Equirectangular approximation to distance on a great circle

Average Error: 38.9 → 19.5

Time: 22.3s

Precision: binary64

Cost: 63304

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

Math

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

↓

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_2 := R \cdot \sqrt{t_1 \cdot t_1 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+263}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (* R (- phi2 phi1)))
        (t_1 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
        (t_2 (* R (sqrt (+ (* t_1 t_1) (* (- phi1 phi2) (- phi1 phi2)))))))
   (if (<= t_2 (- INFINITY)) t_0 (if (<= t_2 5e+263) t_2 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 = R * (phi2 - phi1);
	double t_1 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double t_2 = R * sqrt(((t_1 * t_1) + ((phi1 - phi2) * (phi1 - phi2))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= 5e+263) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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 = R * (phi2 - phi1);
	double t_1 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	double t_2 = R * Math.sqrt(((t_1 * t_1) + ((phi1 - phi2) * (phi1 - phi2))));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_2 <= 5e+263) {
		tmp = t_2;
	} else {
		tmp = 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 = R * (phi2 - phi1)
	t_1 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	t_2 = R * math.sqrt(((t_1 * t_1) + ((phi1 - phi2) * (phi1 - phi2))))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_0
	elif t_2 <= 5e+263:
		tmp = t_2
	else:
		tmp = 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(R * Float64(phi2 - phi1))
	t_1 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	t_2 = Float64(R * sqrt(Float64(Float64(t_1 * t_1) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= 5e+263)
		tmp = t_2;
	else
		tmp = 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 = R * (phi2 - phi1);
	t_1 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	t_2 = R * sqrt(((t_1 * t_1) + ((phi1 - phi2) * (phi1 - phi2))));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_0;
	elseif (t_2 <= 5e+263)
		tmp = t_2;
	else
		tmp = 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[(R * N[(phi2 - phi1), $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[(R * N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, 5e+263], t$95$2, t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 R (sqrt.f64 (+.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))))) < -inf.0 or 5.00000000000000022e263 < (*.f64 R (sqrt.f64 (+.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 62.5

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

   2. Simplified 62.5

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

      [Start] 62.5	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-2 [=>] 62.5	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [=>] 62.5	\[ R \cdot \sqrt{\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [<=] 62.5	\[ R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around -inf 30.9

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

   4. Simplified 30.9

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

      [Start] 30.9	\[ R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]
      rational.json-simplify-1 [=>] 30.9	\[ R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
      rational.json-simplify-2 [=>] 30.9	\[ R \cdot \left(\phi_2 + \color{blue}{\phi_1 \cdot -1}\right) \]
      rational.json-simplify-9 [=>] 30.9	\[ R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

   5. Taylor expanded in R around 0 30.9

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

   if -inf.0 < (*.f64 R (sqrt.f64 (+.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))))) < 5.00000000000000022e263

   1. Initial program 1.8

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

4. Final simplification 19.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \leq -\infty:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \leq 5 \cdot 10^{+263}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	29.2
Cost	21452

\[\begin{array}{l} t_0 := \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -4 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.65 \cdot 10^{-67}:\\ \;\;\;\;R \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \left(\phi_1 \cdot -0.5\right) + \phi_2 \cdot \left(-0.5 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{+109}:\\ \;\;\;\;R \cdot \sqrt{t_0 \cdot \left(t_0 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	29.3
Cost	21004

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{-250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.45 \cdot 10^{-67}:\\ \;\;\;\;R \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \left(\phi_1 \cdot -0.5\right) + \phi_2 \cdot \left(-0.5 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{+109}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left({\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	31.3
Cost	7768

\[\begin{array}{l} t_0 := R \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1.7 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -6.4 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -2.3 \cdot 10^{-271}:\\ \;\;\;\;R \cdot \left(\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -3.1 \cdot 10^{-293}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 2.7 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 4
Error	31.3
Cost	7768

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.9 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1.08 \cdot 10^{-163}:\\ \;\;\;\;R \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -5.5 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -6.8 \cdot 10^{-271}:\\ \;\;\;\;R \cdot \left(\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -3.1 \cdot 10^{-293}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 2.32 \cdot 10^{-228}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot \left(t_0 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 5
Error	30.8
Cost	7240

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-249}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.9 \cdot 10^{-92}:\\ \;\;\;\;R \cdot \left(\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	32.0
Cost	1112

\[\begin{array}{l} t_0 := \left(\lambda_2 - \lambda_1\right) \cdot R\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -5.3 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -2 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -2.65 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -2.75 \cdot 10^{-271}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -6 \cdot 10^{-298}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 5.2 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 7
Error	37.5
Cost	652

\[\begin{array}{l} t_0 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -4.4 \cdot 10^{-249}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.45 \cdot 10^{-117}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{elif}\;\phi_2 \leq 3.6 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 8
Error	33.6
Cost	584

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5.8 \cdot 10^{-118}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	43.2
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.9 \cdot 10^{-83}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 10
Error	45.9
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023074 
(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))))))