Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.3% → 76.1%

Time: 23.3s

Alternatives: 15

Speedup: 0.5×

• 69.7% 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)))))
  (* 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), $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: 60.3% 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)))))
  (* 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), $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: 76.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := -\mathsf{max}\left(\phi_1, \phi_2\right)\\ t_1 := -\mathsf{min}\left(\phi_1, \phi_2\right)\\ t_2 := \frac{1}{2} + \frac{1}{2} \cdot \left(\cos t\_1 \cdot \cos t\_0 - \sin t\_1 \cdot \sin t\_0\right)\\ t_3 := \sqrt{--1 \cdot t\_2}\\ \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right) \leq -200000000000000015034897383303641725494285812870481642696581820471553185048483040932908220219551607085653191007770505265335500800:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot \left(-1 \cdot \frac{R \cdot \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot t\_2\right)}{\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot t\_3} + R \cdot t\_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (- (fmax phi1 phi2)))
       (t_1 (- (fmin phi1 phi2)))
       (t_2
        (+
         1/2
         (* 1/2 (- (* (cos t_1) (cos t_0)) (* (sin t_1) (sin t_0))))))
       (t_3 (sqrt (- (* -1 t_2)))))
  (if (<=
       (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))
       -200000000000000015034897383303641725494285812870481642696581820471553185048483040932908220219551607085653191007770505265335500800)
    (*
     -1
     (*
      (fmin lambda1 lambda2)
      (+
       (*
        -1
        (/
         (* R (* (fmax lambda1 lambda2) t_2))
         (* (fmin lambda1 lambda2) t_3)))
       (* R t_3))))
    (* (- (fmax phi1 phi2) (fmin phi1 phi2)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = -fmax(phi1, phi2);
	double t_1 = -fmin(phi1, phi2);
	double t_2 = 0.5 + (0.5 * ((cos(t_1) * cos(t_0)) - (sin(t_1) * sin(t_0))));
	double t_3 = sqrt(-(-1.0 * t_2));
	double tmp;
	if ((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_3))) + (R * t_3)));
	} 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) :: tmp
    t_0 = -fmax(phi1, phi2)
    t_1 = -fmin(phi1, phi2)
    t_2 = 0.5d0 + (0.5d0 * ((cos(t_1) * cos(t_0)) - (sin(t_1) * sin(t_0))))
    t_3 = sqrt(-((-1.0d0) * t_2))
    if ((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= (-2d+128)) then
        tmp = (-1.0d0) * (fmin(lambda1, lambda2) * (((-1.0d0) * ((r * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_3))) + (r * t_3)))
    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 = -fmax(phi1, phi2);
	double t_1 = -fmin(phi1, phi2);
	double t_2 = 0.5 + (0.5 * ((Math.cos(t_1) * Math.cos(t_0)) - (Math.sin(t_1) * Math.sin(t_0))));
	double t_3 = Math.sqrt(-(-1.0 * t_2));
	double tmp;
	if ((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_3))) + (R * t_3)));
	} else {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = -fmax(phi1, phi2)
	t_1 = -fmin(phi1, phi2)
	t_2 = 0.5 + (0.5 * ((math.cos(t_1) * math.cos(t_0)) - (math.sin(t_1) * math.sin(t_0))))
	t_3 = math.sqrt(-(-1.0 * t_2))
	tmp = 0
	if (fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128:
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_2)) / (fmin(lambda1, lambda2) * t_3))) + (R * t_3)))
	else:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(-fmax(phi1, phi2))
	t_1 = Float64(-fmin(phi1, phi2))
	t_2 = Float64(0.5 + Float64(0.5 * Float64(Float64(cos(t_1) * cos(t_0)) - Float64(sin(t_1) * sin(t_0)))))
	t_3 = sqrt(Float64(-Float64(-1.0 * t_2)))
	tmp = 0.0
	if (Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128)
		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))));
	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 = -max(phi1, phi2);
	t_1 = -min(phi1, phi2);
	t_2 = 0.5 + (0.5 * ((cos(t_1) * cos(t_0)) - (sin(t_1) * sin(t_0))));
	t_3 = sqrt(-(-1.0 * t_2));
	tmp = 0.0;
	if ((min(lambda1, lambda2) - max(lambda1, lambda2)) <= -2e+128)
		tmp = -1.0 * (min(lambda1, lambda2) * ((-1.0 * ((R * (max(lambda1, lambda2) * t_2)) / (min(lambda1, lambda2) * t_3))) + (R * t_3)));
	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[Max[phi1, phi2], $MachinePrecision])}, Block[{t$95$1 = (-N[Min[phi1, phi2], $MachinePrecision])}, Block[{t$95$2 = N[(1/2 + N[(1/2 * N[(N[(N[Cos[t$95$1], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$1], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-N[(-1 * t$95$2), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision], -200000000000000015034897383303641725494285812870481642696581820471553185048483040932908220219551607085653191007770505265335500800], N[(-1 * N[(N[Min[lambda1, lambda2], $MachinePrecision] * N[(N[(-1 * 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], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -2.0000000000000002e128

   1. Initial program 60.3%

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

   3. Applied rewrites 60.3%

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

   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 27.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\lambda_1 \cdot \left(-1 \cdot \frac{R \cdot \left(\lambda_2 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)}{\lambda_1 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}} + R \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right)\right)} \]
   7. 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. mul-1-neg N/A

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

      7. cos-sum N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. mul-1-neg N/A

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

      21. lower-sin.f64 N/A

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

      22. lower-neg.f64 27.4%

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

   8. Applied rewrites 27.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. cos-sum N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. mul-1-neg N/A

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

      21. lower-sin.f64 N/A

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

      22. lower-neg.f64 28.2%

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

   10. Applied rewrites 28.2%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. mul-1-neg N/A

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

       7. cos-sum N/A

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-cos.f64 N/A

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

       11. lower-neg.f64 N/A

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

       12. *-commutative N/A

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

       13. mul-1-neg N/A

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

       14. lower-cos.f64 N/A

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

       15. lower-neg.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-sin.f64 N/A

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

       18. lower-neg.f64 N/A

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

       19. *-commutative N/A

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

       20. mul-1-neg N/A

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

       21. lower-sin.f64 N/A

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

       22. lower-neg.f64 29.1%

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

   12. Applied rewrites 29.1%

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

   if -2.0000000000000002e128 < (-.f64 lambda1 lambda2)

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

Alternative 2: 73.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := {\cos \left(\frac{1}{2} \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)}^{2}\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right) \leq -200000000000000015034897383303641725494285812870481642696581820471553185048483040932908220219551607085653191007770505265335500800:\\ \;\;\;\;-1 \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot \left(-1 \cdot \frac{R \cdot \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot t\_0\right)}{\mathsf{min}\left(\lambda_1, \lambda_2\right) \cdot t\_1} + R \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0
        (pow (cos (* 1/2 (+ (fmin phi1 phi2) (fmax phi1 phi2)))) 2))
       (t_1 (sqrt t_0)))
  (if (<=
       (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))
       -200000000000000015034897383303641725494285812870481642696581820471553185048483040932908220219551607085653191007770505265335500800)
    (*
     -1
     (*
      (fmin lambda1 lambda2)
      (+
       (*
        -1
        (/
         (* R (* (fmax lambda1 lambda2) t_0))
         (* (fmin lambda1 lambda2) t_1)))
       (* R 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 = pow(cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0);
	double t_1 = sqrt(t_0);
	double tmp;
	if ((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_0)) / (fmin(lambda1, lambda2) * t_1))) + (R * 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) :: tmp
    t_0 = cos((0.5d0 * (fmin(phi1, phi2) + fmax(phi1, phi2)))) ** 2.0d0
    t_1 = sqrt(t_0)
    if ((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= (-2d+128)) then
        tmp = (-1.0d0) * (fmin(lambda1, lambda2) * (((-1.0d0) * ((r * (fmax(lambda1, lambda2) * t_0)) / (fmin(lambda1, lambda2) * t_1))) + (r * 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 = Math.pow(Math.cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0);
	double t_1 = Math.sqrt(t_0);
	double tmp;
	if ((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128) {
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_0)) / (fmin(lambda1, lambda2) * t_1))) + (R * t_1)));
	} else {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * 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)
	tmp = 0
	if (fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128:
		tmp = -1.0 * (fmin(lambda1, lambda2) * ((-1.0 * ((R * (fmax(lambda1, lambda2) * t_0)) / (fmin(lambda1, lambda2) * t_1))) + (R * t_1)))
	else:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * 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)
	tmp = 0.0
	if (Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128)
		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))));
	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 = cos((0.5 * (min(phi1, phi2) + max(phi1, phi2)))) ^ 2.0;
	t_1 = sqrt(t_0);
	tmp = 0.0;
	if ((min(lambda1, lambda2) - max(lambda1, lambda2)) <= -2e+128)
		tmp = -1.0 * (min(lambda1, lambda2) * ((-1.0 * ((R * (max(lambda1, lambda2) * t_0)) / (min(lambda1, lambda2) * t_1))) + (R * 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[Power[N[Cos[N[(1/2 * N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision], -200000000000000015034897383303641725494285812870481642696581820471553185048483040932908220219551607085653191007770505265335500800], N[(-1 * N[(N[Min[lambda1, lambda2], $MachinePrecision] * N[(N[(-1 * 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], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -2.0000000000000002e128

   1. Initial program 60.3%

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

   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 29.4%

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.2%

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

   7. Applied rewrites 28.2%

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

   8. Taylor expanded in lambda1 around -inf

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

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

   10. Applied rewrites 27.4%

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

   if -2.0000000000000002e128 < (-.f64 lambda1 lambda2)

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

Alternative 3: 73.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right) + \mathsf{min}\left(\phi_1, \phi_2\right)\right) + 1\right) \cdot \frac{1}{2}\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right) \leq -200000000000000015034897383303641725494285812870481642696581820471553185048483040932908220219551607085653191007770505265335500800:\\ \;\;\;\;\left(t\_1 \cdot R - \frac{\left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot R\right) \cdot t\_0}{t\_1 \cdot \mathsf{min}\left(\lambda_1, \lambda_2\right)}\right) \cdot \left(-\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (* (+ (cos (+ (fmax phi1 phi2) (fmin phi1 phi2))) 1) 1/2))
       (t_1 (sqrt t_0)))
  (if (<=
       (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))
       -200000000000000015034897383303641725494285812870481642696581820471553185048483040932908220219551607085653191007770505265335500800)
    (*
     (-
      (* t_1 R)
      (/
       (* (* (fmax lambda1 lambda2) R) t_0)
       (* t_1 (fmin lambda1 lambda2))))
     (- (fmin lambda1 lambda2)))
    (* (- (fmax phi1 phi2) (fmin phi1 phi2)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos((fmax(phi1, phi2) + fmin(phi1, phi2))) + 1.0) * 0.5;
	double t_1 = sqrt(t_0);
	double tmp;
	if ((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128) {
		tmp = ((t_1 * R) - (((fmax(lambda1, lambda2) * R) * t_0) / (t_1 * fmin(lambda1, lambda2)))) * -fmin(lambda1, lambda2);
	} 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) :: tmp
    t_0 = (cos((fmax(phi1, phi2) + fmin(phi1, phi2))) + 1.0d0) * 0.5d0
    t_1 = sqrt(t_0)
    if ((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= (-2d+128)) then
        tmp = ((t_1 * r) - (((fmax(lambda1, lambda2) * r) * t_0) / (t_1 * fmin(lambda1, lambda2)))) * -fmin(lambda1, lambda2)
    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 = (Math.cos((fmax(phi1, phi2) + fmin(phi1, phi2))) + 1.0) * 0.5;
	double t_1 = Math.sqrt(t_0);
	double tmp;
	if ((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128) {
		tmp = ((t_1 * R) - (((fmax(lambda1, lambda2) * R) * t_0) / (t_1 * fmin(lambda1, lambda2)))) * -fmin(lambda1, lambda2);
	} else {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.cos((fmax(phi1, phi2) + fmin(phi1, phi2))) + 1.0) * 0.5
	t_1 = math.sqrt(t_0)
	tmp = 0
	if (fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128:
		tmp = ((t_1 * R) - (((fmax(lambda1, lambda2) * R) * t_0) / (t_1 * fmin(lambda1, lambda2)))) * -fmin(lambda1, lambda2)
	else:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(Float64(fmax(phi1, phi2) + fmin(phi1, phi2))) + 1.0) * 0.5)
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2)) <= -2e+128)
		tmp = Float64(Float64(Float64(t_1 * R) - Float64(Float64(Float64(fmax(lambda1, lambda2) * R) * t_0) / Float64(t_1 * fmin(lambda1, lambda2)))) * Float64(-fmin(lambda1, lambda2)));
	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 = (cos((max(phi1, phi2) + min(phi1, phi2))) + 1.0) * 0.5;
	t_1 = sqrt(t_0);
	tmp = 0.0;
	if ((min(lambda1, lambda2) - max(lambda1, lambda2)) <= -2e+128)
		tmp = ((t_1 * R) - (((max(lambda1, lambda2) * R) * t_0) / (t_1 * min(lambda1, lambda2)))) * -min(lambda1, lambda2);
	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[(N[Cos[N[(N[Max[phi1, phi2], $MachinePrecision] + N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1), $MachinePrecision] * 1/2), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision], -200000000000000015034897383303641725494285812870481642696581820471553185048483040932908220219551607085653191007770505265335500800], N[(N[(N[(t$95$1 * R), $MachinePrecision] - N[(N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] * R), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(t$95$1 * N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Min[lambda1, lambda2], $MachinePrecision])), $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 (-.f64 lambda1 lambda2) < -2.0000000000000002e128

   1. Initial program 60.3%

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

   3. Applied rewrites 60.3%

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

   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 27.4%

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

   8. Applied rewrites 27.4%

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

   if -2.0000000000000002e128 < (-.f64 lambda1 lambda2)

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

Alternative 4: 68.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\\ \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq \frac{6430607204255973}{5846006549323611672814739330865132078623730171904}:\\ \;\;\;\;t\_0 \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 39999999999999999274034828753599231458870603857312684991833593479596290217520213192823213697572550705053433856:\\ \;\;\;\;\sqrt{t\_0 \cdot t\_0 - \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\mathsf{max}\left(\phi_1, \phi_2\right) + \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)}^{2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (- (fmax phi1 phi2) (fmin phi1 phi2))))
  (if (<=
       (fmax lambda1 lambda2)
       6430607204255973/5846006549323611672814739330865132078623730171904)
    (* t_0 R)
    (if (<=
         (fmax lambda1 lambda2)
         39999999999999999274034828753599231458870603857312684991833593479596290217520213192823213697572550705053433856)
      (*
       (sqrt
        (-
         (* t_0 t_0)
         (*
          (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))
          (*
           (+
            1/2
            (*
             1/2
             (cos
              (* 2 (* (+ (fmax phi1 phi2) (fmin phi1 phi2)) -1/2)))))
           (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))))))
       R)
      (*
       R
       (*
        (fmax lambda1 lambda2)
        (sqrt
         (pow
          (cos (* 1/2 (+ (fmin phi1 phi2) (fmax phi1 phi2))))
          2))))))))

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 (fmax(lambda1, lambda2) <= 1.1e-33) {
		tmp = t_0 * R;
	} else if (fmax(lambda1, lambda2) <= 4e+109) {
		tmp = sqrt(((t_0 * t_0) - ((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 {
		tmp = R * (fmax(lambda1, lambda2) * sqrt(pow(cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0)));
	}
	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 (fmax(lambda1, lambda2) <= 1.1d-33) then
        tmp = t_0 * r
    else if (fmax(lambda1, lambda2) <= 4d+109) then
        tmp = sqrt(((t_0 * t_0) - ((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
        tmp = r * (fmax(lambda1, lambda2) * sqrt((cos((0.5d0 * (fmin(phi1, phi2) + fmax(phi1, phi2)))) ** 2.0d0)))
    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 (fmax(lambda1, lambda2) <= 1.1e-33) {
		tmp = t_0 * R;
	} else if (fmax(lambda1, lambda2) <= 4e+109) {
		tmp = Math.sqrt(((t_0 * t_0) - ((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 {
		tmp = R * (fmax(lambda1, lambda2) * Math.sqrt(Math.pow(Math.cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = fmax(phi1, phi2) - fmin(phi1, phi2)
	tmp = 0
	if fmax(lambda1, lambda2) <= 1.1e-33:
		tmp = t_0 * R
	elif fmax(lambda1, lambda2) <= 4e+109:
		tmp = math.sqrt(((t_0 * t_0) - ((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:
		tmp = R * (fmax(lambda1, lambda2) * math.sqrt(math.pow(math.cos((0.5 * (fmin(phi1, phi2) + fmax(phi1, phi2)))), 2.0)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(fmax(phi1, phi2) - fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 1.1e-33)
		tmp = Float64(t_0 * R);
	elseif (fmax(lambda1, lambda2) <= 4e+109)
		tmp = Float64(sqrt(Float64(Float64(t_0 * t_0) - 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);
	else
		tmp = Float64(R * Float64(fmax(lambda1, lambda2) * sqrt((cos(Float64(0.5 * Float64(fmin(phi1, phi2) + fmax(phi1, phi2)))) ^ 2.0))));
	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 (max(lambda1, lambda2) <= 1.1e-33)
		tmp = t_0 * R;
	elseif (max(lambda1, lambda2) <= 4e+109)
		tmp = sqrt(((t_0 * t_0) - ((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;
	else
		tmp = R * (max(lambda1, lambda2) * sqrt((cos((0.5 * (min(phi1, phi2) + max(phi1, phi2)))) ^ 2.0)));
	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[Max[lambda1, lambda2], $MachinePrecision], 6430607204255973/5846006549323611672814739330865132078623730171904], N[(t$95$0 * R), $MachinePrecision], If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 39999999999999999274034828753599231458870603857312684991833593479596290217520213192823213697572550705053433856], N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * N[(N[(1/2 + N[(1/2 * N[Cos[N[(2 * N[(N[(N[Max[phi1, phi2], $MachinePrecision] + N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(R * N[(N[Max[lambda1, lambda2], $MachinePrecision] * N[Sqrt[N[Power[N[Cos[N[(1/2 * N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 1.1e-33

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

   if 1.1e-33 < lambda2 < 3.9999999999999999e109

   1. Initial program 60.3%

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

   3. Applied rewrites 60.3%

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

   if 3.9999999999999999e109 < lambda2

   1. Initial program 60.3%

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

   2. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 16.7%

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

   4. Applied rewrites 16.7%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\\ \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq \frac{6430607204255973}{5846006549323611672814739330865132078623730171904}:\\ \;\;\;\;t\_0 \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 39999999999999999274034828753599231458870603857312684991833593479596290217520213192823213697572550705053433856:\\ \;\;\;\;\sqrt{t\_0 \cdot t\_0 - \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\mathsf{max}\left(\phi_1, \phi_2\right) + \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right)}\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (let* ((t_0 (- (fmax phi1 phi2) (fmin phi1 phi2))))
  (if (<=
       (fmax lambda1 lambda2)
       6430607204255973/5846006549323611672814739330865132078623730171904)
    (* t_0 R)
    (if (<=
         (fmax lambda1 lambda2)
         39999999999999999274034828753599231458870603857312684991833593479596290217520213192823213697572550705053433856)
      (*
       (sqrt
        (-
         (* t_0 t_0)
         (*
          (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))
          (*
           (+
            1/2
            (*
             1/2
             (cos
              (* 2 (* (+ (fmax phi1 phi2) (fmin phi1 phi2)) -1/2)))))
           (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))))))
       R)
      (*
       (*
        (fmax lambda1 lambda2)
        (sqrt
         (-
          (*
           -1
           (+
            1/2
            (*
             1/2
             (cos (* -1 (+ (fmin phi1 phi2) (fmax phi1 phi2))))))))))
       R)))))

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 (fmax(lambda1, lambda2) <= 1.1e-33) {
		tmp = t_0 * R;
	} else if (fmax(lambda1, lambda2) <= 4e+109) {
		tmp = sqrt(((t_0 * t_0) - ((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 {
		tmp = (fmax(lambda1, lambda2) * sqrt(-(-1.0 * (0.5 + (0.5 * cos((-1.0 * (fmin(phi1, phi2) + fmax(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) :: tmp
    t_0 = fmax(phi1, phi2) - fmin(phi1, phi2)
    if (fmax(lambda1, lambda2) <= 1.1d-33) then
        tmp = t_0 * r
    else if (fmax(lambda1, lambda2) <= 4d+109) then
        tmp = sqrt(((t_0 * t_0) - ((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
        tmp = (fmax(lambda1, lambda2) * sqrt(-((-1.0d0) * (0.5d0 + (0.5d0 * cos(((-1.0d0) * (fmin(phi1, phi2) + fmax(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 = fmax(phi1, phi2) - fmin(phi1, phi2);
	double tmp;
	if (fmax(lambda1, lambda2) <= 1.1e-33) {
		tmp = t_0 * R;
	} else if (fmax(lambda1, lambda2) <= 4e+109) {
		tmp = Math.sqrt(((t_0 * t_0) - ((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 {
		tmp = (fmax(lambda1, lambda2) * Math.sqrt(-(-1.0 * (0.5 + (0.5 * Math.cos((-1.0 * (fmin(phi1, phi2) + fmax(phi1, phi2))))))))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = fmax(phi1, phi2) - fmin(phi1, phi2)
	tmp = 0
	if fmax(lambda1, lambda2) <= 1.1e-33:
		tmp = t_0 * R
	elif fmax(lambda1, lambda2) <= 4e+109:
		tmp = math.sqrt(((t_0 * t_0) - ((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:
		tmp = (fmax(lambda1, lambda2) * math.sqrt(-(-1.0 * (0.5 + (0.5 * math.cos((-1.0 * (fmin(phi1, phi2) + fmax(phi1, phi2))))))))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(fmax(phi1, phi2) - fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 1.1e-33)
		tmp = Float64(t_0 * R);
	elseif (fmax(lambda1, lambda2) <= 4e+109)
		tmp = Float64(sqrt(Float64(Float64(t_0 * t_0) - 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);
	else
		tmp = Float64(Float64(fmax(lambda1, lambda2) * sqrt(Float64(-Float64(-1.0 * Float64(0.5 + Float64(0.5 * cos(Float64(-1.0 * Float64(fmin(phi1, phi2) + fmax(phi1, phi2)))))))))) * R);
	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 (max(lambda1, lambda2) <= 1.1e-33)
		tmp = t_0 * R;
	elseif (max(lambda1, lambda2) <= 4e+109)
		tmp = sqrt(((t_0 * t_0) - ((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;
	else
		tmp = (max(lambda1, lambda2) * sqrt(-(-1.0 * (0.5 + (0.5 * cos((-1.0 * (min(phi1, phi2) + max(phi1, phi2))))))))) * R;
	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[Max[lambda1, lambda2], $MachinePrecision], 6430607204255973/5846006549323611672814739330865132078623730171904], N[(t$95$0 * R), $MachinePrecision], If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 39999999999999999274034828753599231458870603857312684991833593479596290217520213192823213697572550705053433856], N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] * N[(N[(1/2 + N[(1/2 * N[Cos[N[(2 * N[(N[(N[Max[phi1, phi2], $MachinePrecision] + N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[Max[lambda1, lambda2], $MachinePrecision] * N[Sqrt[(-N[(-1 * N[(1/2 + N[(1/2 * N[Cos[N[(-1 * N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 1.1e-33

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

   if 1.1e-33 < lambda2 < 3.9999999999999999e109

   1. Initial program 60.3%

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

   3. Applied rewrites 60.3%

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

   if 3.9999999999999999e109 < lambda2

   1. Initial program 60.3%

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

   3. Applied rewrites 60.3%

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

   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 16.7%

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

   6. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right)} \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 55999999999999997437551452319718568527940010391551237206962722620786245759962322606480872185391376147155880923830865710654754390016:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right)}\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (if (<=
     (fmax lambda1 lambda2)
     55999999999999997437551452319718568527940010391551237206962722620786245759962322606480872185391376147155880923830865710654754390016)
  (* (- (fmax phi1 phi2) (fmin phi1 phi2)) R)
  (*
   (*
    (fmax lambda1 lambda2)
    (sqrt
     (-
      (*
       -1
       (+
        1/2
        (*
         1/2
         (cos (* -1 (+ (fmin phi1 phi2) (fmax phi1 phi2))))))))))
   R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(lambda1, lambda2) <= 5.6e+130) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * R;
	} else {
		tmp = (fmax(lambda1, lambda2) * sqrt(-(-1.0 * (0.5 + (0.5 * cos((-1.0 * (fmin(phi1, phi2) + fmax(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 (fmax(lambda1, lambda2) <= 5.6d+130) then
        tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * r
    else
        tmp = (fmax(lambda1, lambda2) * sqrt(-((-1.0d0) * (0.5d0 + (0.5d0 * cos(((-1.0d0) * (fmin(phi1, phi2) + fmax(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 (fmax(lambda1, lambda2) <= 5.6e+130) {
		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 * (fmin(phi1, phi2) + fmax(phi1, phi2))))))))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmax(lambda1, lambda2) <= 5.6e+130:
		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 * (fmin(phi1, phi2) + fmax(phi1, phi2))))))))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 5.6e+130)
		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 * Float64(fmin(phi1, phi2) + fmax(phi1, phi2)))))))))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (max(lambda1, lambda2) <= 5.6e+130)
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * R;
	else
		tmp = (max(lambda1, lambda2) * sqrt(-(-1.0 * (0.5 + (0.5 * cos((-1.0 * (min(phi1, phi2) + max(phi1, phi2))))))))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 55999999999999997437551452319718568527940010391551237206962722620786245759962322606480872185391376147155880923830865710654754390016], 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 * N[(1/2 + N[(1/2 * N[Cos[N[(-1 * N[(N[Min[phi1, phi2], $MachinePrecision] + N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.5999999999999997e130

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

   if 5.5999999999999997e130 < lambda2

   1. Initial program 60.3%

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

   3. Applied rewrites 60.3%

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

   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 16.7%

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

   6. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \sqrt{--1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_1 + \phi_2\right)\right)\right)}\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.4% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, R\right) \cdot \begin{array}{l} \mathbf{if}\;\left|R\right| \leq 53999999999999999224527139428339429095812089010320194521278052320766135407476736:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \left|R\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{max}\left(\phi_1, \phi_2\right) \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \frac{\frac{\left|R\right|}{\mathsf{min}\left(\phi_1, \phi_2\right)} \cdot \mathsf{max}\left(\phi_1, \phi_2\right) + \left(-\left|R\right|\right)}{\mathsf{max}\left(\phi_1, \phi_2\right)}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (*
 (copysign 1 R)
 (if (<=
      (fabs R)
      53999999999999999224527139428339429095812089010320194521278052320766135407476736)
   (* (- (fmax phi1 phi2) (fmin phi1 phi2)) (fabs R))
   (*
    (fmax phi1 phi2)
    (*
     (fmin phi1 phi2)
     (/
      (+
       (* (/ (fabs R) (fmin phi1 phi2)) (fmax phi1 phi2))
       (- (fabs R)))
      (fmax phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fabs(R) <= 5.4e+79) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * fabs(R);
	} else {
		tmp = fmax(phi1, phi2) * (fmin(phi1, phi2) * ((((fabs(R) / fmin(phi1, phi2)) * fmax(phi1, phi2)) + -fabs(R)) / fmax(phi1, phi2)));
	}
	return copysign(1.0, R) * tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.abs(R) <= 5.4e+79) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * Math.abs(R);
	} else {
		tmp = fmax(phi1, phi2) * (fmin(phi1, phi2) * ((((Math.abs(R) / fmin(phi1, phi2)) * fmax(phi1, phi2)) + -Math.abs(R)) / fmax(phi1, phi2)));
	}
	return Math.copySign(1.0, R) * tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.fabs(R) <= 5.4e+79:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * math.fabs(R)
	else:
		tmp = fmax(phi1, phi2) * (fmin(phi1, phi2) * ((((math.fabs(R) / fmin(phi1, phi2)) * fmax(phi1, phi2)) + -math.fabs(R)) / fmax(phi1, phi2)))
	return math.copysign(1.0, R) * tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (abs(R) <= 5.4e+79)
		tmp = Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * abs(R));
	else
		tmp = Float64(fmax(phi1, phi2) * Float64(fmin(phi1, phi2) * Float64(Float64(Float64(Float64(abs(R) / fmin(phi1, phi2)) * fmax(phi1, phi2)) + Float64(-abs(R))) / fmax(phi1, phi2))));
	end
	return Float64(copysign(1.0, R) * tmp)
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (abs(R) <= 5.4e+79)
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * abs(R);
	else
		tmp = max(phi1, phi2) * (min(phi1, phi2) * ((((abs(R) / min(phi1, phi2)) * max(phi1, phi2)) + -abs(R)) / max(phi1, phi2)));
	end
	tmp_2 = (sign(R) * abs(1.0)) * tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[R], $MachinePrecision], 53999999999999999224527139428339429095812089010320194521278052320766135407476736], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Abs[R], $MachinePrecision]), $MachinePrecision], N[(N[Max[phi1, phi2], $MachinePrecision] * N[(N[Min[phi1, phi2], $MachinePrecision] * N[(N[(N[(N[(N[Abs[R], $MachinePrecision] / N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision] + (-N[Abs[R], $MachinePrecision])), $MachinePrecision] / N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if R < 5.3999999999999999e79

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

   if 5.3999999999999999e79 < R

   1. Initial program 60.3%

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

   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 29.4%

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.2%

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

   7. Applied rewrites 28.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. add-to-fraction N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 27.6%

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

   9. Applied rewrites 27.6%

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

Alternative 8: 60.7% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, R\right) \cdot \begin{array}{l} \mathbf{if}\;\left|R\right| \leq 53999999999999999224527139428339429095812089010320194521278052320766135407476736:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \left|R\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left|R\right|}{\mathsf{min}\left(\phi_1, \phi_2\right)} \cdot \mathsf{max}\left(\phi_1, \phi_2\right) - \left|R\right|\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right)}{\mathsf{max}\left(\phi_1, \phi_2\right)} \cdot \mathsf{max}\left(\phi_1, \phi_2\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (*
 (copysign 1 R)
 (if (<=
      (fabs R)
      53999999999999999224527139428339429095812089010320194521278052320766135407476736)
   (* (- (fmax phi1 phi2) (fmin phi1 phi2)) (fabs R))
   (*
    (/
     (*
      (- (* (/ (fabs R) (fmin phi1 phi2)) (fmax phi1 phi2)) (fabs R))
      (fmin phi1 phi2))
     (fmax phi1 phi2))
    (fmax phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fabs(R) <= 5.4e+79) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * fabs(R);
	} else {
		tmp = (((((fabs(R) / fmin(phi1, phi2)) * fmax(phi1, phi2)) - fabs(R)) * fmin(phi1, phi2)) / fmax(phi1, phi2)) * fmax(phi1, phi2);
	}
	return copysign(1.0, R) * tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.abs(R) <= 5.4e+79) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * Math.abs(R);
	} else {
		tmp = (((((Math.abs(R) / fmin(phi1, phi2)) * fmax(phi1, phi2)) - Math.abs(R)) * fmin(phi1, phi2)) / fmax(phi1, phi2)) * fmax(phi1, phi2);
	}
	return Math.copySign(1.0, R) * tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.fabs(R) <= 5.4e+79:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * math.fabs(R)
	else:
		tmp = (((((math.fabs(R) / fmin(phi1, phi2)) * fmax(phi1, phi2)) - math.fabs(R)) * fmin(phi1, phi2)) / fmax(phi1, phi2)) * fmax(phi1, phi2)
	return math.copysign(1.0, R) * tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (abs(R) <= 5.4e+79)
		tmp = Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * abs(R));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(abs(R) / fmin(phi1, phi2)) * fmax(phi1, phi2)) - abs(R)) * fmin(phi1, phi2)) / fmax(phi1, phi2)) * fmax(phi1, phi2));
	end
	return Float64(copysign(1.0, R) * tmp)
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (abs(R) <= 5.4e+79)
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * abs(R);
	else
		tmp = (((((abs(R) / min(phi1, phi2)) * max(phi1, phi2)) - abs(R)) * min(phi1, phi2)) / max(phi1, phi2)) * max(phi1, phi2);
	end
	tmp_2 = (sign(R) * abs(1.0)) * tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[R], $MachinePrecision], 53999999999999999224527139428339429095812089010320194521278052320766135407476736], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Abs[R], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Abs[R], $MachinePrecision] / N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision] - N[Abs[R], $MachinePrecision]), $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] / N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if R < 5.3999999999999999e79

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

   if 5.3999999999999999e79 < R

   1. Initial program 60.3%

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

   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 29.4%

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.2%

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

   7. Applied rewrites 28.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 28.2%

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

   9. Applied rewrites 28.2%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. sub-to-fraction N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 28.2%

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

   11. Applied rewrites 28.2%

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

Alternative 9: 60.4% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, R\right) \cdot \begin{array}{l} \mathbf{if}\;\left|R\right| \leq 69999999999999996069415967724962592404255642689199354410126306696851159204757504:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \left|R\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{max}\left(\phi_1, \phi_2\right) \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \left(\left|R\right| \cdot \left(\frac{1}{\mathsf{min}\left(\phi_1, \phi_2\right)} - \frac{1}{\mathsf{max}\left(\phi_1, \phi_2\right)}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (*
 (copysign 1 R)
 (if (<=
      (fabs R)
      69999999999999996069415967724962592404255642689199354410126306696851159204757504)
   (* (- (fmax phi1 phi2) (fmin phi1 phi2)) (fabs R))
   (*
    (fmax phi1 phi2)
    (*
     (fmin phi1 phi2)
     (*
      (fabs R)
      (- (/ 1 (fmin phi1 phi2)) (/ 1 (fmax phi1 phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fabs(R) <= 7e+79) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * fabs(R);
	} else {
		tmp = fmax(phi1, phi2) * (fmin(phi1, phi2) * (fabs(R) * ((1.0 / fmin(phi1, phi2)) - (1.0 / fmax(phi1, phi2)))));
	}
	return copysign(1.0, R) * tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.abs(R) <= 7e+79) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * Math.abs(R);
	} else {
		tmp = fmax(phi1, phi2) * (fmin(phi1, phi2) * (Math.abs(R) * ((1.0 / fmin(phi1, phi2)) - (1.0 / fmax(phi1, phi2)))));
	}
	return Math.copySign(1.0, R) * tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.fabs(R) <= 7e+79:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * math.fabs(R)
	else:
		tmp = fmax(phi1, phi2) * (fmin(phi1, phi2) * (math.fabs(R) * ((1.0 / fmin(phi1, phi2)) - (1.0 / fmax(phi1, phi2)))))
	return math.copysign(1.0, R) * tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (abs(R) <= 7e+79)
		tmp = Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * abs(R));
	else
		tmp = Float64(fmax(phi1, phi2) * Float64(fmin(phi1, phi2) * Float64(abs(R) * Float64(Float64(1.0 / fmin(phi1, phi2)) - Float64(1.0 / fmax(phi1, phi2))))));
	end
	return Float64(copysign(1.0, R) * tmp)
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (abs(R) <= 7e+79)
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * abs(R);
	else
		tmp = max(phi1, phi2) * (min(phi1, phi2) * (abs(R) * ((1.0 / min(phi1, phi2)) - (1.0 / max(phi1, phi2)))));
	end
	tmp_2 = (sign(R) * abs(1.0)) * tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[R], $MachinePrecision], 69999999999999996069415967724962592404255642689199354410126306696851159204757504], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Abs[R], $MachinePrecision]), $MachinePrecision], N[(N[Max[phi1, phi2], $MachinePrecision] * N[(N[Min[phi1, phi2], $MachinePrecision] * N[(N[Abs[R], $MachinePrecision] * N[(N[(1 / N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] - N[(1 / N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if R < 6.9999999999999996e79

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

   if 6.9999999999999996e79 < R

   1. Initial program 60.3%

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

   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 29.4%

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.2%

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

   7. Applied rewrites 28.2%

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

   8. Taylor expanded in R around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 28.9%

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

   10. Applied rewrites 28.9%

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

Alternative 10: 60.1% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, R\right) \cdot \begin{array}{l} \mathbf{if}\;\left|R\right| \leq 69999999999999996069415967724962592404255642689199354410126306696851159204757504:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \left|R\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\left|R\right|}{\mathsf{min}\left(\phi_1, \phi_2\right)} - \frac{\left|R\right|}{\mathsf{max}\left(\phi_1, \phi_2\right)}\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \mathsf{max}\left(\phi_1, \phi_2\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (*
 (copysign 1 R)
 (if (<=
      (fabs R)
      69999999999999996069415967724962592404255642689199354410126306696851159204757504)
   (* (- (fmax phi1 phi2) (fmin phi1 phi2)) (fabs R))
   (*
    (*
     (- (/ (fabs R) (fmin phi1 phi2)) (/ (fabs R) (fmax phi1 phi2)))
     (fmin phi1 phi2))
    (fmax phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fabs(R) <= 7e+79) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * fabs(R);
	} else {
		tmp = (((fabs(R) / fmin(phi1, phi2)) - (fabs(R) / fmax(phi1, phi2))) * fmin(phi1, phi2)) * fmax(phi1, phi2);
	}
	return copysign(1.0, R) * tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.abs(R) <= 7e+79) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * Math.abs(R);
	} else {
		tmp = (((Math.abs(R) / fmin(phi1, phi2)) - (Math.abs(R) / fmax(phi1, phi2))) * fmin(phi1, phi2)) * fmax(phi1, phi2);
	}
	return Math.copySign(1.0, R) * tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.fabs(R) <= 7e+79:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * math.fabs(R)
	else:
		tmp = (((math.fabs(R) / fmin(phi1, phi2)) - (math.fabs(R) / fmax(phi1, phi2))) * fmin(phi1, phi2)) * fmax(phi1, phi2)
	return math.copysign(1.0, R) * tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (abs(R) <= 7e+79)
		tmp = Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * abs(R));
	else
		tmp = Float64(Float64(Float64(Float64(abs(R) / fmin(phi1, phi2)) - Float64(abs(R) / fmax(phi1, phi2))) * fmin(phi1, phi2)) * fmax(phi1, phi2));
	end
	return Float64(copysign(1.0, R) * tmp)
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (abs(R) <= 7e+79)
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * abs(R);
	else
		tmp = (((abs(R) / min(phi1, phi2)) - (abs(R) / max(phi1, phi2))) * min(phi1, phi2)) * max(phi1, phi2);
	end
	tmp_2 = (sign(R) * abs(1.0)) * tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[R], $MachinePrecision], 69999999999999996069415967724962592404255642689199354410126306696851159204757504], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Abs[R], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Abs[R], $MachinePrecision] / N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Abs[R], $MachinePrecision] / N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if R < 6.9999999999999996e79

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

   if 6.9999999999999996e79 < R

   1. Initial program 60.3%

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

   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 29.4%

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.2%

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

   7. Applied rewrites 28.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 28.2%

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

   9. Applied rewrites 28.2%

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

Alternative 11: 60.0% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, R\right) \cdot \begin{array}{l} \mathbf{if}\;\left|R\right| \leq 9999999999999999827367757839185598317239782875580932278577147150336:\\ \;\;\;\;\left(\mathsf{max}\left(\phi_1, \phi_2\right) - \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \left|R\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left|R\right| - \frac{\left|R\right|}{\mathsf{max}\left(\phi_1, \phi_2\right)} \cdot \mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \mathsf{max}\left(\phi_1, \phi_2\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (*
 (copysign 1 R)
 (if (<=
      (fabs R)
      9999999999999999827367757839185598317239782875580932278577147150336)
   (* (- (fmax phi1 phi2) (fmin phi1 phi2)) (fabs R))
   (*
    (- (fabs R) (* (/ (fabs R) (fmax phi1 phi2)) (fmin phi1 phi2)))
    (fmax phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fabs(R) <= 1e+67) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * fabs(R);
	} else {
		tmp = (fabs(R) - ((fabs(R) / fmax(phi1, phi2)) * fmin(phi1, phi2))) * fmax(phi1, phi2);
	}
	return copysign(1.0, R) * tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.abs(R) <= 1e+67) {
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * Math.abs(R);
	} else {
		tmp = (Math.abs(R) - ((Math.abs(R) / fmax(phi1, phi2)) * fmin(phi1, phi2))) * fmax(phi1, phi2);
	}
	return Math.copySign(1.0, R) * tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.fabs(R) <= 1e+67:
		tmp = (fmax(phi1, phi2) - fmin(phi1, phi2)) * math.fabs(R)
	else:
		tmp = (math.fabs(R) - ((math.fabs(R) / fmax(phi1, phi2)) * fmin(phi1, phi2))) * fmax(phi1, phi2)
	return math.copysign(1.0, R) * tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (abs(R) <= 1e+67)
		tmp = Float64(Float64(fmax(phi1, phi2) - fmin(phi1, phi2)) * abs(R));
	else
		tmp = Float64(Float64(abs(R) - Float64(Float64(abs(R) / fmax(phi1, phi2)) * fmin(phi1, phi2))) * fmax(phi1, phi2));
	end
	return Float64(copysign(1.0, R) * tmp)
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (abs(R) <= 1e+67)
		tmp = (max(phi1, phi2) - min(phi1, phi2)) * abs(R);
	else
		tmp = (abs(R) - ((abs(R) / max(phi1, phi2)) * min(phi1, phi2))) * max(phi1, phi2);
	end
	tmp_2 = (sign(R) * abs(1.0)) * tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[R], $MachinePrecision], 9999999999999999827367757839185598317239782875580932278577147150336], N[(N[(N[Max[phi1, phi2], $MachinePrecision] - N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Abs[R], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[R], $MachinePrecision] - N[(N[(N[Abs[R], $MachinePrecision] / N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if R < 9.9999999999999998e66

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   3. Applied rewrites 57.9%

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

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

      3. cos-neg-rev N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-frac-neg N/A

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

      7. frac-2neg N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. div-add N/A

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

      11. cos-sum N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.4%

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

   8. Applied rewrites 27.4%

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

   10. Applied rewrites 30.1%

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

   if 9.9999999999999998e66 < R

   1. Initial program 60.3%

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

   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 29.4%

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

   4. Applied rewrites 29.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 29.4%

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

   6. Applied rewrites 29.4%

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

   8. Applied rewrites 29.4%

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

Alternative 12: 57.9% 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 60.3%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. distribute-frac-neg N/A

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

   7. frac-2neg N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. div-add N/A

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

   11. cos-sum N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. mult-flip N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 N/A

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

   22. lower-*.f64 N/A

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

3. Applied rewrites 57.9%

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

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

   3. cos-neg-rev N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. distribute-frac-neg N/A

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

   7. frac-2neg N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. div-add N/A

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

   11. cos-sum N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

       \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} - \sin \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   19. mult-flip N/A

       \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} - \sin \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   20. metadata-eval N/A

       \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   21. lower-*.f64 N/A

       \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} - \sin \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

5. Applied rewrites 60.9%

   \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

6. Taylor expanded in phi1 around -inf

   \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(-1 \cdot \color{blue}{\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto R \cdot \left(-1 \cdot \left(\phi_1 \cdot \color{blue}{\left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto R \cdot \left(-1 \cdot \left(\phi_1 \cdot \left(1 + \color{blue}{-1 \cdot \frac{\phi_2}{\phi_1}}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto R \cdot \left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right)\right) \]

   5. lower-/.f64 27.4%

      \[\leadsto R \cdot \left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\color{blue}{\phi_1}}\right)\right)\right) \]

8. Applied rewrites 27.4%

   \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]

10. Applied rewrites 30.1%

    \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R} \]
11. Add Preprocessing

Alternative 13: 52.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq \frac{-6230756230241793}{2596148429267413814265248164610048}:\\ \;\;\;\;\left(-\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{max}\left(\phi_1, \phi_2\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
  :precision binary64
  (if (<=
     (fmin phi1 phi2)
     -6230756230241793/2596148429267413814265248164610048)
  (* (- (fmin phi1 phi2)) R)
  (* R (fmax phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -2.4e-18) {
		tmp = -fmin(phi1, phi2) * R;
	} else {
		tmp = R * 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 (fmin(phi1, phi2) <= (-2.4d-18)) then
        tmp = -fmin(phi1, phi2) * r
    else
        tmp = r * 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 (fmin(phi1, phi2) <= -2.4e-18) {
		tmp = -fmin(phi1, phi2) * R;
	} else {
		tmp = R * fmax(phi1, phi2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmin(phi1, phi2) <= -2.4e-18:
		tmp = -fmin(phi1, phi2) * R
	else:
		tmp = R * fmax(phi1, phi2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -2.4e-18)
		tmp = Float64(Float64(-fmin(phi1, phi2)) * R);
	else
		tmp = Float64(R * fmax(phi1, phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (min(phi1, phi2) <= -2.4e-18)
		tmp = -min(phi1, phi2) * R;
	else
		tmp = R * max(phi1, phi2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -6230756230241793/2596148429267413814265248164610048], N[((-N[Min[phi1, phi2], $MachinePrecision]) * R), $MachinePrecision], N[(R * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.3999999999999999e-18

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in lambda1 around inf

      \[\leadsto R \cdot \color{blue}{\left(\lambda_1 \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\lambda_1 \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto R \cdot \left(\lambda_1 \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(\lambda_1 \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(\lambda_1 \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\lambda_1 \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right) \]

      6. lower-+.f64 16.1%

         \[\leadsto R \cdot \left(\lambda_1 \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right) \]

   4. Applied rewrites 16.1%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_1 \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}\right)} \]

   5. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 17.6%

         \[\leadsto R \cdot \left(-1 \cdot \color{blue}{\phi_1}\right) \]

   7. Applied rewrites 17.6%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{R \cdot \left(-1 \cdot \phi_1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      3. lower-*.f64 17.6%

         \[\leadsto \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\phi_1}\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]

      6. lower-neg.f64 17.6%

         \[\leadsto \left(-\phi_1\right) \cdot R \]

   9. Applied rewrites 17.6%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot R} \]

   if -2.3999999999999999e-18 < phi1

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   2. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   3. Step-by-step derivation

      1. lower-*.f64 17.8%

         \[\leadsto R \cdot \color{blue}{\phi_2} \]

   4. Applied rewrites 17.8%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 31.9% 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 60.3%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

2. Taylor expanded in phi2 around inf

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Step-by-step derivation

   1. lower-*.f64 17.8%

      \[\leadsto R \cdot \color{blue}{\phi_2} \]

4. Applied rewrites 17.8%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
5. Add Preprocessing

Alternative 15: 17.7% 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 60.3%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

2. Taylor expanded in phi1 around inf

   \[\leadsto \color{blue}{R \cdot \phi_1} \]
3. Step-by-step derivation

   1. lower-*.f64 17.7%

      \[\leadsto R \cdot \color{blue}{\phi_1} \]

4. Applied rewrites 17.7%

   \[\leadsto \color{blue}{R \cdot \phi_1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(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))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))