Equirectangular approximation to distance on a great circle

Average Error: 39.3 → 24.5

Time: 15.1s

Precision: binary64

Cost: 21320

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

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_2 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_3 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_4 := R \cdot \left(t_0 \cdot \left(-\lambda_1\right) + \lambda_2 \cdot t_0\right)\\ \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1.1 \cdot 10^{+37}:\\ \;\;\;\;R \cdot \sqrt{t_3 \cdot t_3 + t_2}\\ \mathbf{elif}\;\phi_1 \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -5 \cdot 10^{-32}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_1 \leq -6 \cdot 10^{-102}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left({t_0}^{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) + t_2}\\ \mathbf{elif}\;\phi_1 \leq 7 \cdot 10^{-184}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;R \cdot \left(t_0 \cdot \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (cos (* 0.5 phi2)))
        (t_1 (* R (- phi2 phi1)))
        (t_2 (* (- phi1 phi2) (- phi1 phi2)))
        (t_3 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
        (t_4 (* R (+ (* t_0 (- lambda1)) (* lambda2 t_0)))))
   (if (<= phi1 -6e+86)
     t_1
     (if (<= phi1 -1.1e+37)
       (* R (sqrt (+ (* t_3 t_3) t_2)))
       (if (<= phi1 -1.7e-9)
         t_1
         (if (<= phi1 -5e-32)
           t_4
           (if (<= phi1 -6e-102)
             (*
              R
              (sqrt
               (+
                (* (- lambda1 lambda2) (* (pow t_0 2.0) (- lambda1 lambda2)))
                t_2)))
             (if (<= phi1 7e-184)
               t_4
               (if (<= phi1 7.2e-142)
                 (* R phi2)
                 (if (<= phi1 6.8e-96) (* R (* t_0 lambda2)) 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 = cos((0.5 * phi2));
	double t_1 = R * (phi2 - phi1);
	double t_2 = (phi1 - phi2) * (phi1 - phi2);
	double t_3 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double t_4 = R * ((t_0 * -lambda1) + (lambda2 * t_0));
	double tmp;
	if (phi1 <= -6e+86) {
		tmp = t_1;
	} else if (phi1 <= -1.1e+37) {
		tmp = R * sqrt(((t_3 * t_3) + t_2));
	} else if (phi1 <= -1.7e-9) {
		tmp = t_1;
	} else if (phi1 <= -5e-32) {
		tmp = t_4;
	} else if (phi1 <= -6e-102) {
		tmp = R * sqrt((((lambda1 - lambda2) * (pow(t_0, 2.0) * (lambda1 - lambda2))) + t_2));
	} else if (phi1 <= 7e-184) {
		tmp = t_4;
	} else if (phi1 <= 7.2e-142) {
		tmp = R * phi2;
	} else if (phi1 <= 6.8e-96) {
		tmp = R * (t_0 * lambda2);
	} 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = cos((0.5d0 * phi2))
    t_1 = r * (phi2 - phi1)
    t_2 = (phi1 - phi2) * (phi1 - phi2)
    t_3 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    t_4 = r * ((t_0 * -lambda1) + (lambda2 * t_0))
    if (phi1 <= (-6d+86)) then
        tmp = t_1
    else if (phi1 <= (-1.1d+37)) then
        tmp = r * sqrt(((t_3 * t_3) + t_2))
    else if (phi1 <= (-1.7d-9)) then
        tmp = t_1
    else if (phi1 <= (-5d-32)) then
        tmp = t_4
    else if (phi1 <= (-6d-102)) then
        tmp = r * sqrt((((lambda1 - lambda2) * ((t_0 ** 2.0d0) * (lambda1 - lambda2))) + t_2))
    else if (phi1 <= 7d-184) then
        tmp = t_4
    else if (phi1 <= 7.2d-142) then
        tmp = r * phi2
    else if (phi1 <= 6.8d-96) then
        tmp = r * (t_0 * lambda2)
    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 = Math.cos((0.5 * phi2));
	double t_1 = R * (phi2 - phi1);
	double t_2 = (phi1 - phi2) * (phi1 - phi2);
	double t_3 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	double t_4 = R * ((t_0 * -lambda1) + (lambda2 * t_0));
	double tmp;
	if (phi1 <= -6e+86) {
		tmp = t_1;
	} else if (phi1 <= -1.1e+37) {
		tmp = R * Math.sqrt(((t_3 * t_3) + t_2));
	} else if (phi1 <= -1.7e-9) {
		tmp = t_1;
	} else if (phi1 <= -5e-32) {
		tmp = t_4;
	} else if (phi1 <= -6e-102) {
		tmp = R * Math.sqrt((((lambda1 - lambda2) * (Math.pow(t_0, 2.0) * (lambda1 - lambda2))) + t_2));
	} else if (phi1 <= 7e-184) {
		tmp = t_4;
	} else if (phi1 <= 7.2e-142) {
		tmp = R * phi2;
	} else if (phi1 <= 6.8e-96) {
		tmp = R * (t_0 * lambda2);
	} 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 = math.cos((0.5 * phi2))
	t_1 = R * (phi2 - phi1)
	t_2 = (phi1 - phi2) * (phi1 - phi2)
	t_3 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	t_4 = R * ((t_0 * -lambda1) + (lambda2 * t_0))
	tmp = 0
	if phi1 <= -6e+86:
		tmp = t_1
	elif phi1 <= -1.1e+37:
		tmp = R * math.sqrt(((t_3 * t_3) + t_2))
	elif phi1 <= -1.7e-9:
		tmp = t_1
	elif phi1 <= -5e-32:
		tmp = t_4
	elif phi1 <= -6e-102:
		tmp = R * math.sqrt((((lambda1 - lambda2) * (math.pow(t_0, 2.0) * (lambda1 - lambda2))) + t_2))
	elif phi1 <= 7e-184:
		tmp = t_4
	elif phi1 <= 7.2e-142:
		tmp = R * phi2
	elif phi1 <= 6.8e-96:
		tmp = R * (t_0 * lambda2)
	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 = cos(Float64(0.5 * phi2))
	t_1 = Float64(R * Float64(phi2 - phi1))
	t_2 = Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2))
	t_3 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	t_4 = Float64(R * Float64(Float64(t_0 * Float64(-lambda1)) + Float64(lambda2 * t_0)))
	tmp = 0.0
	if (phi1 <= -6e+86)
		tmp = t_1;
	elseif (phi1 <= -1.1e+37)
		tmp = Float64(R * sqrt(Float64(Float64(t_3 * t_3) + t_2)));
	elseif (phi1 <= -1.7e-9)
		tmp = t_1;
	elseif (phi1 <= -5e-32)
		tmp = t_4;
	elseif (phi1 <= -6e-102)
		tmp = Float64(R * sqrt(Float64(Float64(Float64(lambda1 - lambda2) * Float64((t_0 ^ 2.0) * Float64(lambda1 - lambda2))) + t_2)));
	elseif (phi1 <= 7e-184)
		tmp = t_4;
	elseif (phi1 <= 7.2e-142)
		tmp = Float64(R * phi2);
	elseif (phi1 <= 6.8e-96)
		tmp = Float64(R * Float64(t_0 * lambda2));
	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 = cos((0.5 * phi2));
	t_1 = R * (phi2 - phi1);
	t_2 = (phi1 - phi2) * (phi1 - phi2);
	t_3 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	t_4 = R * ((t_0 * -lambda1) + (lambda2 * t_0));
	tmp = 0.0;
	if (phi1 <= -6e+86)
		tmp = t_1;
	elseif (phi1 <= -1.1e+37)
		tmp = R * sqrt(((t_3 * t_3) + t_2));
	elseif (phi1 <= -1.7e-9)
		tmp = t_1;
	elseif (phi1 <= -5e-32)
		tmp = t_4;
	elseif (phi1 <= -6e-102)
		tmp = R * sqrt((((lambda1 - lambda2) * ((t_0 ^ 2.0) * (lambda1 - lambda2))) + t_2));
	elseif (phi1 <= 7e-184)
		tmp = t_4;
	elseif (phi1 <= 7.2e-142)
		tmp = R * phi2;
	elseif (phi1 <= 6.8e-96)
		tmp = R * (t_0 * lambda2);
	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[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(R * N[(N[(t$95$0 * (-lambda1)), $MachinePrecision] + N[(lambda2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -6e+86], t$95$1, If[LessEqual[phi1, -1.1e+37], N[(R * N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -1.7e-9], t$95$1, If[LessEqual[phi1, -5e-32], t$95$4, If[LessEqual[phi1, -6e-102], N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 7e-184], t$95$4, If[LessEqual[phi1, 7.2e-142], N[(R * phi2), $MachinePrecision], If[LessEqual[phi1, 6.8e-96], N[(R * N[(t$95$0 * lambda2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if phi1 < -5.99999999999999954e86 or -1.1e37 < phi1 < -1.6999999999999999e-9 or 6.8000000000000002e-96 < phi1

   1. Initial program 49.0

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

   2. Simplified 49.0

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

      [Start] 49.0	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-31 [=>] 49.0	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around -inf 20.7

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

   4. Simplified 20.7

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

      [Start] 20.7	\[ R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]
      rational.json-simplify-2 [=>] 20.7	\[ R \cdot \left(\color{blue}{\phi_1 \cdot -1} + \phi_2\right) \]
      rational.json-simplify-9 [<=] 20.7	\[ R \cdot \left(\color{blue}{\left(-\phi_1\right)} + \phi_2\right) \]
      rational.json-simplify-41 [<=] 20.7	\[ R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)} \]

   if -5.99999999999999954e86 < phi1 < -1.1e37

   1. Initial program 30.6

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

   if -1.6999999999999999e-9 < phi1 < -5e-32 or -6e-102 < phi1 < 6.99999999999999962e-184

   1. Initial program 31.1

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

   2. Simplified 31.1

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

      [Start] 31.1	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-31 [=>] 31.1	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in lambda1 around -inf 24.9

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

   4. Simplified 24.9

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

      [Start] 24.9	\[ R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) + -1 \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right) \]
      rational.json-simplify-31 [=>] 24.9	\[ R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) + \color{blue}{\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(-1 \cdot \lambda_1\right)}\right) \]
      rational.json-simplify-2 [=>] 24.9	\[ R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) + \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \color{blue}{\left(\lambda_1 \cdot -1\right)}\right) \]
      rational.json-simplify-9 [<=] 24.9	\[ R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) + \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \color{blue}{\left(-\lambda_1\right)}\right) \]

   5. Taylor expanded in phi1 around 0 24.9

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

   6. Simplified 24.9

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

      [Start] 24.9	\[ \left(-1 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right) + \lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right) \cdot R \]
      rational.json-simplify-2 [=>] 24.9	\[ \color{blue}{R \cdot \left(-1 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right) + \lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)} \]
      rational.json-simplify-31 [=>] 24.9	\[ R \cdot \left(\color{blue}{\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-1 \cdot \lambda_1\right)} + \lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right) \]
      rational.json-simplify-2 [=>] 24.9	\[ R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \color{blue}{\left(\lambda_1 \cdot -1\right)} + \lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right) \]
      rational.json-simplify-9 [<=] 24.9	\[ R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \color{blue}{\left(-\lambda_1\right)} + \lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right) \]

   if -5e-32 < phi1 < -6e-102

   1. Initial program 28.9

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

   2. Simplified 28.9

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

      [Start] 28.9	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-31 [=>] 28.9	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around 0 28.9

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

   if 6.99999999999999962e-184 < phi1 < 7.20000000000000001e-142

   1. Initial program 33.7

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

   2. Simplified 33.7

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

      [Start] 33.7	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-31 [=>] 33.7	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi2 around inf 30.7

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

   if 7.20000000000000001e-142 < phi1 < 6.8000000000000002e-96

   1. Initial program 37.9

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

   2. Simplified 37.9

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

      [Start] 37.9	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-31 [=>] 37.9	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in lambda2 around inf 50.4

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

   4. Taylor expanded in phi1 around 0 50.4

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)} \]

   5. Simplified 50.4

      \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2\right)} \]
      Proof

      [Start] 50.4	\[ R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right) \]
      rational.json-simplify-2 [=>] 50.4	\[ R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2\right)} \]

3. Recombined 6 regimes into one program.

4. Final simplification 24.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{+86}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.1 \cdot 10^{+37}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -5 \cdot 10^{-32}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_1\right) + \lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -6 \cdot 10^{-102}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left({\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 7 \cdot 10^{-184}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_1\right) + \lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	26.2
Cost	21908

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_2 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ t_3 := R \cdot \left(\lambda_2 \cdot t_2 + t_2 \cdot \left(-\lambda_1\right)\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+215}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -6 \cdot 10^{+126}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+73}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left({\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) + t_1}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -10000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-51}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	27.4
Cost	14996

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := \cos \left(0.5 \cdot \phi_2\right)\\ t_2 := R \cdot \left(t_1 \cdot \left(-\lambda_1\right) + \lambda_2 \cdot t_1\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -6 \cdot 10^{+126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -10000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-51}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	26.7
Cost	14996

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_2 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ t_3 := R \cdot \left(\lambda_2 \cdot t_2 + t_2 \cdot \left(-\lambda_1\right)\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+215}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -6 \cdot 10^{+126}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+73}:\\ \;\;\;\;R \cdot \left(t_0 \cdot \left(-\lambda_1\right) + \lambda_2 \cdot t_0\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -10000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-51}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	25.2
Cost	7376

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 5.3 \cdot 10^{-183}:\\ \;\;\;\;R \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 9.6 \cdot 10^{-144}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 10^{-98}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	29.2
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_1 \leq -1.8 \cdot 10^{+188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq -4.5 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq -1.85 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 5 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]

Alternative 6
Error	34.9
Cost	784

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -4.65 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -3.7 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -3.4 \cdot 10^{-218}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{elif}\;\phi_1 \leq 4.3 \cdot 10^{-183}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 7
Error	24.9
Cost	648

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.000115:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 4.6 \cdot 10^{-183}:\\ \;\;\;\;R \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	39.4
Cost	388

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 9
Error	39.2
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 10
Error	46.0
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023053 
(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))))))