Equirectangular approximation to distance on a great circle

Average Error: 39.0 → 19.3

Time: 21.6s

Precision: binary64

Cost: 77000

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

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_1 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_2 := \left(\lambda_1 - \lambda_2\right) \cdot t_0\\ t_3 := R \cdot \sqrt{t_2 \cdot t_2 + t_1}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(t_0 \cdot \frac{1}{t_0}\right) \cdot t_0\right) \cdot t_0\right)\right) + t_1}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

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 (cos (/ (+ phi1 phi2) 2.0)))
        (t_1 (* (- phi1 phi2) (- phi1 phi2)))
        (t_2 (* (- lambda1 lambda2) t_0))
        (t_3 (* R (sqrt (+ (* t_2 t_2) t_1)))))
   (if (<= t_3 (- INFINITY))
     (- (* R phi2) (* R phi1))
     (if (<= t_3 2e+307)
       (*
        R
        (sqrt
         (+
          (*
           (- lambda1 lambda2)
           (* (- lambda1 lambda2) (* (* (* t_0 (/ 1.0 t_0)) t_0) t_0)))
          t_1)))
       (* R (- phi2 phi1))))))

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 = (lambda1 - lambda2) * t_0;
	double t_3 = R * sqrt(((t_2 * t_2) + t_1));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (R * phi2) - (R * phi1);
	} else if (t_3 <= 2e+307) {
		tmp = R * sqrt((((lambda1 - lambda2) * ((lambda1 - lambda2) * (((t_0 * (1.0 / t_0)) * t_0) * t_0))) + t_1));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 = (lambda1 - lambda2) * t_0;
	double t_3 = R * Math.sqrt(((t_2 * t_2) + t_1));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (R * phi2) - (R * phi1);
	} else if (t_3 <= 2e+307) {
		tmp = R * Math.sqrt((((lambda1 - lambda2) * ((lambda1 - lambda2) * (((t_0 * (1.0 / t_0)) * t_0) * t_0))) + t_1));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 = (lambda1 - lambda2) * t_0
	t_3 = R * math.sqrt(((t_2 * t_2) + t_1))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (R * phi2) - (R * phi1)
	elif t_3 <= 2e+307:
		tmp = R * math.sqrt((((lambda1 - lambda2) * ((lambda1 - lambda2) * (((t_0 * (1.0 / t_0)) * t_0) * t_0))) + t_1))
	else:
		tmp = R * (phi2 - phi1)
	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(Float64(lambda1 - lambda2) * t_0)
	t_3 = Float64(R * sqrt(Float64(Float64(t_2 * t_2) + t_1)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	elseif (t_3 <= 2e+307)
		tmp = Float64(R * sqrt(Float64(Float64(Float64(lambda1 - lambda2) * Float64(Float64(lambda1 - lambda2) * Float64(Float64(Float64(t_0 * Float64(1.0 / t_0)) * t_0) * t_0))) + t_1)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	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 = (lambda1 - lambda2) * t_0;
	t_3 = R * sqrt(((t_2 * t_2) + t_1));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (R * phi2) - (R * phi1);
	elseif (t_3 <= 2e+307)
		tmp = R * sqrt((((lambda1 - lambda2) * ((lambda1 - lambda2) * (((t_0 * (1.0 / t_0)) * t_0) * t_0))) + t_1));
	else
		tmp = R * (phi2 - phi1);
	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[(N[(lambda1 - lambda2), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+307], N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[(t$95$0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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

   1. Initial program 64.0

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

   2. Taylor expanded in phi1 around -inf 30.8

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

   3. Simplified 30.8

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

      [Start] 30.8	\[ R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 30.8	\[ R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 30.8	\[ R \cdot \left(\phi_2 + \color{blue}{\phi_1 \cdot -1}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 30.8	\[ R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

   4. Applied egg-rr 30.8

      \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   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))))) < 1.99999999999999997e307

   1. Initial program 2.0

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

   2. Simplified 2.0

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

      [Start] 2.0	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 2.0	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.0	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 2.0	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied egg-rr 2.0

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

   if 1.99999999999999997e307 < (*.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 64.0

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

   2. Taylor expanded in phi1 around -inf 31.0

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

   3. Simplified 31.0

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

      [Start] 31.0	\[ R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 31.0	\[ R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 31.0	\[ R \cdot \left(\phi_2 + \color{blue}{\phi_1 \cdot -1}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 31.0	\[ R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

   4. Taylor expanded in R around 0 31.0

      \[\leadsto \color{blue}{R \cdot \left(\phi_2 - \phi_1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 19.3

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

Alternatives

Alternative 1
Error	19.3
Cost	63304

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

Alternative 2
Error	27.9
Cost	21388

\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+66}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left({\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]

Alternative 3
Error	28.0
Cost	8268

\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+66}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]

Alternative 4
Error	28.2
Cost	7820

\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+90}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+66}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]

Alternative 5
Error	32.4
Cost	7568

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_1 \leq -2.1 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -2.7 \cdot 10^{-190}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 3.5 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 6
Error	30.9
Cost	7568

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_1 \leq -2.2 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.9 \cdot 10^{-149}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 2 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 7
Error	32.5
Cost	7376

\[\begin{array}{l} t_0 := R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-32}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_1 \leq -2.9 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -2.6 \cdot 10^{-190}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 1.6 \cdot 10^{-276}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 8
Error	32.3
Cost	7376

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.4 \cdot 10^{-41}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_1 \leq -2.9 \cdot 10^{-74}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.45 \cdot 10^{-190}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 1.45 \cdot 10^{-276}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 9
Error	34.2
Cost	7240

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ \mathbf{if}\;\lambda_2 \leq -1.3 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \left(-t_0 \cdot \lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.3 \cdot 10^{+137}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \end{array} \]

Alternative 10
Error	29.2
Cost	7108

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.8 \cdot 10^{+136}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	29.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.2 \cdot 10^{+137}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \end{array} \]

Alternative 12
Error	30.4
Cost	448

\[R \cdot \phi_2 - R \cdot \phi_1 \]

Alternative 13
Error	34.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.98 \cdot 10^{+39}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14
Error	30.4
Cost	320

\[R \cdot \left(\phi_2 - \phi_1\right) \]

Alternative 15
Error	57.3
Cost	192

\[R \cdot \lambda_1 \]

Alternative 16
Error	46.6
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023090 
(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))))))