Equirectangular approximation to distance on a great circle

Average Error: 39.2 → 18.9

Time: 1.9min

Precision: binary64

Cost: 76104

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_2 := R \cdot \sqrt{t_1 \cdot t_1 + t_0}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + \frac{\cos \phi_1 \cdot \cos \phi_2 - \sin \phi_1 \cdot \sin \phi_2}{2}\right)\right) + t_0}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

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 (* (- phi1 phi2) (- phi1 phi2)))
        (t_1 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
        (t_2 (* R (sqrt (+ (* t_1 t_1) t_0)))))
   (if (<= t_2 (- INFINITY))
     (- (* R (+ phi2 phi2)) (* R (+ phi1 phi2)))
     (if (<= t_2 5e+273)
       (*
        R
        (sqrt
         (+
          (*
           (- lambda1 lambda2)
           (*
            (- lambda1 lambda2)
            (+
             0.5
             (/ (- (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))) 2.0))))
          t_0)))
       (* 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 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double t_2 = R * sqrt(((t_1 * t_1) + t_0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (R * (phi2 + phi2)) - (R * (phi1 + phi2));
	} else if (t_2 <= 5e+273) {
		tmp = R * sqrt((((lambda1 - lambda2) * ((lambda1 - lambda2) * (0.5 + (((cos(phi1) * cos(phi2)) - (sin(phi1) * sin(phi2))) / 2.0)))) + t_0));
	} 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 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	double t_2 = R * Math.sqrt(((t_1 * t_1) + t_0));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (R * (phi2 + phi2)) - (R * (phi1 + phi2));
	} else if (t_2 <= 5e+273) {
		tmp = R * Math.sqrt((((lambda1 - lambda2) * ((lambda1 - lambda2) * (0.5 + (((Math.cos(phi1) * Math.cos(phi2)) - (Math.sin(phi1) * Math.sin(phi2))) / 2.0)))) + t_0));
	} 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 = (phi1 - phi2) * (phi1 - phi2)
	t_1 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	t_2 = R * math.sqrt(((t_1 * t_1) + t_0))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (R * (phi2 + phi2)) - (R * (phi1 + phi2))
	elif t_2 <= 5e+273:
		tmp = R * math.sqrt((((lambda1 - lambda2) * ((lambda1 - lambda2) * (0.5 + (((math.cos(phi1) * math.cos(phi2)) - (math.sin(phi1) * math.sin(phi2))) / 2.0)))) + t_0))
	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 = Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2))
	t_1 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	t_2 = Float64(R * sqrt(Float64(Float64(t_1 * t_1) + t_0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(R * Float64(phi2 + phi2)) - Float64(R * Float64(phi1 + phi2)));
	elseif (t_2 <= 5e+273)
		tmp = Float64(R * sqrt(Float64(Float64(Float64(lambda1 - lambda2) * Float64(Float64(lambda1 - lambda2) * Float64(0.5 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) - Float64(sin(phi1) * sin(phi2))) / 2.0)))) + t_0)));
	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 = (phi1 - phi2) * (phi1 - phi2);
	t_1 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	t_2 = R * sqrt(((t_1 * t_1) + t_0));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (R * (phi2 + phi2)) - (R * (phi1 + phi2));
	elseif (t_2 <= 5e+273)
		tmp = R * sqrt((((lambda1 - lambda2) * ((lambda1 - lambda2) * (0.5 + (((cos(phi1) * cos(phi2)) - (sin(phi1) * sin(phi2))) / 2.0)))) + t_0));
	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[(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[(R * N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(R * N[(phi2 + phi2), $MachinePrecision]), $MachinePrecision] - N[(R * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+273], N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(0.5 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $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. Simplified 64.0

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

      [Start] 64.0	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi2 around 0 64.0

      \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + \frac{\color{blue}{\cos \phi_1 + -1 \cdot \left(\sin \phi_1 \cdot \phi_2\right)}}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   4. Simplified 64.0

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

      [Start] 64.0	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + \frac{\cos \phi_1 + -1 \cdot \left(\sin \phi_1 \cdot \phi_2\right)}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-50 [=>] 64.0	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + \frac{\cos \phi_1 + \color{blue}{\phi_2 \cdot \left(\sin \phi_1 \cdot -1\right)}}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-10 [=>] 64.0	\[ R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + \frac{\cos \phi_1 + \phi_2 \cdot \color{blue}{\left(-\sin \phi_1\right)}}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   5. Taylor expanded in phi1 around -inf 29.8

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

   6. Simplified 29.8

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

      [Start] 29.8	\[ R \cdot \phi_2 + -1 \cdot \left(\phi_1 \cdot R\right) \]
      rational_best-simplify-1 [=>] 29.8	\[ \color{blue}{\phi_2 \cdot R} + -1 \cdot \left(\phi_1 \cdot R\right) \]
      rational_best-simplify-1 [=>] 29.8	\[ \phi_2 \cdot R + -1 \cdot \color{blue}{\left(R \cdot \phi_1\right)} \]
      rational_best-simplify-50 [=>] 29.8	\[ \phi_2 \cdot R + \color{blue}{\phi_1 \cdot \left(R \cdot -1\right)} \]
      rational_best-simplify-10 [=>] 29.8	\[ \phi_2 \cdot R + \phi_1 \cdot \color{blue}{\left(-R\right)} \]

   7. Applied egg-rr 29.9

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

   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))))) < 4.99999999999999961e273

   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 2.0

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

      [Start] 1.9	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Applied egg-rr 0.8

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

   if 4.99999999999999961e273 < (*.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 61.8

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

   2. Simplified 61.8

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

      [Start] 61.8	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-1 [=>] 61.8	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational_best-simplify-50 [=>] 61.8	\[ R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around -inf 30.8

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

   4. 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-simplify-3 [=>] 30.8	\[ R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
      rational_best-simplify-1 [=>] 30.8	\[ R \cdot \left(\phi_2 + \color{blue}{\phi_1 \cdot -1}\right) \]
      rational_best-simplify-10 [=>] 30.8	\[ R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

   5. Taylor expanded in R around 0 30.8

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

4. Final simplification 18.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \leq -\infty:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \leq 5 \cdot 10^{+273}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + \frac{\cos \phi_1 \cdot \cos \phi_2 - \sin \phi_1 \cdot \sin \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.3
Cost	70472

\[\begin{array}{l} t_0 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ t_2 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_3 := R \cdot \sqrt{t_2 \cdot t_2 + t_0}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{\left(1 + t_1\right) - \left(1 - t_1\right)}{2} \cdot t_2\right) + t_0}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 2
Error	19.3
Cost	63304

\[\begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_1 := R \cdot \sqrt{t_0 \cdot t_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 3
Error	31.6
Cost	14216

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq 10^{-189}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0 + \lambda_1 \cdot \left(-t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 5.8 \cdot 10^{-167}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-78}:\\ \;\;\;\;t_0 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 4
Error	33.6
Cost	7772

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -0.00043:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -3.4 \cdot 10^{-72}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -4.5 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -4.2 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1.75 \cdot 10^{-246}:\\ \;\;\;\;R \cdot \left(0.25 \cdot \left(\phi_2 \cdot \left(\phi_1 \cdot \lambda_1\right)\right) - \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq 3.8 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	33.6
Cost	7772

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00043:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -2.1 \cdot 10^{-72}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -9.5 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -4.2 \cdot 10^{-136}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -4.2 \cdot 10^{-246}:\\ \;\;\;\;R \cdot \left(0.25 \cdot \left(\phi_2 \cdot \left(\phi_1 \cdot \lambda_1\right)\right) - \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq 4.2 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 3.3 \cdot 10^{-193}:\\ \;\;\;\;\lambda_2 \cdot \left(t_0 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	33.6
Cost	7772

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00075:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -7.8 \cdot 10^{-78}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -2.85 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -6.2 \cdot 10^{-136}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -1.4 \cdot 10^{-246}:\\ \;\;\;\;R \cdot \left(0.25 \cdot \left(\phi_2 \cdot \left(\phi_1 \cdot \lambda_1\right)\right) - \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq 4.8 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 7.4 \cdot 10^{-190}:\\ \;\;\;\;\lambda_2 \cdot \left(t_0 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	33.5
Cost	7772

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00075:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -3.5 \cdot 10^{-72}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -4.2 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -8.5 \cdot 10^{-136}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -3.6 \cdot 10^{-246}:\\ \;\;\;\;R \cdot \left(0.25 \cdot \left(\phi_2 \cdot \left(\phi_1 \cdot \lambda_1\right)\right) - \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq 3.9 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-192}:\\ \;\;\;\;\lambda_2 \cdot \left(t_0 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	33.4
Cost	7772

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00043:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -2.4 \cdot 10^{-72}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -6 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -6 \cdot 10^{-136}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -5.4 \cdot 10^{-246}:\\ \;\;\;\;\left(t_0 \cdot \lambda_1\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq 4.6 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 8.5 \cdot 10^{-190}:\\ \;\;\;\;\lambda_2 \cdot \left(t_0 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	33.4
Cost	7772

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00043:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -7.8 \cdot 10^{-72}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -5 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -5.1 \cdot 10^{-136}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -7.5 \cdot 10^{-246}:\\ \;\;\;\;\left(t_0 \cdot \lambda_1\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq 4.5 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 6 \cdot 10^{-190}:\\ \;\;\;\;\lambda_2 \cdot \left(t_0 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	33.2
Cost	7772

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ t_1 := \cos \left(0.5 \cdot \phi_2\right)\\ t_2 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.0052:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{-77}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -8.2 \cdot 10^{-92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -1.8 \cdot 10^{-135}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -4.5 \cdot 10^{-246}:\\ \;\;\;\;\left(t_1 \cdot \lambda_1\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq 4.5 \cdot 10^{-224}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 3.1 \cdot 10^{-193}:\\ \;\;\;\;\lambda_2 \cdot \left(t_1 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	31.6
Cost	7632

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq 1.35 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 4.2 \cdot 10^{-167}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 12
Error	33.0
Cost	7304

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{+28}:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq 10^{-189}:\\ \;\;\;\;R \cdot \left(\lambda_2 + \cos \left(\phi_1 \cdot 0.5\right) \cdot \left(-\lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 13
Error	35.0
Cost	916

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 5.2 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.2 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 2.06 \cdot 10^{-16}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq 0.000105:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14
Error	32.2
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00043:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -3.5 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -4.5 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -4.2 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	32.2
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00043:\\ \;\;\;\;\phi_2 \cdot R - R \cdot \phi_1\\ \mathbf{elif}\;\phi_1 \leq -7.5 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -2.5 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -6.8 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	32.2
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00043:\\ \;\;\;\;\left(2 \cdot \phi_2 - \left(\phi_2 + \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.3 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -2.7 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -4 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	32.2
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00048:\\ \;\;\;\;R \cdot \left(\phi_2 + \phi_2\right) - R \cdot \left(\phi_1 + \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -2.3 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.55 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -6.2 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	43.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.000105:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 19
Error	45.9
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023099 
(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))))))