Equirectangular approximation to distance on a great circle

Average Error: 39.1 → 19.4

Time: 34.0s

Precision: binary64

Cost: 63304

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

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_1 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_2 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_3 := \left(\lambda_1 - \lambda_2\right) \cdot t_0\\ t_4 := R \cdot \sqrt{t_3 \cdot t_3 + t_1}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;R \cdot \sqrt{t_0 \cdot \left(t_0 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (/ (+ phi1 phi2) 2.0)))
        (t_1 (* (- phi1 phi2) (- phi1 phi2)))
        (t_2 (* R (+ phi2 (- phi1))))
        (t_3 (* (- lambda1 lambda2) t_0))
        (t_4 (* R (sqrt (+ (* t_3 t_3) t_1)))))
   (if (<= t_4 (- INFINITY))
     t_2
     (if (<= t_4 2e+304)
       (*
        R
        (sqrt
         (+ (* t_0 (* t_0 (* (- lambda1 lambda2) (- lambda1 lambda2)))) t_1)))
       t_2))))

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(((phi1 + phi2) / 2.0));
	double t_1 = (phi1 - phi2) * (phi1 - phi2);
	double t_2 = R * (phi2 + -phi1);
	double t_3 = (lambda1 - lambda2) * t_0;
	double t_4 = R * sqrt(((t_3 * t_3) + t_1));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_4 <= 2e+304) {
		tmp = R * sqrt(((t_0 * (t_0 * ((lambda1 - lambda2) * (lambda1 - lambda2)))) + t_1));
	} else {
		tmp = t_2;
	}
	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 = Math.cos(((phi1 + phi2) / 2.0));
	double t_1 = (phi1 - phi2) * (phi1 - phi2);
	double t_2 = R * (phi2 + -phi1);
	double t_3 = (lambda1 - lambda2) * t_0;
	double t_4 = R * Math.sqrt(((t_3 * t_3) + t_1));
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_4 <= 2e+304) {
		tmp = R * Math.sqrt(((t_0 * (t_0 * ((lambda1 - lambda2) * (lambda1 - lambda2)))) + t_1));
	} else {
		tmp = t_2;
	}
	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(((phi1 + phi2) / 2.0))
	t_1 = (phi1 - phi2) * (phi1 - phi2)
	t_2 = R * (phi2 + -phi1)
	t_3 = (lambda1 - lambda2) * t_0
	t_4 = R * math.sqrt(((t_3 * t_3) + t_1))
	tmp = 0
	if t_4 <= -math.inf:
		tmp = t_2
	elif t_4 <= 2e+304:
		tmp = R * math.sqrt(((t_0 * (t_0 * ((lambda1 - lambda2) * (lambda1 - lambda2)))) + t_1))
	else:
		tmp = t_2
	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(Float64(phi1 + phi2) / 2.0))
	t_1 = Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2))
	t_2 = Float64(R * Float64(phi2 + Float64(-phi1)))
	t_3 = Float64(Float64(lambda1 - lambda2) * t_0)
	t_4 = Float64(R * sqrt(Float64(Float64(t_3 * t_3) + t_1)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_4 <= 2e+304)
		tmp = Float64(R * sqrt(Float64(Float64(t_0 * Float64(t_0 * Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)))) + t_1)));
	else
		tmp = t_2;
	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(((phi1 + phi2) / 2.0));
	t_1 = (phi1 - phi2) * (phi1 - phi2);
	t_2 = R * (phi2 + -phi1);
	t_3 = (lambda1 - lambda2) * t_0;
	t_4 = R * sqrt(((t_3 * t_3) + t_1));
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = t_2;
	elseif (t_4 <= 2e+304)
		tmp = R * sqrt(((t_0 * (t_0 * ((lambda1 - lambda2) * (lambda1 - lambda2)))) + t_1));
	else
		tmp = t_2;
	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[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(phi2 + (-phi1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(lambda1 - lambda2), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(R * N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$2, If[LessEqual[t$95$4, 2e+304], N[(R * N[Sqrt[N[(N[(t$95$0 * N[(t$95$0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

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 1.9999999999999999e304 < (*.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 63.9

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

   2. Simplified 63.9

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

      [Start] 63.9	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-2 [=>] 63.9	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [=>] 63.9	\[ R \cdot \sqrt{\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [<=] 63.9	\[ R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around -inf 31.1

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

   4. Simplified 31.1

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

      [Start] 31.1	\[ R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]
      rational.json-simplify-1 [=>] 31.1	\[ R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
      rational.json-simplify-2 [=>] 31.1	\[ R \cdot \left(\phi_2 + \color{blue}{\phi_1 \cdot -1}\right) \]
      rational.json-simplify-9 [=>] 31.1	\[ R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

   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))))) < 1.9999999999999999e304

   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. Simplified 1.9

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

      [Start] 1.9	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-2 [=>] 1.9	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [=>] 1.9	\[ R \cdot \sqrt{\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [<=] 1.9	\[ R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 19.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \leq -\infty:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \mathbf{elif}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	33.1
Cost	21324

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ t_1 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_2 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(t_2 \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 3.5 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 8.5 \cdot 10^{+144}:\\ \;\;\;\;R \cdot \sqrt{t_0 \cdot \frac{t_0}{\frac{1}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 6.2 \cdot 10^{+184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 4.2 \cdot 10^{+270}:\\ \;\;\;\;\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(R \cdot \left(\left(-\lambda_1\right) + \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(-t_2\right)\right)\\ \end{array} \]

Alternative 2
Error	29.3
Cost	21064

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ t_1 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_2 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_3 := \cos \left(\phi_1 \cdot -0.5\right)\\ \mathbf{if}\;\phi_1 \leq -9.5 \cdot 10^{+130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -7.5 \cdot 10^{+53}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(t_3 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot t_3\right)\right) + t_1}\\ \mathbf{elif}\;\phi_1 \leq -4.1 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -8.8 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -6.8 \cdot 10^{-284}:\\ \;\;\;\;R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2} + t_1}\\ \mathbf{elif}\;\phi_1 \leq 3.6 \cdot 10^{-261}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 3
Error	29.9
Cost	14356

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ t_1 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1.12 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -2.8 \cdot 10^{+54}:\\ \;\;\;\;\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(R \cdot \left(\left(-\lambda_1\right) + \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -0.00072:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -9 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -6.2 \cdot 10^{-284}:\\ \;\;\;\;R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.9 \cdot 10^{-261}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 4
Error	35.8
Cost	7704

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_1 := \cos \left(\phi_1 \cdot -0.5\right)\\ t_2 := R \cdot \left(\lambda_2 \cdot t_1\right)\\ \mathbf{if}\;\lambda_2 \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;\lambda_1 \cdot \left(-\cos \left(0.5 \cdot \phi_2\right) \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 1.6 \cdot 10^{+186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 2.5 \cdot 10^{+271}:\\ \;\;\;\;t_1 \cdot \left(\lambda_2 \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.5 \cdot 10^{+298}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	35.7
Cost	7704

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_1 := \cos \left(\phi_1 \cdot -0.5\right)\\ \mathbf{if}\;\lambda_2 \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;\lambda_1 \cdot \left(-\cos \left(0.5 \cdot \phi_2\right) \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 9.5 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 6 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 10^{+186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 2.3 \cdot 10^{+271}:\\ \;\;\;\;t_1 \cdot \left(\lambda_2 \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.65 \cdot 10^{+298}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_1\right)\\ \end{array} \]

Alternative 6
Error	35.5
Cost	7704

\[\begin{array}{l} t_0 := \cos \left(\phi_1 \cdot -0.5\right)\\ t_1 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_2 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-t_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{+136}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 6.2 \cdot 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.4 \cdot 10^{+271}:\\ \;\;\;\;t_0 \cdot \left(\lambda_2 \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 3.4 \cdot 10^{+298}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \end{array} \]

Alternative 7
Error	35.4
Cost	7700

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_1 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ t_2 := -t_1\\ \mathbf{if}\;\lambda_2 \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot t_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 9.6 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(\lambda_2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_2\right)\\ \end{array} \]

Alternative 8
Error	35.4
Cost	7700

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_1 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;\lambda_1 \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.12 \cdot 10^{+136}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.8 \cdot 10^{+186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{+271}:\\ \;\;\;\;\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(\lambda_2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(-t_1\right)\right)\\ \end{array} \]

Alternative 9
Error	35.0
Cost	7700

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_1 := \cos \left(\phi_1 \cdot -0.5\right) \cdot \left(R \cdot \left(\left(-\lambda_1\right) + \lambda_2\right)\right)\\ \mathbf{if}\;\lambda_2 \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;\lambda_1 \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.46 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 5 \cdot 10^{+185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.25 \cdot 10^{+271}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(-\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	32.9
Cost	7700

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ t_1 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ t_2 := R \cdot \left(t_1 \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{if}\;\lambda_2 \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 9.5 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 7.5 \cdot 10^{+137}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 2.5 \cdot 10^{+271}:\\ \;\;\;\;\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(R \cdot \left(\left(-\lambda_1\right) + \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(-t_1\right)\right)\\ \end{array} \]

Alternative 11
Error	31.5
Cost	7572

\[\begin{array}{l} t_0 := \cos \left(\phi_1 \cdot -0.5\right)\\ t_1 := R \cdot \left(\lambda_2 \cdot t_0\right)\\ t_2 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 4.6 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 8 \cdot 10^{+185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 2.3 \cdot 10^{+271}:\\ \;\;\;\;t_0 \cdot \left(\lambda_2 \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.22 \cdot 10^{+299}:\\ \;\;\;\;-\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	31.5
Cost	7572

\[\begin{array}{l} t_0 := \cos \left(\phi_1 \cdot -0.5\right)\\ t_1 := R \cdot \left(\lambda_2 \cdot t_0\right)\\ t_2 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq 1.45 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 9.5 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 2 \cdot 10^{+187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 2.7 \cdot 10^{+271}:\\ \;\;\;\;t_0 \cdot \left(\lambda_2 \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.3 \cdot 10^{+298}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	31.1
Cost	7244

\[\begin{array}{l} t_0 := R \cdot \left(\lambda_2 \cdot \cos \left(\phi_1 \cdot -0.5\right)\right)\\ t_1 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 6.4 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 5.2 \cdot 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	31.2
Cost	7244

\[\begin{array}{l} t_0 := \cos \left(\phi_1 \cdot -0.5\right)\\ t_1 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 3.1 \cdot 10^{+145}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.8 \cdot 10^{+187}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\lambda_2 \cdot R\right)\\ \end{array} \]

Alternative 15
Error	31.3
Cost	780

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 + \left(-\phi_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 4 \cdot 10^{+145}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.25 \cdot 10^{+186}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]

Alternative 16
Error	35.5
Cost	720

\[\begin{array}{l} t_0 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.42 \cdot 10^{-136}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{elif}\;\phi_2 \leq 3.05 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 17
Error	43.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.6 \cdot 10^{-10}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 18
Error	46.1
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023063 
(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))))))