Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.4% → 90.5%

Time: 8.5s

Alternatives: 16

Speedup: 8.5×

• 30.9% 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: 90.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \end{array} \]

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 (<= phi1 -2.1e-6)
   (* (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) phi1) R)
   (* (hypot (* (cos (* 0.5 phi2)) (- lambda1 lambda2)) 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 (phi1 <= -2.1e-6) {
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2) * R;
	}
	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 (phi1 <= -2.1e-6) {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = Math.hypot((Math.cos((0.5 * phi2)) * (lambda1 - lambda2)), 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 phi1 <= -2.1e-6:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), phi1) * R
	else:
		tmp = math.hypot((math.cos((0.5 * phi2)) * (lambda1 - lambda2)), 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 (phi1 <= -2.1e-6)
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2)), 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 (phi1 <= -2.1e-6)
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	else
		tmp = hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), 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[phi1, -2.1e-6], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.0999999999999998e-6

   1. Initial program 59.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -2.0999999999999998e-6 < phi1

   1. Initial program 65.2%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 77.9

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

   5. Applied rewrites 77.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-278}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 7.2 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (hypot (* (cos (* 0.5 phi1)) lambda1) phi1) R)))
   (if (<= phi2 8.5e-278)
     t_0
     (if (<= phi2 1.75e-99)
       (*
        (hypot (* (fma -0.125 (* phi1 phi1) 1.0) (- lambda1 lambda2)) phi1)
        R)
       (if (<= phi2 7.2e+30)
         t_0
         (* (hypot (* (cos (* 0.5 phi2)) lambda2) 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 t_0 = hypot((cos((0.5 * phi1)) * lambda1), phi1) * R;
	double tmp;
	if (phi2 <= 8.5e-278) {
		tmp = t_0;
	} else if (phi2 <= 1.75e-99) {
		tmp = hypot((fma(-0.125, (phi1 * phi1), 1.0) * (lambda1 - lambda2)), phi1) * R;
	} else if (phi2 <= 7.2e+30) {
		tmp = t_0;
	} else {
		tmp = hypot((cos((0.5 * phi2)) * lambda2), phi2) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * lambda1), phi1) * R)
	tmp = 0.0
	if (phi2 <= 8.5e-278)
		tmp = t_0;
	elseif (phi2 <= 1.75e-99)
		tmp = Float64(hypot(Float64(fma(-0.125, Float64(phi1 * phi1), 1.0) * Float64(lambda1 - lambda2)), phi1) * R);
	elseif (phi2 <= 7.2e+30)
		tmp = t_0;
	else
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * lambda2), phi2) * 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_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, 8.5e-278], t$95$0, If[LessEqual[phi2, 1.75e-99], N[(N[Sqrt[N[(N[(-0.125 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 7.2e+30], t$95$0, N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 8.49999999999999955e-278 or 1.7499999999999999e-99 < phi2 < 7.2000000000000004e30

   1. Initial program 65.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 58.2%

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

   if 8.49999999999999955e-278 < phi2 < 1.7499999999999999e-99

   1. Initial program 63.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(1 + \frac{-1}{8} \cdot {\phi_1}^{2}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

   if 7.2000000000000004e30 < phi2

   1. Initial program 56.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 73.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 7.2 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-278}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 55000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (hypot (* (cos (* 0.5 phi1)) lambda1) phi1) R)))
   (if (<= phi2 8.5e-278)
     t_0
     (if (<= phi2 1.75e-99)
       (*
        (hypot (* (fma -0.125 (* phi1 phi1) 1.0) (- lambda1 lambda2)) phi1)
        R)
       (if (<= phi2 55000.0) t_0 (* (- 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 t_0 = hypot((cos((0.5 * phi1)) * lambda1), phi1) * R;
	double tmp;
	if (phi2 <= 8.5e-278) {
		tmp = t_0;
	} else if (phi2 <= 1.75e-99) {
		tmp = hypot((fma(-0.125, (phi1 * phi1), 1.0) * (lambda1 - lambda2)), phi1) * R;
	} else if (phi2 <= 55000.0) {
		tmp = t_0;
	} else {
		tmp = (phi2 - phi1) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * lambda1), phi1) * R)
	tmp = 0.0
	if (phi2 <= 8.5e-278)
		tmp = t_0;
	elseif (phi2 <= 1.75e-99)
		tmp = Float64(hypot(Float64(fma(-0.125, Float64(phi1 * phi1), 1.0) * Float64(lambda1 - lambda2)), phi1) * R);
	elseif (phi2 <= 55000.0)
		tmp = t_0;
	else
		tmp = Float64(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_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, 8.5e-278], t$95$0, If[LessEqual[phi2, 1.75e-99], N[(N[Sqrt[N[(N[(-0.125 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 55000.0], t$95$0, N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 8.49999999999999955e-278 or 1.7499999999999999e-99 < phi2 < 55000

   1. Initial program 64.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 58.4%

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

   if 8.49999999999999955e-278 < phi2 < 1.7499999999999999e-99

   1. Initial program 63.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(1 + \frac{-1}{8} \cdot {\phi_1}^{2}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

   if 55000 < phi2

   1. Initial program 60.7%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 55.9%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 55000:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

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 (<= phi1 -2.3e+44)
   (* (hypot (* (cos (* 0.5 phi1)) lambda1) phi1) R)
   (if (<= phi1 -4.8e-155)
     (* (hypot (* (fma -0.125 (* phi1 phi1) 1.0) (- lambda1 lambda2)) phi1) R)
     (* (hypot (* (cos (* 0.5 phi2)) lambda1) 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 (phi1 <= -2.3e+44) {
		tmp = hypot((cos((0.5 * phi1)) * lambda1), phi1) * R;
	} else if (phi1 <= -4.8e-155) {
		tmp = hypot((fma(-0.125, (phi1 * phi1), 1.0) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = hypot((cos((0.5 * phi2)) * lambda1), 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 (phi1 <= -2.3e+44)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * lambda1), phi1) * R);
	elseif (phi1 <= -4.8e-155)
		tmp = Float64(hypot(Float64(fma(-0.125, Float64(phi1 * phi1), 1.0) * Float64(lambda1 - lambda2)), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * lambda1), phi2) * 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[phi1, -2.3e+44], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, -4.8e-155], N[(N[Sqrt[N[(N[(-0.125 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.30000000000000004e44

   1. Initial program 58.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 70.2%

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

   if -2.30000000000000004e44 < phi1 < -4.8e-155

   1. Initial program 71.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(1 + \frac{-1}{8} \cdot {\phi_1}^{2}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 64.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

   if -4.8e-155 < phi1

   1. Initial program 63.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 59.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

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 2.2e+35)
   (* (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) phi1) R)
   (* (hypot (* (cos (* 0.5 phi2)) lambda2) 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 <= 2.2e+35) {
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = hypot((cos((0.5 * phi2)) * lambda2), phi2) * R;
	}
	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 <= 2.2e+35) {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = Math.hypot((Math.cos((0.5 * phi2)) * lambda2), 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 <= 2.2e+35:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), phi1) * R
	else:
		tmp = math.hypot((math.cos((0.5 * phi2)) * lambda2), 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 <= 2.2e+35)
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * lambda2), 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 <= 2.2e+35)
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	else
		tmp = hypot((cos((0.5 * phi2)) * lambda2), 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, 2.2e+35], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.1999999999999999e35

   1. Initial program 65.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if 2.1999999999999999e35 < phi2

   1. Initial program 54.2%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 74.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+73}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \end{array} \]

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 (<= (- lambda1 lambda2) -2e+216)
   (* (hypot (* (fma -0.125 (* phi1 phi1) 1.0) (- lambda1 lambda2)) phi1) R)
   (if (<= (- lambda1 lambda2) -4e+73)
     (* (- phi1) (fma (- phi2) (/ 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 ((lambda1 - lambda2) <= -2e+216) {
		tmp = hypot((fma(-0.125, (phi1 * phi1), 1.0) * (lambda1 - lambda2)), phi1) * R;
	} else if ((lambda1 - lambda2) <= -4e+73) {
		tmp = -phi1 * fma(-phi2, (R / phi1), R);
	} else {
		tmp = (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 (Float64(lambda1 - lambda2) <= -2e+216)
		tmp = Float64(hypot(Float64(fma(-0.125, Float64(phi1 * phi1), 1.0) * Float64(lambda1 - lambda2)), phi1) * R);
	elseif (Float64(lambda1 - lambda2) <= -4e+73)
		tmp = Float64(Float64(-phi1) * fma(Float64(-phi2), Float64(R / phi1), R));
	else
		tmp = Float64(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[N[(lambda1 - lambda2), $MachinePrecision], -2e+216], N[(N[Sqrt[N[(N[(-0.125 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -4e+73], N[((-phi1) * N[((-phi2) * N[(R / phi1), $MachinePrecision] + R), $MachinePrecision]), $MachinePrecision], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -2e216

   1. Initial program 56.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 81.5

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(1 + \frac{-1}{8} \cdot {\phi_1}^{2}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 75.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

   if -2e216 < (-.f64 lambda1 lambda2) < -3.99999999999999993e73

   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

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 30.7

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

   8. Applied rewrites 30.7%

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

   10. Applied rewrites 30.7%

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

   if -3.99999999999999993e73 < (-.f64 lambda1 lambda2)

   1. Initial program 65.8%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 29.9

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 32.6%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(-0.125, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+73}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.55 \cdot 10^{-30}:\\ \;\;\;\;\left(\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \left(-\lambda_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\\ \end{array} \end{array} \]

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 (<= lambda2 -2.55e-30)
   (* (* (cos (* (+ phi2 phi1) 0.5)) (- lambda1)) R)
   (if (<= lambda2 2.4e+73)
     (* (- phi2 phi1) R)
     (if (<= lambda2 1.8e+216)
       (* (- phi1) (fma (- phi2) (/ R phi1) R))
       (* (* lambda2 R) (cos (* 0.5 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 (lambda2 <= -2.55e-30) {
		tmp = (cos(((phi2 + phi1) * 0.5)) * -lambda1) * R;
	} else if (lambda2 <= 2.4e+73) {
		tmp = (phi2 - phi1) * R;
	} else if (lambda2 <= 1.8e+216) {
		tmp = -phi1 * fma(-phi2, (R / phi1), R);
	} else {
		tmp = (lambda2 * R) * cos((0.5 * 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 (lambda2 <= -2.55e-30)
		tmp = Float64(Float64(cos(Float64(Float64(phi2 + phi1) * 0.5)) * Float64(-lambda1)) * R);
	elseif (lambda2 <= 2.4e+73)
		tmp = Float64(Float64(phi2 - phi1) * R);
	elseif (lambda2 <= 1.8e+216)
		tmp = Float64(Float64(-phi1) * fma(Float64(-phi2), Float64(R / phi1), R));
	else
		tmp = Float64(Float64(lambda2 * R) * cos(Float64(0.5 * 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[lambda2, -2.55e-30], N[(N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * (-lambda1)), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 2.4e+73], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 1.8e+216], N[((-phi1) * N[((-phi2) * N[(R / phi1), $MachinePrecision] + R), $MachinePrecision]), $MachinePrecision], N[(N[(lambda2 * R), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if lambda2 < -2.54999999999999986e-30

   1. Initial program 54.5%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 16.1

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

   5. Applied rewrites 16.1%

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

   if -2.54999999999999986e-30 < lambda2 < 2.40000000000000002e73

   1. Initial program 70.5%

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

   3. Taylor expanded in phi2 around 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 32.8

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

   5. Applied rewrites 32.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 35.8%

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

   if 2.40000000000000002e73 < lambda2 < 1.8000000000000001e216

   1. Initial program 59.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 29.9

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

   8. Applied rewrites 29.9%

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

   10. Applied rewrites 40.7%

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

   if 1.8000000000000001e216 < lambda2

   1. Initial program 56.7%

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

   3. Taylor expanded in lambda2 around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 69.0

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot 0.5\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.0%

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.55 \cdot 10^{-30}:\\ \;\;\;\;\left(\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \left(-\lambda_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.55 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(-\lambda_1\right) \cdot R\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\\ \end{array} \end{array} \]

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 (<= lambda2 -2.55e-30)
   (* (* (- lambda1) R) (cos (* (+ phi2 phi1) 0.5)))
   (if (<= lambda2 2.4e+73)
     (* (- phi2 phi1) R)
     (if (<= lambda2 1.8e+216)
       (* (- phi1) (fma (- phi2) (/ R phi1) R))
       (* (* lambda2 R) (cos (* 0.5 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 (lambda2 <= -2.55e-30) {
		tmp = (-lambda1 * R) * cos(((phi2 + phi1) * 0.5));
	} else if (lambda2 <= 2.4e+73) {
		tmp = (phi2 - phi1) * R;
	} else if (lambda2 <= 1.8e+216) {
		tmp = -phi1 * fma(-phi2, (R / phi1), R);
	} else {
		tmp = (lambda2 * R) * cos((0.5 * 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 (lambda2 <= -2.55e-30)
		tmp = Float64(Float64(Float64(-lambda1) * R) * cos(Float64(Float64(phi2 + phi1) * 0.5)));
	elseif (lambda2 <= 2.4e+73)
		tmp = Float64(Float64(phi2 - phi1) * R);
	elseif (lambda2 <= 1.8e+216)
		tmp = Float64(Float64(-phi1) * fma(Float64(-phi2), Float64(R / phi1), R));
	else
		tmp = Float64(Float64(lambda2 * R) * cos(Float64(0.5 * 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[lambda2, -2.55e-30], N[(N[((-lambda1) * R), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 2.4e+73], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 1.8e+216], N[((-phi1) * N[((-phi2) * N[(R / phi1), $MachinePrecision] + R), $MachinePrecision]), $MachinePrecision], N[(N[(lambda2 * R), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if lambda2 < -2.54999999999999986e-30

   1. Initial program 54.5%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 16.1

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

   5. Applied rewrites 16.1%

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

   if -2.54999999999999986e-30 < lambda2 < 2.40000000000000002e73

   1. Initial program 70.5%

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

   3. Taylor expanded in phi2 around 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 32.8

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

   5. Applied rewrites 32.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 35.8%

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

   if 2.40000000000000002e73 < lambda2 < 1.8000000000000001e216

   1. Initial program 59.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 29.9

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

   8. Applied rewrites 29.9%

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

   10. Applied rewrites 40.7%

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

   if 1.8000000000000001e216 < lambda2

   1. Initial program 56.7%

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

   3. Taylor expanded in lambda2 around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 69.0

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot 0.5\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.0%

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.55 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(-\lambda_1\right) \cdot R\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq -2.55 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(-\lambda_1\right) \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot t\_0\\ \end{array} \end{array} \]

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
 (let* ((t_0 (cos (* 0.5 phi1))))
   (if (<= lambda2 -2.55e-30)
     (* (* (- lambda1) t_0) R)
     (if (<= lambda2 2.4e+73)
       (* (- phi2 phi1) R)
       (if (<= lambda2 1.8e+216)
         (* (- phi1) (fma (- phi2) (/ R phi1) R))
         (* (* lambda2 R) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double tmp;
	if (lambda2 <= -2.55e-30) {
		tmp = (-lambda1 * t_0) * R;
	} else if (lambda2 <= 2.4e+73) {
		tmp = (phi2 - phi1) * R;
	} else if (lambda2 <= 1.8e+216) {
		tmp = -phi1 * fma(-phi2, (R / phi1), R);
	} else {
		tmp = (lambda2 * R) * t_0;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	tmp = 0.0
	if (lambda2 <= -2.55e-30)
		tmp = Float64(Float64(Float64(-lambda1) * t_0) * R);
	elseif (lambda2 <= 2.4e+73)
		tmp = Float64(Float64(phi2 - phi1) * R);
	elseif (lambda2 <= 1.8e+216)
		tmp = Float64(Float64(-phi1) * fma(Float64(-phi2), Float64(R / phi1), R));
	else
		tmp = Float64(Float64(lambda2 * R) * t_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_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -2.55e-30], N[(N[((-lambda1) * t$95$0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 2.4e+73], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 1.8e+216], N[((-phi1) * N[((-phi2) * N[(R / phi1), $MachinePrecision] + R), $MachinePrecision]), $MachinePrecision], N[(N[(lambda2 * R), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda2 < -2.54999999999999986e-30

   1. Initial program 54.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in lambda1 around -inf

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

   8. Applied rewrites 11.9%

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

   if -2.54999999999999986e-30 < lambda2 < 2.40000000000000002e73

   1. Initial program 70.5%

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

   3. Taylor expanded in phi2 around 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 32.8

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

   5. Applied rewrites 32.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 35.8%

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

   if 2.40000000000000002e73 < lambda2 < 1.8000000000000001e216

   1. Initial program 59.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 29.9

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

   8. Applied rewrites 29.9%

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

   10. Applied rewrites 40.7%

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

   if 1.8000000000000001e216 < lambda2

   1. Initial program 56.7%

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

   3. Taylor expanded in lambda2 around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 69.0

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot 0.5\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.0%

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.55 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(-\lambda_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\\ \end{array} \end{array} \]

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 (<= lambda2 2.4e+73)
   (* (- phi2 phi1) R)
   (if (<= lambda2 1.8e+216)
     (* (- phi1) (fma (- phi2) (/ R phi1) R))
     (* (* lambda2 R) (cos (* 0.5 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 (lambda2 <= 2.4e+73) {
		tmp = (phi2 - phi1) * R;
	} else if (lambda2 <= 1.8e+216) {
		tmp = -phi1 * fma(-phi2, (R / phi1), R);
	} else {
		tmp = (lambda2 * R) * cos((0.5 * 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 (lambda2 <= 2.4e+73)
		tmp = Float64(Float64(phi2 - phi1) * R);
	elseif (lambda2 <= 1.8e+216)
		tmp = Float64(Float64(-phi1) * fma(Float64(-phi2), Float64(R / phi1), R));
	else
		tmp = Float64(Float64(lambda2 * R) * cos(Float64(0.5 * 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[lambda2, 2.4e+73], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 1.8e+216], N[((-phi1) * N[((-phi2) * N[(R / phi1), $MachinePrecision] + R), $MachinePrecision]), $MachinePrecision], N[(N[(lambda2 * R), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 2.40000000000000002e73

   1. Initial program 64.8%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 28.3

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 30.7%

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

   if 2.40000000000000002e73 < lambda2 < 1.8000000000000001e216

   1. Initial program 59.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 29.9

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

   8. Applied rewrites 29.9%

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

   10. Applied rewrites 40.7%

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

   if 1.8000000000000001e216 < lambda2

   1. Initial program 56.7%

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

   3. Taylor expanded in lambda2 around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 69.0

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot 0.5\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.0%

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.5 \cdot 10^{+211}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\\ \end{array} \end{array} \]

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 (<= lambda2 2.4e+73)
   (* (- phi2 phi1) R)
   (if (<= lambda2 1.5e+211)
     (* (- phi1) (fma (- phi2) (/ R phi1) R))
     (* (* lambda2 R) (cos (* 0.5 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 (lambda2 <= 2.4e+73) {
		tmp = (phi2 - phi1) * R;
	} else if (lambda2 <= 1.5e+211) {
		tmp = -phi1 * fma(-phi2, (R / phi1), R);
	} else {
		tmp = (lambda2 * R) * cos((0.5 * 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 (lambda2 <= 2.4e+73)
		tmp = Float64(Float64(phi2 - phi1) * R);
	elseif (lambda2 <= 1.5e+211)
		tmp = Float64(Float64(-phi1) * fma(Float64(-phi2), Float64(R / phi1), R));
	else
		tmp = Float64(Float64(lambda2 * R) * cos(Float64(0.5 * 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[lambda2, 2.4e+73], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 1.5e+211], N[((-phi1) * N[((-phi2) * N[(R / phi1), $MachinePrecision] + R), $MachinePrecision]), $MachinePrecision], N[(N[(lambda2 * R), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 2.40000000000000002e73

   1. Initial program 64.8%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 28.3

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 30.7%

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

   if 2.40000000000000002e73 < lambda2 < 1.5e211

   1. Initial program 59.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 29.9

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

   8. Applied rewrites 29.9%

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

   10. Applied rewrites 40.7%

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

   if 1.5e211 < lambda2

   1. Initial program 56.7%

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

   3. Taylor expanded in lambda2 around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 69.0

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot 0.5\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \]

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 62.6%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.5 \cdot 10^{+211}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \end{array} \end{array} \]

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 1.25e+34)
   (* (- phi1) (fma (- phi2) (/ R phi1) R))
   (* (fma R (/ (- phi1) phi2) R) 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 <= 1.25e+34) {
		tmp = -phi1 * fma(-phi2, (R / phi1), R);
	} else {
		tmp = fma(R, (-phi1 / phi2), R) * 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 <= 1.25e+34)
		tmp = Float64(Float64(-phi1) * fma(Float64(-phi2), Float64(R / phi1), R));
	else
		tmp = Float64(fma(R, Float64(Float64(-phi1) / phi2), R) * 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, 1.25e+34], N[((-phi1) * N[((-phi2) * N[(R / phi1), $MachinePrecision] + R), $MachinePrecision]), $MachinePrecision], N[(N[(R * N[((-phi1) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.25e34

   1. Initial program 65.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 24.9

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

   8. Applied rewrites 24.9%

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

   10. Applied rewrites 27.4%

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

   if 1.25e34 < phi2

   1. Initial program 54.2%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 56.4

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

   5. Applied rewrites 56.4%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(-\phi_2, \frac{R}{\phi_1}, R\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{\phi_2}{\phi_1}, -R\right) \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \end{array} \]

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 7e+42) (* (fma R (/ phi2 phi1) (- R)) phi1) (* (- 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 (phi2 <= 7e+42) {
		tmp = fma(R, (phi2 / phi1), -R) * phi1;
	} else {
		tmp = (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 (phi2 <= 7e+42)
		tmp = Float64(fma(R, Float64(phi2 / phi1), Float64(-R)) * phi1);
	else
		tmp = Float64(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[phi2, 7e+42], N[(N[(R * N[(phi2 / phi1), $MachinePrecision] + (-R)), $MachinePrecision] * phi1), $MachinePrecision], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.00000000000000047e42

   1. Initial program 65.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 24.8

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

   8. Applied rewrites 24.8%

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

   9. Taylor expanded in phi1 around inf

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

   11. Applied rewrites 26.2%

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

   if 7.00000000000000047e42 < phi2

   1. Initial program 55.3%

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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 57.6%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{\phi_2}{\phi_1}, -R\right) \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.7% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{+28}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

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 8e+28) (* (- 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 <= 8e+28) {
		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 <= 8d+28) 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 <= 8e+28) {
		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 <= 8e+28:
		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 <= 8e+28)
		tmp = Float64(Float64(-phi1) * 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 <= 8e+28)
		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, 8e+28], N[((-phi1) * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.99999999999999967e28

   1. Initial program 65.4%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 24.1

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

   5. Applied rewrites 24.1%

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

   if 7.99999999999999967e28 < phi2

   1. Initial program 56.0%

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

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{+28}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.3% accurate, 31.0× speedup

Math

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

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 phi1) 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 - phi1) * 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 - phi1) * 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 - phi1) * R;
}

Python

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

Julia

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in phi2 around 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

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 28.2

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

5. Applied rewrites 28.2%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 30.2%

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

9. Final simplification 30.2%

   \[\leadsto \left(\phi_2 - \phi_1\right) \cdot R \]
10. Add Preprocessing

Alternative 16: 30.8% 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 63.5%

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

3. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 15.8

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

5. Applied rewrites 15.8%

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

6. Final simplification 15.8%

   \[\leadsto \phi_2 \cdot R \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(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))))))