Equirectangular approximation to distance on a great circle

Average Error: 39.4 → 20.9

Time: 34.0s

Precision: binary64

Cost: 46404

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -3 \cdot 10^{+127}:\\ \;\;\;\;R \cdot \left(\sqrt{1 + \left(\cos \phi_2 \cdot \cos \phi_1 - \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \left(\lambda_2 \cdot \sqrt{0.5} + -0.5 \cdot \frac{\lambda_1}{\sqrt{0.5}}\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-152}:\\ \;\;\;\;R \cdot \sqrt{\frac{1 + \cos \left(2 \cdot \frac{\phi_1 + \phi_2}{2}\right)}{2} \cdot t_1 + t_0}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{-210}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{1 \cdot t_1 + t_0}\\ \end{array} \]

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 (* (- phi1 phi2) (- phi1 phi2)))
        (t_1 (* (- lambda1 lambda2) (- lambda1 lambda2))))
   (if (<= (- lambda1 lambda2) -3e+127)
     (*
      R
      (*
       (sqrt (+ 1.0 (- (* (cos phi2) (cos phi1)) (* (sin phi2) (sin phi1)))))
       (+ (* lambda2 (sqrt 0.5)) (* -0.5 (/ lambda1 (sqrt 0.5))))))
     (if (<= (- lambda1 lambda2) -1e-152)
       (*
        R
        (sqrt
         (+ (* (/ (+ 1.0 (cos (* 2.0 (/ (+ phi1 phi2) 2.0)))) 2.0) t_1) t_0)))
       (if (<= (- lambda1 lambda2) -4e-210)
         (* R (+ phi1 (- phi2)))
         (* R (sqrt (+ (* 1.0 t_1) t_0))))))))

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 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = (lambda1 - lambda2) * (lambda1 - lambda2);
	double tmp;
	if ((lambda1 - lambda2) <= -3e+127) {
		tmp = R * (sqrt((1.0 + ((cos(phi2) * cos(phi1)) - (sin(phi2) * sin(phi1))))) * ((lambda2 * sqrt(0.5)) + (-0.5 * (lambda1 / sqrt(0.5)))));
	} else if ((lambda1 - lambda2) <= -1e-152) {
		tmp = R * sqrt(((((1.0 + cos((2.0 * ((phi1 + phi2) / 2.0)))) / 2.0) * t_1) + t_0));
	} else if ((lambda1 - lambda2) <= -4e-210) {
		tmp = R * (phi1 + -phi2);
	} else {
		tmp = R * sqrt(((1.0 * t_1) + t_0));
	}
	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 = (phi1 - phi2) * (phi1 - phi2)
    t_1 = (lambda1 - lambda2) * (lambda1 - lambda2)
    if ((lambda1 - lambda2) <= (-3d+127)) then
        tmp = r * (sqrt((1.0d0 + ((cos(phi2) * cos(phi1)) - (sin(phi2) * sin(phi1))))) * ((lambda2 * sqrt(0.5d0)) + ((-0.5d0) * (lambda1 / sqrt(0.5d0)))))
    else if ((lambda1 - lambda2) <= (-1d-152)) then
        tmp = r * sqrt(((((1.0d0 + cos((2.0d0 * ((phi1 + phi2) / 2.0d0)))) / 2.0d0) * t_1) + t_0))
    else if ((lambda1 - lambda2) <= (-4d-210)) then
        tmp = r * (phi1 + -phi2)
    else
        tmp = r * sqrt(((1.0d0 * t_1) + t_0))
    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 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = (lambda1 - lambda2) * (lambda1 - lambda2);
	double tmp;
	if ((lambda1 - lambda2) <= -3e+127) {
		tmp = R * (Math.sqrt((1.0 + ((Math.cos(phi2) * Math.cos(phi1)) - (Math.sin(phi2) * Math.sin(phi1))))) * ((lambda2 * Math.sqrt(0.5)) + (-0.5 * (lambda1 / Math.sqrt(0.5)))));
	} else if ((lambda1 - lambda2) <= -1e-152) {
		tmp = R * Math.sqrt(((((1.0 + Math.cos((2.0 * ((phi1 + phi2) / 2.0)))) / 2.0) * t_1) + t_0));
	} else if ((lambda1 - lambda2) <= -4e-210) {
		tmp = R * (phi1 + -phi2);
	} else {
		tmp = R * Math.sqrt(((1.0 * t_1) + t_0));
	}
	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 = (phi1 - phi2) * (phi1 - phi2)
	t_1 = (lambda1 - lambda2) * (lambda1 - lambda2)
	tmp = 0
	if (lambda1 - lambda2) <= -3e+127:
		tmp = R * (math.sqrt((1.0 + ((math.cos(phi2) * math.cos(phi1)) - (math.sin(phi2) * math.sin(phi1))))) * ((lambda2 * math.sqrt(0.5)) + (-0.5 * (lambda1 / math.sqrt(0.5)))))
	elif (lambda1 - lambda2) <= -1e-152:
		tmp = R * math.sqrt(((((1.0 + math.cos((2.0 * ((phi1 + phi2) / 2.0)))) / 2.0) * t_1) + t_0))
	elif (lambda1 - lambda2) <= -4e-210:
		tmp = R * (phi1 + -phi2)
	else:
		tmp = R * math.sqrt(((1.0 * t_1) + t_0))
	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(phi1 - phi2) * Float64(phi1 - phi2))
	t_1 = Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -3e+127)
		tmp = Float64(R * Float64(sqrt(Float64(1.0 + Float64(Float64(cos(phi2) * cos(phi1)) - Float64(sin(phi2) * sin(phi1))))) * Float64(Float64(lambda2 * sqrt(0.5)) + Float64(-0.5 * Float64(lambda1 / sqrt(0.5))))));
	elseif (Float64(lambda1 - lambda2) <= -1e-152)
		tmp = Float64(R * sqrt(Float64(Float64(Float64(Float64(1.0 + cos(Float64(2.0 * Float64(Float64(phi1 + phi2) / 2.0)))) / 2.0) * t_1) + t_0)));
	elseif (Float64(lambda1 - lambda2) <= -4e-210)
		tmp = Float64(R * Float64(phi1 + Float64(-phi2)));
	else
		tmp = Float64(R * sqrt(Float64(Float64(1.0 * t_1) + t_0)));
	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 = (phi1 - phi2) * (phi1 - phi2);
	t_1 = (lambda1 - lambda2) * (lambda1 - lambda2);
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -3e+127)
		tmp = R * (sqrt((1.0 + ((cos(phi2) * cos(phi1)) - (sin(phi2) * sin(phi1))))) * ((lambda2 * sqrt(0.5)) + (-0.5 * (lambda1 / sqrt(0.5)))));
	elseif ((lambda1 - lambda2) <= -1e-152)
		tmp = R * sqrt(((((1.0 + cos((2.0 * ((phi1 + phi2) / 2.0)))) / 2.0) * t_1) + t_0));
	elseif ((lambda1 - lambda2) <= -4e-210)
		tmp = R * (phi1 + -phi2);
	else
		tmp = R * sqrt(((1.0 * t_1) + t_0));
	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[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -3e+127], N[(R * N[(N[Sqrt[N[(1.0 + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(lambda2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(lambda1 / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1e-152], N[(R * N[Sqrt[N[(N[(N[(N[(1.0 + N[Cos[N[(2.0 * N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -4e-210], N[(R * N[(phi1 + (-phi2)), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(1.0 * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 lambda1 lambda2) < -3.0000000000000002e127

   1. Initial program 58.5

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

   2. Simplified 58.6

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

      [Start] 58.5	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 58.5	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-2 [=>] 58.5	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 58.5	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-2 [=>] 58.5	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 58.6	\[ R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      trig-simplify-6 [=>] 58.6	\[ R \cdot \sqrt{\color{blue}{\frac{1 + \cos \left(2 \cdot \frac{\phi_1 + \phi_2}{2}\right)}{2}} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in lambda2 around inf 22.1

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

   4. Simplified 22.1

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

      [Start] 22.1	\[ R \cdot \left(\left(\sqrt{0.5} \cdot \lambda_2\right) \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} + -0.5 \cdot \left(\frac{\lambda_1}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      rational_best-simplify-2 [=>] 22.1	\[ R \cdot \left(\color{blue}{\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \left(\sqrt{0.5} \cdot \lambda_2\right)} + -0.5 \cdot \left(\frac{\lambda_1}{\sqrt{0.5}} \cdot \sqrt{1 + \cos \left(\phi_1 + \phi_2\right)}\right)\right) \]
      rational_best-simplify-2 [=>] 22.1	\[ R \cdot \left(\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \left(\sqrt{0.5} \cdot \lambda_2\right) + -0.5 \cdot \color{blue}{\left(\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \frac{\lambda_1}{\sqrt{0.5}}\right)}\right) \]
      rational_best-simplify-44 [=>] 22.1	\[ R \cdot \left(\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \left(\sqrt{0.5} \cdot \lambda_2\right) + \color{blue}{\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \left(-0.5 \cdot \frac{\lambda_1}{\sqrt{0.5}}\right)}\right) \]
      rational_best-simplify-47 [=>] 22.1	\[ R \cdot \color{blue}{\left(\sqrt{1 + \cos \left(\phi_1 + \phi_2\right)} \cdot \left(-0.5 \cdot \frac{\lambda_1}{\sqrt{0.5}} + \sqrt{0.5} \cdot \lambda_2\right)\right)} \]
      rational_best-simplify-1 [=>] 22.1	\[ R \cdot \left(\sqrt{1 + \cos \color{blue}{\left(\phi_2 + \phi_1\right)}} \cdot \left(-0.5 \cdot \frac{\lambda_1}{\sqrt{0.5}} + \sqrt{0.5} \cdot \lambda_2\right)\right) \]
      rational_best-simplify-1 [<=] 22.1	\[ R \cdot \left(\sqrt{1 + \cos \left(\phi_2 + \phi_1\right)} \cdot \color{blue}{\left(\sqrt{0.5} \cdot \lambda_2 + -0.5 \cdot \frac{\lambda_1}{\sqrt{0.5}}\right)}\right) \]
      rational_best-simplify-2 [=>] 22.1	\[ R \cdot \left(\sqrt{1 + \cos \left(\phi_2 + \phi_1\right)} \cdot \left(\color{blue}{\lambda_2 \cdot \sqrt{0.5}} + -0.5 \cdot \frac{\lambda_1}{\sqrt{0.5}}\right)\right) \]

   5. Applied egg-rr 15.6

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

   if -3.0000000000000002e127 < (-.f64 lambda1 lambda2) < -1.00000000000000007e-152

   1. Initial program 24.4

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

   2. Simplified 24.4

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

      [Start] 24.4	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 24.4	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-2 [=>] 24.4	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 24.4	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-2 [=>] 24.4	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 24.4	\[ R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      trig-simplify-6 [=>] 24.4	\[ R \cdot \sqrt{\color{blue}{\frac{1 + \cos \left(2 \cdot \frac{\phi_1 + \phi_2}{2}\right)}{2}} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   if -1.00000000000000007e-152 < (-.f64 lambda1 lambda2) < -4.0000000000000002e-210

   1. Initial program 24.1

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

   2. Simplified 24.1

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

      [Start] 24.1	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 24.1	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-2 [=>] 24.1	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 24.1	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around inf 31.7

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

   4. Simplified 31.7

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

      [Start] 31.7	\[ R \cdot \left(-1 \cdot \phi_2 + \phi_1\right) \]
      rational_best-simplify-1 [=>] 31.7	\[ R \cdot \color{blue}{\left(\phi_1 + -1 \cdot \phi_2\right)} \]
      rational_best-simplify-2 [=>] 31.7	\[ R \cdot \left(\phi_1 + \color{blue}{\phi_2 \cdot -1}\right) \]
      rational_best-simplify-12 [=>] 31.7	\[ R \cdot \left(\phi_1 + \color{blue}{\left(-\phi_2\right)}\right) \]

   if -4.0000000000000002e-210 < (-.f64 lambda1 lambda2)

   1. Initial program 24.6

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

   2. Simplified 24.6

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

      [Start] 24.6	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 24.6	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-2 [=>] 24.6	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 24.6	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-2 [=>] 24.6	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-44 [=>] 24.6	\[ R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      trig-simplify-6 [=>] 24.6	\[ R \cdot \sqrt{\color{blue}{\frac{1 + \cos \left(2 \cdot \frac{\phi_1 + \phi_2}{2}\right)}{2}} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi2 around 0 24.6

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

   4. Taylor expanded in phi1 around 0 24.6

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

4. Final simplification 20.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -3 \cdot 10^{+127}:\\ \;\;\;\;R \cdot \left(\sqrt{1 + \left(\cos \phi_2 \cdot \cos \phi_1 - \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \left(\lambda_2 \cdot \sqrt{0.5} + -0.5 \cdot \frac{\lambda_1}{\sqrt{0.5}}\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-152}:\\ \;\;\;\;R \cdot \sqrt{\frac{1 + \cos \left(2 \cdot \frac{\phi_1 + \phi_2}{2}\right)}{2} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{-210}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{1 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.6
Cost	27084

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -5 \cdot 10^{+57}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -5.5 \cdot 10^{-133}:\\ \;\;\;\;R \cdot \sqrt{\frac{1 + \cos \left(2 \cdot \frac{\phi_1 + \phi_2}{2}\right)}{2} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 2.65 \cdot 10^{+39}:\\ \;\;\;\;R \cdot \left(\sqrt{1 + \cos \left(\phi_2 + \phi_1\right)} \cdot \left(\lambda_2 \cdot \sqrt{0.5} + -0.5 \cdot \frac{\lambda_1}{\sqrt{0.5}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternative 2
Error	24.8
Cost	26956

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.2 \cdot 10^{+57}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -2.5 \cdot 10^{-133}:\\ \;\;\;\;R \cdot \sqrt{\frac{1 + \cos \left(2 \cdot \frac{\phi_1 + \phi_2}{2}\right)}{2} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 0.23:\\ \;\;\;\;R \cdot \left(\sqrt{\cos \phi_1 + 1} \cdot \left(\lambda_2 \cdot \sqrt{0.5} + -0.5 \cdot \frac{\lambda_1}{\sqrt{0.5}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternative 3
Error	27.3
Cost	14920

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.8 \cdot 10^{+56}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -4.6 \cdot 10^{-133}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1 + \cos \left(2 \cdot \frac{\phi_1 + \phi_2}{2}\right)}{2}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 3.25 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternative 4
Error	27.2
Cost	14920

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{+57}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -2.1 \cdot 10^{-134}:\\ \;\;\;\;R \cdot \sqrt{\frac{1 + \cos \left(2 \cdot \frac{\phi_1 + \phi_2}{2}\right)}{2} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 3.25 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternative 5
Error	27.7
Cost	14536

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.6 \cdot 10^{+56}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -3.9 \cdot 10^{-135}:\\ \;\;\;\;R \cdot \sqrt{\frac{1 + \cos \phi_1}{2} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 3.25 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternative 6
Error	36.9
Cost	7504

\[\begin{array}{l} t_0 := R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -2.1 \cdot 10^{-134}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -1.16 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5.1 \cdot 10^{-195}:\\ \;\;\;\;R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 3.3 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternative 7
Error	36.1
Cost	7504

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ t_1 := R \cdot \left(\lambda_2 \cdot t_0\right)\\ \mathbf{if}\;\phi_2 \leq -1.1 \cdot 10^{-132}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-155}:\\ \;\;\;\;R \cdot \left(t_0 \cdot \left(-\lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternative 8
Error	27.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00066:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 10^{-24}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternative 9
Error	40.1
Cost	516

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.000295:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 10
Error	38.0
Cost	516

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.6 \cdot 10^{-145}:\\ \;\;\;\;R \cdot \left(\phi_1 + \left(-\phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternative 11
Error	41.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-22}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 0.19:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_1\\ \end{array} \]

Alternative 12
Error	41.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-33}:\\ \;\;\;\;R \cdot \left(-\phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00128:\\ \;\;\;\;R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 13
Error	47.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq 0.125:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_1\\ \end{array} \]

Alternative 14
Error	61.9
Cost	192

\[R \cdot \lambda_1 \]

Alternative 15
Error	53.8
Cost	192

\[R \cdot \phi_1 \]

Reproduce

herbie shell --seed 2023092 
(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))))))