float64(Float64(t_1 - Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))) / Float64(t_1 * fmax(lambda1, lambda2))) * fmin(lambda1, lambda2))) * fmax(lambda1, lambda2)) * R);
	elseif (t_2 <= -2e+63)
		tmp = Float64(sqrt(Float64(Float64(t_3 * t_3) - Float64(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * t_2)))) * R);
	else
		tmp = Float64(t_3 * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = -0.5 * (min(phi1, phi2) + max(phi1, phi2));
	t_1 = abs(cos(t_0));
	t_2 = min(lambda1, lambda2) - max(lambda1, lambda2);
	t_3 = max(phi1, phi2) - min(phi1, phi2);
	tmp = 0.0;
	if (t_2 <= -1e+167)
		tmp = ((t_1 - (((0.5 + (0.5 * cos((2.0 * t_0)))) / (t_1 * max(lambda1, lambda2))) * min(lambda1, lambda2))) * max(lambda1, lambda2)) * R;
	elseif (t_2 <= -2e+63)
		tmp = sqrt(((t_3 * t_3) - ((max(lambda1, lambda2) - min(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((max(phi1, phi2) + min(phi1, phi2)) * -0.5))))) * t_2)))) * R;
	else
		tmp = t_3 * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(-0.5 * N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+167], N[(N[(N[(t$95$1 - N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[t$95$2, -2e+63], N[(N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[Max[phi1, phi2], $MachinePrecision] + N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(t$95$3 * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -1e167

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   4. Applied rewrites 26.6%

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

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\left(\left(\left|\cos \left(-0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right| - \frac{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\left|\cos \left(-0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right| \cdot \lambda_2} \cdot \lambda_1\right) \cdot \lambda_2\right) \cdot R} \]

   if -1e167 < (-.f64 lambda1 lambda2) < -2.0000000000000001e63

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   if -2.0000000000000001e63 < (-.f64 lambda1 lambda2)

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\\ t_1 := \mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -7.5 \cdot 10^{+132}:\\ \;\;\;\;\left(\left(-\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left|\cos \left(t\_0 \cdot -0.5\right)\right|\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{t\_1 \cdot t\_1 - \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{max}\left(\phi_1, \phi_2\right) + \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot -0.5\right)\right)\right) \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)} \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq 5.7 \cdot 10^{-62}:\\ \;\;\;\;t\_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot t\_0\right)\right)}\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (+ (fmin phi1 phi2) (fmax phi1 phi2)))
       (t_1 (- (fmax phi1 phi2) (fmin phi1 phi2))))
  (if (<= (fmin lambda1 lambda2) -7.5e+132)
    (* (* (- (fmin lambda1 lambda2)) (fabs (cos (* t_0 -0.5)))) R)
    (if (<= (fmin lambda1 lambda2) -2.5e+63)
      (*
       (sqrt
        (-
         (* t_1 t_1)
         (*
          (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))
          (*
           (+
            0.5
            (*
             0.5
             (cos
              (*
               2.0
               (* (+ (fmax phi1 phi2) (fmin phi1 phi2)) -0.5)))))
           (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))))))
       R)
      (if (<= (fmin lambda1 lambda2) 5.7e-62)
        (* t_1 R)
        (*
         (*
          (fmax lambda1 lambda2)
          (sqrt (- (* -1.0 (+ 0.5 (* 0.5 (cos (* -1.0 t_0))))))))
         R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fmin(phi1, phi2) + fmax(phi1, phi2);
	double t_1 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (fmin(lambda1, lambda2) <= -7.5e+132) {
		tmp = (-fmin(lambda1, lambda2) * fabs(cos((t_0 * -0.5)))) * R;
	} else if (fmin(lambda1, lambda2) <= -2.5e+63) {
		tmp = sqrt(((t_1 * t_1) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * (fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R;
	} else if (fmin(lambda1, lambda2) <= 5.7e-62) {
		tmp = t_1 * R;
	} else {
		tmp = (fmax(lambda1, lambda2) * sqrt(-(-1.0 * (0.5 + (0.5 * cos((-1.0 * t_0))))))) * R;
	}
	return tmp;
}

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) :: tmp
    t_0 = fmin(phi1, phi2) + fmax(phi1, phi2)
    t_1 = fmax(phi1, phi2) - fmin(phi1, phi2)
    if (fmin(lambda1, lambda2) <= (-7.5d+132)) then
        tmp = (-fmin(lambda1, lambda2) * abs(cos((t_0 * (-0.5d0))))) * r
    else if (fmin(lambda1, lambda2) <= (-2.5d+63)) then
        tmp = sqrt(((t_1 * t_1) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5d0 + (0.5d0 * cos((2.0d0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * (-0.5d0)))))) * (fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * r
    else if (fmin(lambda1, lambda2) <= 5.7d-62) then
        tmp = t_1 * r
    else
        tmp = (fmax(lambda1, lambda2) * sqrt(-((-1.0d0) * (0.5d0 + (0.5d0 * cos(((-1.0d0) * t_0))))))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fmin(phi1, phi2) + fmax(phi1, phi2);
	double t_1 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (fmin(lambda1, lambda2) <= -7.5e+132) {
		tmp = (-fmin(lambda1, lambda2) * Math.abs(Math.cos((t_0 * -0.5)))) * R;
	} else if (fmin(lambda1, lambda2) <= -2.5e+63) {
		tmp = Math.sqrt(((t_1 * t_1) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * Math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * (fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R;
	} else if (fmin(lambda1, lambda2) <= 5.7e-62) {
		tmp = t_1 * R;
	} else {
		tmp = (fmax(lambda1, lambda2) * Math.sqrt(-(-1.0 * (0.5 + (0.5 * Math.cos((-1.0 * t_0))))))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = fmin(phi1, phi2) + fmax(phi1, phi2)
	t_1 = fmax(phi1, phi2) - fmin(phi1, phi2)
	tmp = 0
	if fmin(lambda1, lambda2) <= -7.5e+132:
		tmp = (-fmin(lambda1, lambda2) * math.fabs(math.cos((t_0 * -0.5)))) * R
	elif fmin(lambda1, lambda2) <= -2.5e+63:
		tmp = math.sqrt(((t_1 * t_1) - ((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * ((0.5 + (0.5 * math.cos((2.0 * ((fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * (fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R
	elif fmin(lambda1, lambda2) <= 5.7e-62:
		tmp = t_1 * R
	else:
		tmp = (fmax(lambda1, lambda2) * math.sqrt(-(-1.0 * (0.5 + (0.5 * math.cos((-1.0 * t_0))))))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(fmin(phi1, phi2) + fmax(phi1, phi2))
	t_1 = Float64(fmax(phi1, phi2) - fmin(phi1, phi2))
	tmp = 0.0
	if (fmin(lambda1, lambda2) <= -7.5e+132)
		tmp = Float64(Float64(Float64(-fmin(lambda1, lambda2)) * abs(cos(Float64(t_0 * -0.5)))) * R);
	elseif (fmin(lambda1, lambda2) <= -2.5e+63)
		tmp = Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2)) * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(fmax(phi1, phi2) + fmin(phi1, phi2)) * -0.5))))) * Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R);
	elseif (fmin(lambda1, lambda2) <= 5.7e-62)
		tmp = Float64(t_1 * R);
	else
		tmp = Float64(Float64(fmax(lambda1, lambda2) * sqrt(Float64(-Float64(-1.0 * Float64(0.5 + Float64(0.5 * cos(Float64(-1.0 * t_0)))))))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = min(phi1, phi2) + max(phi1, phi2);
	t_1 = max(phi1, phi2) - min(phi1, phi2);
	tmp = 0.0;
	if (min(lambda1, lambda2) <= -7.5e+132)
		tmp = (-min(lambda1, lambda2) * abs(cos((t_0 * -0.5)))) * R;
	elseif (min(lambda1, lambda2) <= -2.5e+63)
		tmp = sqrt(((t_1 * t_1) - ((max(lambda1, lambda2) - min(lambda1, lambda2)) * ((0.5 + (0.5 * cos((2.0 * ((max(phi1, phi2) + min(phi1, phi2)) * -0.5))))) * (min(lambda1, lambda2) - max(lambda1, lambda2)))))) * R;
	elseif (min(lambda1, lambda2) <= 5.7e-62)
		tmp = t_1 * R;
	else
		tmp = (max(lambda1, lambda2) * sqrt(-(-1.0 * (0.5 + (0.5 * cos((-1.0 * t_0))))))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -7.5e+132], N[(N[((-N[Min[lambda1, lambda2], $MachinePrecision]) * N[Abs[N[Cos[N[(t$95$0 * -0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -2.5e+63], N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[Max[phi1, phi2], $MachinePrecision] + N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], 5.7e-62], N[(t$95$1 * R), $MachinePrecision], N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] * N[Sqrt[(-N[(-1.0 * N[(0.5 + N[(0.5 * N[Cos[N[(-1.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -7.5000000000000002e132

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

   5. Taylor expanded in lambda1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 17.4%

         \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_1 \cdot \sqrt{{\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right)\right) \]

   7. Applied rewrites 17.4%

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

   9. Applied rewrites 17.4%

      \[\leadsto \color{blue}{\left(\left(-\lambda_1\right) \cdot \left|\cos \left(\left(\phi_1 + \phi_2\right) \cdot -0.5\right)\right|\right) \cdot R} \]

   if -7.5000000000000002e132 < lambda1 < -2.5000000000000001e63

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   if -2.5000000000000001e63 < lambda1 < 5.6999999999999999e-62

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]

   if 5.6999999999999999e-62 < lambda1

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   4. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \sqrt{\mathsf{neg}\left(-1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)}\right)} \cdot R \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{\mathsf{neg}\left(-1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)}\right) \cdot R \]

      3. lower-neg.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      9. lower-+.f64 16.9%

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

   6. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right)} \cdot R \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -2.6 \cdot 10^{+63}:\\ \;\;\;\;\left(\left(-\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left|\cos \left(t\_0 \cdot -0.5\right)\right|\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq 5.7 \cdot 10^{-62}:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot t\_0\right)\right)}\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (+ (fmin phi1 phi2) (fmax phi1 phi2))))
  (if (<= (fmin lambda1 lambda2) -2.6e+63)
    (* (* (- (fmin lambda1 lambda2)) (fabs (cos (* t_0 -0.5)))) R)
    (if (<= (fmin lambda1 lambda2) 5.7e-62)
      (* (- (fmax phi1 phi2) (fmin phi1 phi2)) R)
      (*
       (*
        (fmax lambda1 lambda2)
        (sqrt (- (* -1.0 (+ 0.5 (* 0.5 (cos (* -1.0 t_0))))))))
       R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fmin(phi1, phi2) + fmax(phi1, phi2);
	double tmp;
	if (fmin(lambda1, lambda2) <= -2.6e+63) {
		tmp = (-fmin(lambda1, lambda2) * fabs(cos((t_0 * -0.5)))) * R;
	} else if (fmin(lambda1, lambda2) <= 5.7e-62) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	} else {
		tmp = (fmax(lambda1, lambda2) * sqrt(-(-1.0 * (0.5 + (0.5 * cos((-1.0 * t_0))))))) * R;
	}
	return tmp;
}

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) :: tmp
    t_0 = fmin(phi1, phi2) + fmax(phi1, phi2)
    if (fmin(lambda1, lambda2) <= (-2.6d+63)) then
        tmp = (-fmin(lambda1, lambda2) * abs(cos((t_0 * (-0.5d0))))) * r
    else if (fmin(lambda1, lambda2) <= 5.7d-62) then
        tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * r
    else
        tmp = (fmax(lambda1, lambda2) * sqrt(-((-1.0d0) * (0.5d0 + (0.5d0 * cos(((-1.0d0) * t_0))))))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fmin(phi1, phi2) + fmax(phi1, phi2);
	double tmp;
	if (fmin(lambda1, lambda2) <= -2.6e+63) {
		tmp = (-fmin(lambda1, lambda2) * Math.abs(Math.cos((t_0 * -0.5)))) * R;
	} else if (fmin(lambda1, lambda2) <= 5.7e-62) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	} else {
		tmp = (fmax(lambda1, lambda2) * Math.sqrt(-(-1.0 * (0.5 + (0.5 * Math.cos((-1.0 * t_0))))))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = fmin(phi1, phi2) + fmax(phi1, phi2)
	tmp = 0
	if fmin(lambda1, lambda2) <= -2.6e+63:
		tmp = (-fmin(lambda1, lambda2) * math.fabs(math.cos((t_0 * -0.5)))) * R
	elif fmin(lambda1, lambda2) <= 5.7e-62:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R
	else:
		tmp = (fmax(lambda1, lambda2) * math.sqrt(-(-1.0 * (0.5 + (0.5 * math.cos((-1.0 * t_0))))))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(fmin(phi1, phi2) + fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(lambda1, lambda2) <= -2.6e+63)
		tmp = Float64(Float64(Float64(-fmin(lambda1, lambda2)) * abs(cos(Float64(t_0 * -0.5)))) * R);
	elseif (fmin(lambda1, lambda2) <= 5.7e-62)
		tmp = Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * R);
	else
		tmp = Float64(Float64(fmax(lambda1, lambda2) * sqrt(Float64(-Float64(-1.0 * Float64(0.5 + Float64(0.5 * cos(Float64(-1.0 * t_0)))))))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = min(phi1, phi2) + max(phi1, phi2);
	tmp = 0.0;
	if (min(lambda1, lambda2) <= -2.6e+63)
		tmp = (-min(lambda1, lambda2) * abs(cos((t_0 * -0.5)))) * R;
	elseif (min(lambda1, lambda2) <= 5.7e-62)
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * R;
	else
		tmp = (max(lambda1, lambda2) * sqrt(-(-1.0 * (0.5 + (0.5 * cos((-1.0 * t_0))))))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -2.6e+63], N[(N[((-N[Min[lambda1, lambda2], $MachinePrecision]) * N[Abs[N[Cos[N[(t$95$0 * -0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], 5.7e-62], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] * N[Sqrt[(-N[(-1.0 * N[(0.5 + N[(0.5 * N[Cos[N[(-1.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -2.6000000000000001e63

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

   5. Taylor expanded in lambda1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 17.4%

         \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_1 \cdot \sqrt{{\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right)\right) \]

   7. Applied rewrites 17.4%

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

   9. Applied rewrites 17.4%

      \[\leadsto \color{blue}{\left(\left(-\lambda_1\right) \cdot \left|\cos \left(\left(\phi_1 + \phi_2\right) \cdot -0.5\right)\right|\right) \cdot R} \]

   if -2.6000000000000001e63 < lambda1 < 5.6999999999999999e-62

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]

   if 5.6999999999999999e-62 < lambda1

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   4. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \sqrt{\mathsf{neg}\left(-1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)}\right)} \cdot R \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{\mathsf{neg}\left(-1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)}\right) \cdot R \]

      3. lower-neg.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

      9. lower-+.f64 16.9%

         \[\leadsto \left(\lambda_2 \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right) \cdot R \]

   6. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \sqrt{--1 \cdot \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right)} \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -2.6 \cdot 10^{+63}:\\ \;\;\;\;\left(\left(-\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left|\cos \left(\left(\mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot -0.5\right)\right|\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (if (<= (fmin lambda1 lambda2) -2.6e+63)
  (*
   (*
    (- (fmin lambda1 lambda2))
    (fabs (cos (* (+ (fmin phi1 phi2) (fmax phi1 phi2)) -0.5))))
   R)
  (* (- (fmax phi1 phi2) (fmin phi1 phi2)) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(lambda1, lambda2) <= -2.6e+63) {
		tmp = (-fmin(lambda1, lambda2) * fabs(cos(((fmin(phi1, phi2) + fmax(phi1, phi2)) * -0.5)))) * R;
	} else {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	}
	return tmp;
}

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) :: tmp
    if (fmin(lambda1, lambda2) <= (-2.6d+63)) then
        tmp = (-fmin(lambda1, lambda2) * abs(cos(((fmin(phi1, phi2) + fmax(phi1, phi2)) * (-0.5d0))))) * r
    else
        tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(lambda1, lambda2) <= -2.6e+63) {
		tmp = (-fmin(lambda1, lambda2) * Math.abs(Math.cos(((fmin(phi1, phi2) + fmax(phi1, phi2)) * -0.5)))) * R;
	} else {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmin(lambda1, lambda2) <= -2.6e+63:
		tmp = (-fmin(lambda1, lambda2) * math.fabs(math.cos(((fmin(phi1, phi2) + fmax(phi1, phi2)) * -0.5)))) * R
	else:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(lambda1, lambda2) <= -2.6e+63)
		tmp = Float64(Float64(Float64(-fmin(lambda1, lambda2)) * abs(cos(Float64(Float64(fmin(phi1, phi2) + fmax(phi1, phi2)) * -0.5)))) * R);
	else
		tmp = Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (min(lambda1, lambda2) <= -2.6e+63)
		tmp = (-min(lambda1, lambda2) * abs(cos(((min(phi1, phi2) + max(phi1, phi2)) * -0.5)))) * R;
	else
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -2.6e+63], N[(N[((-N[Min[lambda1, lambda2], $MachinePrecision]) * N[Abs[N[Cos[N[(N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.6000000000000001e63

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

   5. Taylor expanded in lambda1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 17.4%

         \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_1 \cdot \sqrt{{\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right)\right) \]

   7. Applied rewrites 17.4%

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

   9. Applied rewrites 17.4%

      \[\leadsto \color{blue}{\left(\left(-\lambda_1\right) \cdot \left|\cos \left(\left(\phi_1 + \phi_2\right) \cdot -0.5\right)\right|\right) \cdot R} \]

   if -2.6000000000000001e63 < lambda1

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1.12 \cdot 10^{+154}:\\ \;\;\;\;t\_0 \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 1.5 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{t\_0 \cdot t\_0 - \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_2 - \lambda_1\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \left(R + -1 \cdot \frac{R \cdot \mathsf{max}\left(\phi_1, \phi_2\right)}{\mathsf{min}\left(\phi_1, \phi_2\right)}\right)\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (- (fmax phi1 phi2) (fmin phi1 phi2))))
  (if (<= (fmin phi1 phi2) -1.12e+154)
    (* t_0 R)
    (if (<= (fmin phi1 phi2) 1.5e-206)
      (*
       (sqrt
        (- (* t_0 t_0) (* (- lambda1 lambda2) (- lambda2 lambda1))))
       R)
      (*
       -1.0
       (*
        (fmin phi1 phi2)
        (+
         R
         (* -1.0 (/ (* R (fmax phi1 phi2)) (fmin phi1 phi2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (fmin(phi1, phi2) <= -1.12e+154) {
		tmp = t_0 * R;
	} else if (fmin(phi1, phi2) <= 1.5e-206) {
		tmp = sqrt(((t_0 * t_0) - ((lambda1 - lambda2) * (lambda2 - lambda1)))) * R;
	} else {
		tmp = -1.0 * (fmin(phi1, phi2) * (R + (-1.0 * ((R * fmax(phi1, phi2)) / fmin(phi1, phi2)))));
	}
	return tmp;
}

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) :: tmp
    t_0 = fmax(phi1, phi2) - fmin(phi1, phi2)
    if (fmin(phi1, phi2) <= (-1.12d+154)) then
        tmp = t_0 * r
    else if (fmin(phi1, phi2) <= 1.5d-206) then
        tmp = sqrt(((t_0 * t_0) - ((lambda1 - lambda2) * (lambda2 - lambda1)))) * r
    else
        tmp = (-1.0d0) * (fmin(phi1, phi2) * (r + ((-1.0d0) * ((r * fmax(phi1, phi2)) / fmin(phi1, phi2)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (fmin(phi1, phi2) <= -1.12e+154) {
		tmp = t_0 * R;
	} else if (fmin(phi1, phi2) <= 1.5e-206) {
		tmp = Math.sqrt(((t_0 * t_0) - ((lambda1 - lambda2) * (lambda2 - lambda1)))) * R;
	} else {
		tmp = -1.0 * (fmin(phi1, phi2) * (R + (-1.0 * ((R * fmax(phi1, phi2)) / fmin(phi1, phi2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = fmax(phi1, phi2) - fmin(phi1, phi2)
	tmp = 0
	if fmin(phi1, phi2) <= -1.12e+154:
		tmp = t_0 * R
	elif fmin(phi1, phi2) <= 1.5e-206:
		tmp = math.sqrt(((t_0 * t_0) - ((lambda1 - lambda2) * (lambda2 - lambda1)))) * R
	else:
		tmp = -1.0 * (fmin(phi1, phi2) * (R + (-1.0 * ((R * fmax(phi1, phi2)) / fmin(phi1, phi2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(fmax(phi1, phi2) - fmin(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1.12e+154)
		tmp = Float64(t_0 * R);
	elseif (fmin(phi1, phi2) <= 1.5e-206)
		tmp = Float64(sqrt(Float64(Float64(t_0 * t_0) - Float64(Float64(lambda1 - lambda2) * Float64(lambda2 - lambda1)))) * R);
	else
		tmp = Float64(-1.0 * Float64(fmin(phi1, phi2) * Float64(R + Float64(-1.0 * Float64(Float64(R * fmax(phi1, phi2)) / fmin(phi1, phi2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = max(phi1, phi2) - min(phi1, phi2);
	tmp = 0.0;
	if (min(phi1, phi2) <= -1.12e+154)
		tmp = t_0 * R;
	elseif (min(phi1, phi2) <= 1.5e-206)
		tmp = sqrt(((t_0 * t_0) - ((lambda1 - lambda2) * (lambda2 - lambda1)))) * R;
	else
		tmp = -1.0 * (min(phi1, phi2) * (R + (-1.0 * ((R * max(phi1, phi2)) / min(phi1, phi2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1.12e+154], N[(t$95$0 * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 1.5e-206], N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(-1.0 * N[(N[Min[phi1, phi2], $MachinePrecision] * N[(R + N[(-1.0 * N[(N[(R * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision] / N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.1199999999999999e154

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]

   if -1.1199999999999999e154 < phi1 < 1.5000000000000001e-206

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_2 - \lambda_1\right) \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R} \]

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 57.2%

         \[\leadsto \sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \phi_1\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_2 - \color{blue}{\lambda_1}\right)\right)} \cdot R \]

   6. Applied rewrites 57.2%

      \[\leadsto \sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(-1 \cdot \phi_1\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)}} \cdot R \]

   7. Taylor expanded in phi1 around 0

      \[\leadsto \sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\lambda_2 - \lambda_1\right)}} \cdot R \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_2 - \color{blue}{\lambda_1}\right)} \cdot R \]

      2. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_2 - \lambda_1\right)} \cdot R \]

      3. lower--.f64 55.9%

         \[\leadsto \sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_2 - \lambda_1\right)} \cdot R \]

   9. Applied rewrites 55.9%

      \[\leadsto \sqrt{\left(\phi_2 - \phi_1\right) \cdot \left(\phi_2 - \phi_1\right) - \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\lambda_2 - \lambda_1\right)}} \cdot R \]

   if 1.5000000000000001e-206 < phi1

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\phi_1 \cdot \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(\phi_1 \cdot \left(R + \color{blue}{-1 \cdot \frac{R \cdot \phi_2}{\phi_1}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \color{blue}{\frac{R \cdot \phi_2}{\phi_1}}\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\color{blue}{\phi_1}}\right)\right) \]

      6. lower-*.f64 29.1%

         \[\leadsto -1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right) \]

   4. Applied rewrites 29.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 10^{+207}:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{max}\left(\phi_1, \phi_2\right) \cdot \left(R + -1 \cdot \frac{R \cdot \mathsf{min}\left(\phi_1, \phi_2\right)}{\mathsf{max}\left(\phi_1, \phi_2\right)}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (if (<= (fmax lambda1 lambda2) 1e+207)
  (* (- (fmax phi1 phi2) (fmin phi1 phi2)) R)
  (*
   (fmax phi1 phi2)
   (+ R (* -1.0 (/ (* R (fmin phi1 phi2)) (fmax phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(lambda1, lambda2) <= 1e+207) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	} else {
		tmp = fmax(phi1, phi2) * (R + (-1.0 * ((R * fmin(phi1, phi2)) / fmax(phi1, phi2))));
	}
	return tmp;
}

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) :: tmp
    if (fmax(lambda1, lambda2) <= 1d+207) then
        tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * r
    else
        tmp = fmax(phi1, phi2) * (r + ((-1.0d0) * ((r * fmin(phi1, phi2)) / fmax(phi1, phi2))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(lambda1, lambda2) <= 1e+207) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	} else {
		tmp = fmax(phi1, phi2) * (R + (-1.0 * ((R * fmin(phi1, phi2)) / fmax(phi1, phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmax(lambda1, lambda2) <= 1e+207:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R
	else:
		tmp = fmax(phi1, phi2) * (R + (-1.0 * ((R * fmin(phi1, phi2)) / fmax(phi1, phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 1e+207)
		tmp = Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * R);
	else
		tmp = Float64(fmax(phi1, phi2) * Float64(R + Float64(-1.0 * Float64(Float64(R * fmin(phi1, phi2)) / fmax(phi1, phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (max(lambda1, lambda2) <= 1e+207)
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * R;
	else
		tmp = max(phi1, phi2) * (R + (-1.0 * ((R * min(phi1, phi2)) / max(phi1, phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 1e+207], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], N[(N[Max[phi1, phi2], $MachinePrecision] * N[(R + N[(-1.0 * N[(N[(R * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] / N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1e207

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

      4. lower-/.f64 27.9%

         \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

   4. Applied rewrites 27.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

      3. lower-*.f64 27.9%

         \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]

   if 1e207 < lambda2

   1. Initial program 58.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \phi_2 \cdot \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \phi_2 \cdot \left(R + \color{blue}{-1 \cdot \frac{R \cdot \phi_1}{\phi_2}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \phi_2 \cdot \left(R + -1 \cdot \color{blue}{\frac{R \cdot \phi_1}{\phi_2}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\color{blue}{\phi_2}}\right) \]

      5. lower-*.f64 29.5%

         \[\leadsto \phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right) \]

   4. Applied rewrites 29.5%

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.7% accurate, 1.3× speedup

Math

\[\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (* (- (fmax phi1 phi2) (fmin phi1 phi2)) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
}

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
    code = (fmax(phi1, phi2) - fmin(phi1, phi2)) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return (fmax(phi1, phi2) - fmin(phi1, phi2)) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = (max(phi1, phi2) - min(phi1, phi2)) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 58.6%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

2. Taylor expanded in phi2 around inf

   \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_1}{\phi_2}}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right) \]

   4. lower-/.f64 27.9%

      \[\leadsto R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\color{blue}{\phi_2}}\right)\right) \]

4. Applied rewrites 27.9%

   \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{R \cdot \left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

   3. lower-*.f64 27.9%

      \[\leadsto \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right) \cdot R} \]

6. Applied rewrites 29.9%

   \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]
7. Add Preprocessing

Alternative 13: 31.6% accurate, 2.6× speedup

Math

\[R \cdot \mathsf{max}\left(\phi_1, \phi_2\right) \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (* R (fmax phi1 phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * fmax(phi1, phi2);
}

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
    code = r * fmax(phi1, phi2)
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * fmax(phi1, phi2);
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * fmax(phi1, phi2)

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * fmax(phi1, phi2))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * max(phi1, phi2);
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.6%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

2. Taylor expanded in phi2 around inf

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Step-by-step derivation

   1. lower-*.f64 18.0%

      \[\leadsto R \cdot \color{blue}{\phi_2} \]

4. Applied rewrites 18.0%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
5. Add Preprocessing

Alternative 14: 17.3% accurate, 46.5× speedup

Math

\[R \cdot \phi_1 \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (* R phi1))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi1;
}

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
    code = r * phi1
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi1;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * phi1

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * phi1)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * phi1;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * phi1), $MachinePrecision]

Derivation

1. Initial program 58.6%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

2. Taylor expanded in phi1 around inf

   \[\leadsto \color{blue}{R \cdot \phi_1} \]
3. Step-by-step derivation

   1. lower-*.f64 17.3%

      \[\leadsto R \cdot \color{blue}{\phi_1} \]

4. Applied rewrites 17.3%

   \[\leadsto \color{blue}{R \cdot \phi_1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))