Equirectangular approximation to distance on a great circle

Average Error: 38.8 → 18.9

Time: 21.0s

Precision: binary64

Cost: 35908

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_2 := \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\\ \mathbf{if}\;t_1 \cdot t_1 + t_0 \leq 10^{+307}:\\ \;\;\;\;R \cdot \sqrt{\frac{1}{\frac{1}{t_2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot t_2\right)\right)}} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \end{array} \]

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) (cos (/ (+ phi1 phi2) 2.0))))
        (t_2 (cos (* (+ phi1 phi2) 0.5))))
   (if (<= (+ (* t_1 t_1) t_0) 1e+307)
     (*
      R
      (sqrt
       (+
        (/
         1.0
         (/ 1.0 (* t_2 (* (- lambda1 lambda2) (* (- lambda1 lambda2) t_2)))))
        t_0)))
     (+ (* phi2 R) (* phi1 (- R))))))

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) * cos(((phi1 + phi2) / 2.0));
	double t_2 = cos(((phi1 + phi2) * 0.5));
	double tmp;
	if (((t_1 * t_1) + t_0) <= 1e+307) {
		tmp = R * sqrt(((1.0 / (1.0 / (t_2 * ((lambda1 - lambda2) * ((lambda1 - lambda2) * t_2))))) + t_0));
	} else {
		tmp = (phi2 * R) + (phi1 * -R);
	}
	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) :: tmp
    t_0 = (phi1 - phi2) * (phi1 - phi2)
    t_1 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    t_2 = cos(((phi1 + phi2) * 0.5d0))
    if (((t_1 * t_1) + t_0) <= 1d+307) then
        tmp = r * sqrt(((1.0d0 / (1.0d0 / (t_2 * ((lambda1 - lambda2) * ((lambda1 - lambda2) * t_2))))) + t_0))
    else
        tmp = (phi2 * r) + (phi1 * -r)
    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) * Math.cos(((phi1 + phi2) / 2.0));
	double t_2 = Math.cos(((phi1 + phi2) * 0.5));
	double tmp;
	if (((t_1 * t_1) + t_0) <= 1e+307) {
		tmp = R * Math.sqrt(((1.0 / (1.0 / (t_2 * ((lambda1 - lambda2) * ((lambda1 - lambda2) * t_2))))) + t_0));
	} else {
		tmp = (phi2 * R) + (phi1 * -R);
	}
	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) * math.cos(((phi1 + phi2) / 2.0))
	t_2 = math.cos(((phi1 + phi2) * 0.5))
	tmp = 0
	if ((t_1 * t_1) + t_0) <= 1e+307:
		tmp = R * math.sqrt(((1.0 / (1.0 / (t_2 * ((lambda1 - lambda2) * ((lambda1 - lambda2) * t_2))))) + t_0))
	else:
		tmp = (phi2 * R) + (phi1 * -R)
	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) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	t_2 = cos(Float64(Float64(phi1 + phi2) * 0.5))
	tmp = 0.0
	if (Float64(Float64(t_1 * t_1) + t_0) <= 1e+307)
		tmp = Float64(R * sqrt(Float64(Float64(1.0 / Float64(1.0 / Float64(t_2 * Float64(Float64(lambda1 - lambda2) * Float64(Float64(lambda1 - lambda2) * t_2))))) + t_0)));
	else
		tmp = Float64(Float64(phi2 * R) + Float64(phi1 * Float64(-R)));
	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) * cos(((phi1 + phi2) / 2.0));
	t_2 = cos(((phi1 + phi2) * 0.5));
	tmp = 0.0;
	if (((t_1 * t_1) + t_0) <= 1e+307)
		tmp = R * sqrt(((1.0 / (1.0 / (t_2 * ((lambda1 - lambda2) * ((lambda1 - lambda2) * t_2))))) + t_0));
	else
		tmp = (phi2 * R) + (phi1 * -R);
	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[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision], 1e+307], N[(R * N[Sqrt[N[(N[(1.0 / N[(1.0 / N[(t$95$2 * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(phi2 * R), $MachinePrecision] + N[(phi1 * (-R)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2))) < 9.99999999999999986e306

   1. Initial program 1.9

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

   2. Applied egg-rr 1.9

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

   if 9.99999999999999986e306 < (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))

   1. Initial program 63.8

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

   2. Taylor expanded in phi1 around 0 63.9

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

   3. Simplified 63.8

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

      [Start] 63.9	\[ R \cdot \sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      exponential.json-simplify-27 [=>] 63.8	\[ R \cdot \sqrt{\color{blue}{{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around inf 30.3

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

   5. Simplified 30.3

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

      [Start] 30.3	\[ -1 \cdot \left(\phi_1 \cdot R\right) + R \cdot \phi_2 \]
      rational.json-simplify-1 [=>] 30.3	\[ \color{blue}{R \cdot \phi_2 + -1 \cdot \left(\phi_1 \cdot R\right)} \]
      rational.json-simplify-2 [=>] 30.3	\[ \color{blue}{\phi_2 \cdot R} + -1 \cdot \left(\phi_1 \cdot R\right) \]
      rational.json-simplify-43 [=>] 30.3	\[ \phi_2 \cdot R + \color{blue}{\phi_1 \cdot \left(R \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 30.3	\[ \phi_2 \cdot R + \phi_1 \cdot \color{blue}{\left(-R\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 18.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \leq 10^{+307}:\\ \;\;\;\;R \cdot \sqrt{\frac{1}{\frac{1}{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\right)}} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.8
Cost	35652

\[\begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_1 := t_0 \cdot t_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{if}\;t_1 \leq 10^{+307}:\\ \;\;\;\;R \cdot \sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \end{array} \]

Alternative 2
Error	30.9
Cost	14216

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1650000:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -1.2 \cdot 10^{-235}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_1 + \lambda_1 \cdot \left(-t_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.02 \cdot 10^{-295}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-206}:\\ \;\;\;\;R \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	31.4
Cost	7568

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.5 \cdot 10^{+17}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -7.8 \cdot 10^{-237}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.38 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	34.6
Cost	7240

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\left(-\lambda_1\right) \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.35 \cdot 10^{+176}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	34.3
Cost	7240

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;-\lambda_1 \cdot \left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot -0.5\right) \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.4 \cdot 10^{+176}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	35.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.35 \cdot 10^{+176}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7
Error	35.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.4 \cdot 10^{+176}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot R\right)\\ \end{array} \]

Alternative 8
Error	35.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.4 \cdot 10^{+176}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\phi_2 \cdot -0.5\right) \cdot \left(\lambda_2 \cdot R\right)\\ \end{array} \]

Alternative 9
Error	34.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-\cos \left(\phi_2 \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.4 \cdot 10^{+176}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\phi_2 \cdot -0.5\right) \cdot \left(\lambda_2 \cdot R\right)\\ \end{array} \]

Alternative 10
Error	34.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\left(-\lambda_1\right) \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.4 \cdot 10^{+176}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\phi_2 \cdot -0.5\right) \cdot \left(\lambda_2 \cdot R\right)\\ \end{array} \]

Alternative 11
Error	35.3
Cost	776

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.35 \cdot 10^{+176}:\\ \;\;\;\;\phi_2 \cdot R + \phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]

Alternative 12
Error	43.1
Cost	720

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 2.35 \cdot 10^{-253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5.5 \cdot 10^{-52}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{elif}\;\phi_2 \leq 2.95 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 13
Error	38.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.5 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -3.9 \cdot 10^{-234}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -9.4 \cdot 10^{-296}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 8.5 \cdot 10^{-210}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14
Error	35.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.4 \cdot 10^{+176}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]

Alternative 15
Error	43.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.5 \cdot 10^{-11}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 16
Error	57.1
Cost	192

\[R \cdot \lambda_1 \]

Alternative 17
Error	45.6
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023068 
(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))))))