Equirectangular approximation to distance on a great circle

Average Error: 38.6 → 18.1

Time: 46.9s

Precision: binary64

Cost: 53252

\[ \begin{array}{c}[lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+121}:\\ \;\;\;\;R \cdot \left(\left(-\lambda_1 \cdot \sqrt{0.5 \cdot \left(1 + \cos \left(\phi_1 + \phi_2\right)\right)}\right) + \sqrt{\left(1 + \cos \phi_1 \cdot \cos \phi_2\right) - \sin \phi_1 \cdot \sin \phi_2} \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot 0.5\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;R \cdot \sqrt{{\phi_1}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \sqrt{t_0 \cdot t_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
        (t_1 (* R (- phi2 phi1))))
   (if (<= (- lambda1 lambda2) -1e+121)
     (*
      R
      (+
       (- (* lambda1 (sqrt (* 0.5 (+ 1.0 (cos (+ phi1 phi2)))))))
       (*
        (sqrt (- (+ 1.0 (* (cos phi1) (cos phi2))) (* (sin phi1) (sin phi2))))
        (* (/ lambda2 (sqrt 0.5)) 0.5))))
     (if (<= (- lambda1 lambda2) -4e+102)
       t_1
       (if (<= (- lambda1 lambda2) -5e+79)
         (*
          R
          (sqrt
           (+
            (pow phi1 2.0)
            (* (pow (cos (* 0.5 phi1)) 2.0) (pow (- lambda1 lambda2) 2.0)))))
         (if (<= (- lambda1 lambda2) -5e+75)
           t_1
           (if (<= (- lambda1 lambda2) -2e+51)
             (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))
             t_1)))))))

C

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

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double t_1 = R * (phi2 - phi1);
	double tmp;
	if ((lambda1 - lambda2) <= -1e+121) {
		tmp = R * (-(lambda1 * sqrt((0.5 * (1.0 + cos((phi1 + phi2)))))) + (sqrt(((1.0 + (cos(phi1) * cos(phi2))) - (sin(phi1) * sin(phi2)))) * ((lambda2 / sqrt(0.5)) * 0.5)));
	} else if ((lambda1 - lambda2) <= -4e+102) {
		tmp = t_1;
	} else if ((lambda1 - lambda2) <= -5e+79) {
		tmp = R * sqrt((pow(phi1, 2.0) + (pow(cos((0.5 * phi1)), 2.0) * pow((lambda1 - lambda2), 2.0))));
	} else if ((lambda1 - lambda2) <= -5e+75) {
		tmp = t_1;
	} else if ((lambda1 - lambda2) <= -2e+51) {
		tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    t_1 = r * (phi2 - phi1)
    if ((lambda1 - lambda2) <= (-1d+121)) then
        tmp = r * (-(lambda1 * sqrt((0.5d0 * (1.0d0 + cos((phi1 + phi2)))))) + (sqrt(((1.0d0 + (cos(phi1) * cos(phi2))) - (sin(phi1) * sin(phi2)))) * ((lambda2 / sqrt(0.5d0)) * 0.5d0)))
    else if ((lambda1 - lambda2) <= (-4d+102)) then
        tmp = t_1
    else if ((lambda1 - lambda2) <= (-5d+79)) then
        tmp = r * sqrt(((phi1 ** 2.0d0) + ((cos((0.5d0 * phi1)) ** 2.0d0) * ((lambda1 - lambda2) ** 2.0d0))))
    else if ((lambda1 - lambda2) <= (-5d+75)) then
        tmp = t_1
    else if ((lambda1 - lambda2) <= (-2d+51)) then
        tmp = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

↓

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));
	double t_1 = R * (phi2 - phi1);
	double tmp;
	if ((lambda1 - lambda2) <= -1e+121) {
		tmp = R * (-(lambda1 * Math.sqrt((0.5 * (1.0 + Math.cos((phi1 + phi2)))))) + (Math.sqrt(((1.0 + (Math.cos(phi1) * Math.cos(phi2))) - (Math.sin(phi1) * Math.sin(phi2)))) * ((lambda2 / Math.sqrt(0.5)) * 0.5)));
	} else if ((lambda1 - lambda2) <= -4e+102) {
		tmp = t_1;
	} else if ((lambda1 - lambda2) <= -5e+79) {
		tmp = R * Math.sqrt((Math.pow(phi1, 2.0) + (Math.pow(Math.cos((0.5 * phi1)), 2.0) * Math.pow((lambda1 - lambda2), 2.0))));
	} else if ((lambda1 - lambda2) <= -5e+75) {
		tmp = t_1;
	} else if ((lambda1 - lambda2) <= -2e+51) {
		tmp = R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	t_1 = R * (phi2 - phi1)
	tmp = 0
	if (lambda1 - lambda2) <= -1e+121:
		tmp = R * (-(lambda1 * math.sqrt((0.5 * (1.0 + math.cos((phi1 + phi2)))))) + (math.sqrt(((1.0 + (math.cos(phi1) * math.cos(phi2))) - (math.sin(phi1) * math.sin(phi2)))) * ((lambda2 / math.sqrt(0.5)) * 0.5)))
	elif (lambda1 - lambda2) <= -4e+102:
		tmp = t_1
	elif (lambda1 - lambda2) <= -5e+79:
		tmp = R * math.sqrt((math.pow(phi1, 2.0) + (math.pow(math.cos((0.5 * phi1)), 2.0) * math.pow((lambda1 - lambda2), 2.0))))
	elif (lambda1 - lambda2) <= -5e+75:
		tmp = t_1
	elif (lambda1 - lambda2) <= -2e+51:
		tmp = R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	t_1 = Float64(R * Float64(phi2 - phi1))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1e+121)
		tmp = Float64(R * Float64(Float64(-Float64(lambda1 * sqrt(Float64(0.5 * Float64(1.0 + cos(Float64(phi1 + phi2))))))) + Float64(sqrt(Float64(Float64(1.0 + Float64(cos(phi1) * cos(phi2))) - Float64(sin(phi1) * sin(phi2)))) * Float64(Float64(lambda2 / sqrt(0.5)) * 0.5))));
	elseif (Float64(lambda1 - lambda2) <= -4e+102)
		tmp = t_1;
	elseif (Float64(lambda1 - lambda2) <= -5e+79)
		tmp = Float64(R * sqrt(Float64((phi1 ^ 2.0) + Float64((cos(Float64(0.5 * phi1)) ^ 2.0) * (Float64(lambda1 - lambda2) ^ 2.0)))));
	elseif (Float64(lambda1 - lambda2) <= -5e+75)
		tmp = t_1;
	elseif (Float64(lambda1 - lambda2) <= -2e+51)
		tmp = Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	t_1 = R * (phi2 - phi1);
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -1e+121)
		tmp = R * (-(lambda1 * sqrt((0.5 * (1.0 + cos((phi1 + phi2)))))) + (sqrt(((1.0 + (cos(phi1) * cos(phi2))) - (sin(phi1) * sin(phi2)))) * ((lambda2 / sqrt(0.5)) * 0.5)));
	elseif ((lambda1 - lambda2) <= -4e+102)
		tmp = t_1;
	elseif ((lambda1 - lambda2) <= -5e+79)
		tmp = R * sqrt(((phi1 ^ 2.0) + ((cos((0.5 * phi1)) ^ 2.0) * ((lambda1 - lambda2) ^ 2.0))));
	elseif ((lambda1 - lambda2) <= -5e+75)
		tmp = t_1;
	elseif ((lambda1 - lambda2) <= -2e+51)
		tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1e+121], N[(R * N[((-N[(lambda1 * N[Sqrt[N[(0.5 * N[(1.0 + N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) + N[(N[Sqrt[N[(N[(1.0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(lambda2 / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -4e+102], t$95$1, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -5e+79], N[(R * N[Sqrt[N[(N[Power[phi1, 2.0], $MachinePrecision] + N[(N[Power[N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(lambda1 - lambda2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -5e+75], t$95$1, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -2e+51], 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], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 lambda1 lambda2) < -1.00000000000000004e121

   1. Initial program 57.4

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

   2. Simplified 57.4

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

      [Start] 57.4	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in lambda1 around -inf 22.7

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

   4. Simplified 22.6

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

      [Start] 22.7	\[ R \cdot \left(-1 \cdot \left(\left(\sqrt{0.5} \cdot \lambda_1\right) \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right) + 0.5 \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      rational.json-simplify-2 [=>] 22.7	\[ R \cdot \left(\color{blue}{\left(\left(\sqrt{0.5} \cdot \lambda_1\right) \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right) \cdot -1} + 0.5 \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      rational.json-simplify-9 [=>] 22.7	\[ R \cdot \left(\color{blue}{\left(-\left(\sqrt{0.5} \cdot \lambda_1\right) \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)} + 0.5 \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      rational.json-simplify-2 [=>] 22.7	\[ R \cdot \left(\left(-\color{blue}{\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \left(\sqrt{0.5} \cdot \lambda_1\right)}\right) + 0.5 \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      rational.json-simplify-2 [=>] 22.7	\[ R \cdot \left(\left(-\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \color{blue}{\left(\lambda_1 \cdot \sqrt{0.5}\right)}\right) + 0.5 \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      rational.json-simplify-43 [=>] 22.8	\[ R \cdot \left(\left(-\color{blue}{\lambda_1 \cdot \left(\sqrt{0.5} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)}\right) + 0.5 \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      exponential.json-simplify-20 [=>] 22.6	\[ R \cdot \left(\left(-\lambda_1 \cdot \color{blue}{\sqrt{\left(1 + \cos \left(\phi_1 + \phi_2\right)\right) \cdot 0.5}}\right) + 0.5 \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      rational.json-simplify-2 [<=] 22.6	\[ R \cdot \left(\left(-\lambda_1 \cdot \sqrt{\color{blue}{0.5 \cdot \left(1 + \cos \left(\phi_1 + \phi_2\right)\right)}}\right) + 0.5 \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      rational.json-simplify-2 [=>] 22.6	\[ R \cdot \left(\left(-\lambda_1 \cdot \sqrt{0.5 \cdot \left(1 + \cos \left(\phi_1 + \phi_2\right)\right)}\right) + 0.5 \cdot \color{blue}{\left(\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \frac{\lambda_2}{\sqrt{0.5}}\right)}\right) \]
      rational.json-simplify-43 [=>] 22.6	\[ R \cdot \left(\left(-\lambda_1 \cdot \sqrt{0.5 \cdot \left(1 + \cos \left(\phi_1 + \phi_2\right)\right)}\right) + \color{blue}{\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot 0.5\right)}\right) \]

   5. Applied egg-rr 19.1

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

   if -1.00000000000000004e121 < (-.f64 lambda1 lambda2) < -3.99999999999999991e102 or -5e79 < (-.f64 lambda1 lambda2) < -5.0000000000000002e75 or -2e51 < (-.f64 lambda1 lambda2)

   1. Initial program 22.7

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

   2. Simplified 22.7

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

      [Start] 22.7	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [=>] 22.7	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [=>] 22.7	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around -inf 14.6

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

   4. Simplified 14.6

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

      [Start] 14.6	\[ R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]
      rational.json-simplify-1 [=>] 14.6	\[ R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
      rational.json-simplify-2 [=>] 14.6	\[ R \cdot \left(\phi_2 + \color{blue}{\phi_1 \cdot -1}\right) \]
      rational.json-simplify-9 [=>] 14.6	\[ R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

   5. Taylor expanded in R around 0 14.6

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

   if -3.99999999999999991e102 < (-.f64 lambda1 lambda2) < -5e79

   1. Initial program 25.6

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

   2. Simplified 25.6

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

      [Start] 25.6	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [=>] 25.6	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [=>] 25.6	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi2 around 0 34.6

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

   if -5.0000000000000002e75 < (-.f64 lambda1 lambda2) < -2e51

   1. Initial program 22.4

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+121}:\\ \;\;\;\;R \cdot \left(\left(-\lambda_1 \cdot \sqrt{0.5 \cdot \left(1 + \cos \left(\phi_1 + \phi_2\right)\right)}\right) + \sqrt{\left(1 + \cos \phi_1 \cdot \cos \phi_2\right) - \sin \phi_1 \cdot \sin \phi_2} \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot 0.5\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;R \cdot \sqrt{{\phi_1}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+75}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.7
Cost	33796

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := 1 + \cos \left(\phi_1 + \phi_2\right)\\ t_2 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+121}:\\ \;\;\;\;R \cdot \left(\left(-\lambda_1 \cdot \sqrt{0.5 \cdot t_1}\right) + \sqrt{t_1} \cdot \left(\frac{\lambda_2}{\sqrt{0.5}} \cdot 0.5\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;R \cdot \sqrt{{\phi_1}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \sqrt{t_2 \cdot t_2 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	19.7
Cost	33676

\[\begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+121}:\\ \;\;\;\;R \cdot \left(\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \left(\lambda_2 \cdot \sqrt{0.5} + \frac{\lambda_1}{\sqrt{0.5}} \cdot -0.5\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;R \cdot \sqrt{{\phi_1}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \sqrt{t_0 \cdot t_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	19.7
Cost	27468

\[\begin{array}{l} t_0 := R \cdot \left(\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \left(\lambda_2 \cdot \sqrt{0.5} + \frac{\lambda_1}{\sqrt{0.5}} \cdot -0.5\right)\right)\\ t_1 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_2 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \sqrt{t_1 \cdot t_1 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	24.5
Cost	21320

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_2 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -9.5 \cdot 10^{+16}:\\ \;\;\;\;R \cdot \sqrt{t_1 \cdot t_1 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq -2.75 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.22 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -2.7 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-157}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	25.2
Cost	14668

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -5.3 \cdot 10^{+26}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \sqrt{\left(1 + \cos \phi_1\right) \cdot 0.5}\right)\\ \mathbf{elif}\;\phi_1 \leq -0.85:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 \cdot \left(\left(1 + \cos \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -4.5 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	24.5
Cost	14536

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -3.8 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1 + \cos \phi_1}{2}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq -1.18 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1.35 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -3.6 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	25.0
Cost	13640

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -2.05 \cdot 10^{+28}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \sqrt{\left(1 + \cos \phi_1\right) \cdot 0.5}\right)\\ \mathbf{elif}\;\phi_1 \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -5.2 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	24.6
Cost	7696

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -4.6 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -5 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	32.6
Cost	7572

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{-272}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-239}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 3.3 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.7 \cdot 10^{-39}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	30.7
Cost	7440

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_1\right)\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5.4 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	27.4
Cost	7240

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -2.85 \cdot 10^{+168}:\\ \;\;\;\;R \cdot \left(t_0 \cdot \left(-\lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 6.2 \cdot 10^{-91}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \end{array} \]

Alternative 12
Error	30.9
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -2.1 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 4.6 \cdot 10^{-238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	34.1
Cost	784

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -6.2 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14
Error	39.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.2 \cdot 10^{-39}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 15
Error	45.8
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023077 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))