Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.4% → 90.8%

Time: 14.2s

Alternatives: 12

Speedup: 2.4×

• 29.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 90.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 6.8e-38)
   (* (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))) R)
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.8e-38) {
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.8e-38) {
		tmp = Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5)))) * R;
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 6.8e-38:
		tmp = math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5)))) * R
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 6.8e-38)
		tmp = Float64(hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))) * R);
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 6.8e-38)
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 6.8e-38], N[(N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.8000000000000004e-38

   1. Initial program 62.3%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

         \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      9. *-commutative N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      13. lower-*.f64 79.5

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_1\right)}\right) \cdot R \]

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot R} \]

   if 6.8000000000000004e-38 < phi2

   1. Initial program 48.8%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

         \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_2, \cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_2, \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      12. lower--.f64 84.5

          \[\leadsto \mathsf{hypot}\left(\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   5. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.9 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 5.8 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \cos \left(\phi_2 + \phi_1\right), 0.5\right)} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.9e-5)
   (* (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))) R)
   (if (<= phi2 5.8e+93)
     (* (sqrt (fma 0.5 (cos (+ phi2 phi1)) 0.5)) (* R (- lambda2 lambda1)))
     (* R (- phi2 phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.9e-5) {
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	} else if (phi2 <= 5.8e+93) {
		tmp = sqrt(fma(0.5, cos((phi2 + phi1)), 0.5)) * (R * (lambda2 - lambda1));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.9e-5)
		tmp = Float64(hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))) * R);
	elseif (phi2 <= 5.8e+93)
		tmp = Float64(sqrt(fma(0.5, cos(Float64(phi2 + phi1)), 0.5)) * Float64(R * Float64(lambda2 - lambda1)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.9e-5], N[(N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 5.8e+93], N[(N[Sqrt[N[(0.5 * N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision] * N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 3.8999999999999999e-5

   1. Initial program 63.0%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

         \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      9. *-commutative N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      13. lower-*.f64 79.2

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_1\right)}\right) \cdot R \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot R} \]

   if 3.8999999999999999e-5 < phi2 < 5.7999999999999997e93

   1. Initial program 49.1%

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   7. Applied rewrites 26.0%

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

   8. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. sub-neg N/A

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

      12. distribute-lft-out-- N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 20.3

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

   10. Applied rewrites 20.3%

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

   if 5.7999999999999997e93 < phi2

   1. Initial program 44.3%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 78.4

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

   8. Applied rewrites 78.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.9 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 5.8 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \cos \left(\phi_2 + \phi_1\right), 0.5\right)} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.5 \cdot 10^{-189}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 5.8 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \cos \left(\phi_2 + \phi_1\right), 0.5\right)} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.5e-189)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (if (<= phi2 5.8e+93)
     (* (sqrt (fma 0.5 (cos (+ phi2 phi1)) 0.5)) (* R (- lambda2 lambda1)))
     (* R (- phi2 phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.5e-189) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else if (phi2 <= 5.8e+93) {
		tmp = sqrt(fma(0.5, cos((phi2 + phi1)), 0.5)) * (R * (lambda2 - lambda1));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.5e-189)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	elseif (phi2 <= 5.8e+93)
		tmp = Float64(sqrt(fma(0.5, cos(Float64(phi2 + phi1)), 0.5)) * Float64(R * Float64(lambda2 - lambda1)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.5e-189], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5.8e+93], N[(N[Sqrt[N[(0.5 * N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision] * N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.5e-189

   1. Initial program 61.0%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

         \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      9. *-commutative N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      13. lower-*.f64 75.1

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_1\right)}\right) \cdot R \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot R} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]
   7. Step-by-step derivation

      1. lower--.f64 69.5

         \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]

   8. Applied rewrites 69.5%

      \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]

   if 1.5e-189 < phi2 < 5.7999999999999997e93

   1. Initial program 64.1%

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   7. Applied rewrites 32.6%

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

   8. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. sub-neg N/A

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

      12. distribute-lft-out-- N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 34.4

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

   10. Applied rewrites 34.4%

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

   if 5.7999999999999997e93 < phi2

   1. Initial program 44.3%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 78.4

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

   8. Applied rewrites 78.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.5 \cdot 10^{-189}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 5.8 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \cos \left(\phi_2 + \phi_1\right), 0.5\right)} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 6.5e-38)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.5e-38) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.5e-38) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 6.5e-38:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 6.5e-38)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 6.5e-38)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 6.5e-38], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.49999999999999949e-38

   1. Initial program 62.3%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

         \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      9. *-commutative N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      13. lower-*.f64 79.5

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_1\right)}\right) \cdot R \]

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot R} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]
   7. Step-by-step derivation

      1. lower--.f64 72.3

         \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]

   8. Applied rewrites 72.3%

      \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]

   if 6.49999999999999949e-38 < phi2

   1. Initial program 48.8%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

         \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_2, \cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_2, \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      12. lower--.f64 84.5

          \[\leadsto \mathsf{hypot}\left(\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   5. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]
   7. Step-by-step derivation

      1. lower--.f64 72.1

         \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]

   8. Applied rewrites 72.1%

      \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{+71}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, \frac{\phi_1}{-\phi_2}, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7e+71)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (fma phi2 (/ phi1 (- phi2)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7e+71) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * fma(phi2, (phi1 / -phi2), phi2);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 7e+71)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * fma(phi2, Float64(phi1 / Float64(-phi2)), phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 7e+71], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(phi1 / (-phi2)), $MachinePrecision] + phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.9999999999999998e71

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

         \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      9. *-commutative N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \cdot R \]

      13. lower-*.f64 77.2

          \[\leadsto \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_1\right)}\right) \cdot R \]

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot R} \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]
   7. Step-by-step derivation

      1. lower--.f64 70.5

         \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]

   8. Applied rewrites 70.5%

      \[\leadsto \mathsf{hypot}\left(\phi_1, \color{blue}{\lambda_1 - \lambda_2}\right) \cdot R \]

   if 6.9999999999999998e71 < phi2

   1. Initial program 44.3%

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

   3. Taylor expanded in phi2 around inf

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{fma}\left(\phi_2, -1 \cdot \frac{\phi_1}{\phi_2}, \phi_2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto R \cdot \mathsf{fma}\left(\phi_2, \color{blue}{\mathsf{neg}\left(\frac{\phi_1}{\phi_2}\right)}, \phi_2\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto R \cdot \mathsf{fma}\left(\phi_2, \color{blue}{\frac{\phi_1}{\mathsf{neg}\left(\phi_2\right)}}, \phi_2\right) \]

      7. mul-1-neg N/A

         \[\leadsto R \cdot \mathsf{fma}\left(\phi_2, \frac{\phi_1}{\color{blue}{-1 \cdot \phi_2}}, \phi_2\right) \]

      8. lower-/.f64 N/A

         \[\leadsto R \cdot \mathsf{fma}\left(\phi_2, \color{blue}{\frac{\phi_1}{-1 \cdot \phi_2}}, \phi_2\right) \]

      9. mul-1-neg N/A

         \[\leadsto R \cdot \mathsf{fma}\left(\phi_2, \frac{\phi_1}{\color{blue}{\mathsf{neg}\left(\phi_2\right)}}, \phi_2\right) \]

      10. lower-neg.f64 72.5

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

   5. Applied rewrites 72.5%

      \[\leadsto R \cdot \color{blue}{\mathsf{fma}\left(\phi_2, \frac{\phi_1}{-\phi_2}, \phi_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{+71}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, \frac{\phi_1}{-\phi_2}, \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+259}:\\ \;\;\;\;R \cdot \left(\mathsf{fma}\left(0.5, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(-0.25, \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right), 1\right)}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -2e+259)
   (*
    R
    (-
     (fma
      0.5
      (/
       (*
        (* phi2 phi2)
        (fma -0.25 (* (- lambda1 lambda2) (- lambda1 lambda2)) 1.0))
       (- lambda1 lambda2))
      lambda1)
     lambda2))
   (* R (- phi2 phi1))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -2e+259) {
		tmp = R * (fma(0.5, (((phi2 * phi2) * fma(-0.25, ((lambda1 - lambda2) * (lambda1 - lambda2)), 1.0)) / (lambda1 - lambda2)), lambda1) - lambda2);
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -2e+259)
		tmp = Float64(R * Float64(fma(0.5, Float64(Float64(Float64(phi2 * phi2) * fma(-0.25, Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)), 1.0)) / Float64(lambda1 - lambda2)), lambda1) - lambda2));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -2e+259], N[(R * N[(N[(0.5 * N[(N[(N[(phi2 * phi2), $MachinePrecision] * N[(-0.25 * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] + lambda1), $MachinePrecision] - lambda2), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -2e259

   1. Initial program 45.7%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

         \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_2, \cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_2, \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      12. lower--.f64 85.1

          \[\leadsto \mathsf{hypot}\left(\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{\left(\left(\lambda_1 + \frac{1}{2} \cdot \frac{{\phi_2}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2}\right) - \lambda_2\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\lambda_1 + \frac{1}{2} \cdot \frac{{\phi_2}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2}\right) - \lambda_2\right)} \cdot R \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{{\phi_2}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2} + \lambda_1\right)} - \lambda_2\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{{\phi_2}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2}, \lambda_1\right)} - \lambda_2\right) \cdot R \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{\phi_2}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2}}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{{\phi_2}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\left(\phi_2 \cdot \phi_2\right)} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\left(\phi_2 \cdot \phi_2\right)} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      8. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + 1\right)}}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {\left(\lambda_1 - \lambda_2\right)}^{2}, 1\right)}}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}, 1\right)}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}, 1\right)}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \left(\lambda_1 - \lambda_2\right), 1\right)}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      13. lower--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}, 1\right)}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right) \cdot R \]

      14. lower--.f64 35.5

          \[\leadsto \left(\mathsf{fma}\left(0.5, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(-0.25, \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right), 1\right)}{\color{blue}{\lambda_1 - \lambda_2}}, \lambda_1\right) - \lambda_2\right) \cdot R \]

   8. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(-0.25, \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right), 1\right)}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right)} \cdot R \]

   if -2e259 < (-.f64 lambda1 lambda2)

   1. Initial program 59.7%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 24.8

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

   5. Applied rewrites 24.8%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 29.0

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

   8. Applied rewrites 29.0%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+259}:\\ \;\;\;\;R \cdot \left(\mathsf{fma}\left(0.5, \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(-0.25, \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right), 1\right)}{\lambda_1 - \lambda_2}, \lambda_1\right) - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;R \leq 9.5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(R, \phi_2, \phi_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= R 9.5e+116)
   (fma R phi2 (* phi1 (- R)))
   (* phi1 (* R (+ (/ phi2 phi1) -1.0)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 9.5e+116) {
		tmp = fma(R, phi2, (phi1 * -R));
	} else {
		tmp = phi1 * (R * ((phi2 / phi1) + -1.0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (R <= 9.5e+116)
		tmp = fma(R, phi2, Float64(phi1 * Float64(-R)));
	else
		tmp = Float64(phi1 * Float64(R * Float64(Float64(phi2 / phi1) + -1.0)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[R, 9.5e+116], N[(R * phi2 + N[(phi1 * (-R)), $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(R * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if R < 9.5000000000000004e116

   1. Initial program 52.0%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 24.0

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

   5. Applied rewrites 24.0%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right) + R \cdot \phi_2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(R, \phi_2, \color{blue}{\mathsf{neg}\left(R \cdot \phi_1\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(R, \phi_2, \mathsf{neg}\left(\color{blue}{\phi_1 \cdot R}\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(R, \phi_2, \color{blue}{\phi_1 \cdot \left(\mathsf{neg}\left(R\right)\right)}\right) \]

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(R, \phi_2, \phi_1 \cdot \color{blue}{\left(\mathsf{neg}\left(R\right)\right)}\right) \]

      9. lower-neg.f64 28.9

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

   8. Applied rewrites 28.9%

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

   if 9.5000000000000004e116 < R

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 23.9

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

   5. Applied rewrites 23.9%

      \[\leadsto R \cdot \color{blue}{\left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \left(-\phi_1\right)\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 32.0

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

   7. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\left(R \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right) \cdot \left(-\phi_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \leq 9.5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(R, \phi_2, \phi_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;R \leq 9.5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(R, \phi_2, \phi_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= R 9.5e+116)
   (fma R phi2 (* phi1 (- R)))
   (* phi1 (- (* R (/ phi2 phi1)) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 9.5e+116) {
		tmp = fma(R, phi2, (phi1 * -R));
	} else {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (R <= 9.5e+116)
		tmp = fma(R, phi2, Float64(phi1 * Float64(-R)));
	else
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[R, 9.5e+116], N[(R * phi2 + N[(phi1 * (-R)), $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if R < 9.5000000000000004e116

   1. Initial program 52.0%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 24.0

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

   5. Applied rewrites 24.0%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right) + R \cdot \phi_2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(R, \phi_2, \color{blue}{\mathsf{neg}\left(R \cdot \phi_1\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(R, \phi_2, \mathsf{neg}\left(\color{blue}{\phi_1 \cdot R}\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(R, \phi_2, \color{blue}{\phi_1 \cdot \left(\mathsf{neg}\left(R\right)\right)}\right) \]

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(R, \phi_2, \phi_1 \cdot \color{blue}{\left(\mathsf{neg}\left(R\right)\right)}\right) \]

      9. lower-neg.f64 28.9

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

   8. Applied rewrites 28.9%

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

   if 9.5000000000000004e116 < R

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 23.9

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

   5. Applied rewrites 23.9%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 23.9

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in phi1 around inf

      \[\leadsto \color{blue}{\phi_1 \cdot \left(-1 \cdot R + \frac{R \cdot \phi_2}{\phi_1}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 32.0

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

   11. Applied rewrites 32.0%

       \[\leadsto \color{blue}{\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.7% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 6.5e-38) (* phi1 (- R)) (* phi2 R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.5e-38) {
		tmp = phi1 * -R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 6.5d-38) then
        tmp = phi1 * -r
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.5e-38) {
		tmp = phi1 * -R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 6.5e-38:
		tmp = phi1 * -R
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 6.5e-38)
		tmp = Float64(phi1 * Float64(-R));
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 6.5e-38)
		tmp = phi1 * -R;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 6.5e-38], N[(phi1 * (-R)), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.49999999999999949e-38

   1. Initial program 62.3%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(R \cdot \phi_1\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\phi_1 \cdot R}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\phi_1 \cdot \left(\mathsf{neg}\left(R\right)\right)} \]

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \phi_1 \cdot \color{blue}{\left(\mathsf{neg}\left(R\right)\right)} \]

      7. lower-neg.f64 17.2

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

   5. Applied rewrites 17.2%

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

   if 6.49999999999999949e-38 < phi2

   1. Initial program 48.8%

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

   3. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 62.2

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

   5. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.8% accurate, 31.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \left(\phi_2 - \phi_1\right) \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R (- phi2 phi1)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (phi2 - phi1);
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * (phi2 - phi1)
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (phi2 - phi1);
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * (phi2 - phi1)

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(phi2 - phi1))
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * (phi2 - phi1);
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.6%

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

3. Taylor expanded in phi1 around -inf

   \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f64 24.0

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

5. Applied rewrites 24.0%

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

6. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 28.2

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

8. Applied rewrites 28.2%

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

Alternative 11: 30.7% accurate, 46.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \phi_2 \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* phi2 R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return phi2 * R;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = phi2 * r
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return phi2 * R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return phi2 * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(phi2 * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = phi2 * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(phi2 * R), $MachinePrecision]

Derivation

1. Initial program 58.6%

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

3. Taylor expanded in phi2 around inf

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 19.2

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

5. Applied rewrites 19.2%

   \[\leadsto \color{blue}{\phi_2 \cdot R} \]
6. Add Preprocessing

Alternative 12: 13.6% accurate, 46.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_1 \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* lambda1 R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 * R;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 * r
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 * R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return lambda1 * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = lambda1 * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 * R), $MachinePrecision]

Derivation

1. Initial program 58.6%

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

3. Taylor expanded in phi1 around 0

   \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. unswap-sqr N/A

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

   8. lower-hypot.f64 N/A

      \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_2, \cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   10. lower-cos.f64 N/A

       \[\leadsto \mathsf{hypot}\left(\phi_2, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{hypot}\left(\phi_2, \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   12. lower--.f64 72.9

       \[\leadsto \mathsf{hypot}\left(\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

5. Applied rewrites 72.9%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

6. Taylor expanded in lambda1 around inf

   \[\leadsto \color{blue}{\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)} \cdot R \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 16.3

      \[\leadsto \left(\lambda_1 \cdot \cos \color{blue}{\left(0.5 \cdot \phi_2\right)}\right) \cdot R \]

8. Applied rewrites 16.3%

   \[\leadsto \color{blue}{\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)} \cdot R \]

9. Taylor expanded in phi2 around 0

   \[\leadsto \color{blue}{R \cdot \lambda_1} \]
10. Step-by-step derivation

    1. lower-*.f64 14.9

       \[\leadsto \color{blue}{R \cdot \lambda_1} \]

11. Applied rewrites 14.9%

    \[\leadsto \color{blue}{R \cdot \lambda_1} \]

12. Final simplification 14.9%

    \[\leadsto \lambda_1 \cdot R \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024211 
(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))))))