Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.7% → 91.5%

Time: 13.1s

Alternatives: 12

Speedup: 2.4×

• 30.4% 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: 59.7% 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: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \frac{1}{\frac{1}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.1e-17)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (*
    R
    (hypot phi2 (/ 1.0 (/ 1.0 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.1e-17) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else {
		tmp = R * hypot(phi2, (1.0 / (1.0 / ((lambda1 - lambda2) * cos((phi2 * 0.5))))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.1e-17) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5))));
	} else {
		tmp = R * Math.hypot(phi2, (1.0 / (1.0 / ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.1e-17:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5))))
	else:
		tmp = R * math.hypot(phi2, (1.0 / (1.0 / ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.1e-17)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(1.0 / Float64(1.0 / Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.1e-17)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	else
		tmp = R * hypot(phi2, (1.0 / (1.0 / ((lambda1 - lambda2) * cos((phi2 * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.1e-17], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(1.0 / N[(1.0 / N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.09999999999999992e-17

   1. Initial program 64.4%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 79.5

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

   5. Simplified 79.5%

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

   if 2.09999999999999992e-17 < phi2

   1. Initial program 53.9%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 84.2

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

   5. Simplified 84.2%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 84.2

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

   7. Applied egg-rr 84.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_2, \color{blue}{\frac{\cos \left(0.5 \cdot \phi_2\right)}{\frac{1}{\lambda_1 - \lambda_2}}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 84.3

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

      13. /-rgt-identity N/A

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

      14. lower-*.f64 84.2

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

   9. Applied egg-rr 84.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_2, \color{blue}{\frac{1}{\frac{1}{\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \frac{1}{\frac{1}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.1e-17)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.1e-17) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.1e-17) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.1e-17:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.1e-17)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.1e-17)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.1e-17], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.09999999999999992e-17

   1. Initial program 64.4%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 79.5

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

   5. Simplified 79.5%

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

   if 2.09999999999999992e-17 < phi2

   1. Initial program 53.9%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 84.2

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

   5. Simplified 84.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.65:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{+150}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\lambda_1 - \lambda_2, 0.5, \cos \left(\phi_2 + \phi_1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, -\frac{\phi_1}{\phi_2}, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.65)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (if (<= phi2 2.2e+150)
     (*
      R
      (sqrt
       (+
        (*
         (- lambda1 lambda2)
         (fma
          (- lambda1 lambda2)
          0.5
          (* (cos (+ phi2 phi1)) (* (- lambda1 lambda2) 0.5))))
        (* (- phi1 phi2) (- phi1 phi2)))))
     (* R (fma phi2 (- (/ phi1 phi2)) phi2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.65) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else if (phi2 <= 2.2e+150) {
		tmp = R * sqrt((((lambda1 - lambda2) * fma((lambda1 - lambda2), 0.5, (cos((phi2 + phi1)) * ((lambda1 - lambda2) * 0.5)))) + ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = R * fma(phi2, -(phi1 / phi2), phi2);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.65)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	elseif (phi2 <= 2.2e+150)
		tmp = Float64(R * sqrt(Float64(Float64(Float64(lambda1 - lambda2) * fma(Float64(lambda1 - lambda2), 0.5, Float64(cos(Float64(phi2 + phi1)) * Float64(Float64(lambda1 - lambda2) * 0.5)))) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	else
		tmp = Float64(R * fma(phi2, Float64(-Float64(phi1 / phi2)), phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.65], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.2e+150], N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5 + N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * (-N[(phi1 / phi2), $MachinePrecision]) + phi2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 0.650000000000000022

   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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 79.8

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

   5. Simplified 79.8%

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

   if 0.650000000000000022 < phi2 < 2.19999999999999999e150

   1. Initial program 58.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   4. Applied egg-rr 58.8%

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 58.9

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

   6. Applied egg-rr 58.9%

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

   if 2.19999999999999999e150 < phi2

   1. Initial program 48.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 94.9

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

   5. Simplified 94.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.65:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{+150}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\lambda_1 - \lambda_2, 0.5, \cos \left(\phi_2 + \phi_1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, -\frac{\phi_1}{\phi_2}, \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.65:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{+150}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\lambda_1 - \lambda_2, 0.5, \cos \left(\phi_2 + \phi_1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, -\frac{\phi_1}{\phi_2}, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.65)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (if (<= phi2 2.2e+150)
     (*
      R
      (sqrt
       (+
        (*
         (- lambda1 lambda2)
         (fma
          (- lambda1 lambda2)
          0.5
          (* (cos (+ phi2 phi1)) (* (- lambda1 lambda2) 0.5))))
        (* (- phi1 phi2) (- phi1 phi2)))))
     (* R (fma phi2 (- (/ phi1 phi2)) phi2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.65) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else if (phi2 <= 2.2e+150) {
		tmp = R * sqrt((((lambda1 - lambda2) * fma((lambda1 - lambda2), 0.5, (cos((phi2 + phi1)) * ((lambda1 - lambda2) * 0.5)))) + ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = R * fma(phi2, -(phi1 / phi2), phi2);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.65)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	elseif (phi2 <= 2.2e+150)
		tmp = Float64(R * sqrt(Float64(Float64(Float64(lambda1 - lambda2) * fma(Float64(lambda1 - lambda2), 0.5, Float64(cos(Float64(phi2 + phi1)) * Float64(Float64(lambda1 - lambda2) * 0.5)))) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	else
		tmp = Float64(R * fma(phi2, Float64(-Float64(phi1 / phi2)), phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.65], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.2e+150], N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5 + N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * (-N[(phi1 / phi2), $MachinePrecision]) + phi2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 0.650000000000000022

   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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 72.4

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

   8. Simplified 72.4%

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

   if 0.650000000000000022 < phi2 < 2.19999999999999999e150

   1. Initial program 58.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   4. Applied egg-rr 58.8%

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 58.9

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

   6. Applied egg-rr 58.9%

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

   if 2.19999999999999999e150 < phi2

   1. Initial program 48.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 94.9

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

   5. Simplified 94.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.65:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{+150}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\lambda_1 - \lambda_2, 0.5, \cos \left(\phi_2 + \phi_1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, -\frac{\phi_1}{\phi_2}, \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.65:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{+150}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\phi_1 - \phi_2, \phi_1 - \phi_2, \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, -\frac{\phi_1}{\phi_2}, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.65)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (if (<= phi2 2.2e+150)
     (*
      R
      (sqrt
       (fma
        (- phi1 phi2)
        (- phi1 phi2)
        (*
         (* (- lambda1 lambda2) (- lambda1 lambda2))
         (+ 0.5 (* 0.5 (cos (+ phi2 phi1))))))))
     (* R (fma phi2 (- (/ phi1 phi2)) phi2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.65) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else if (phi2 <= 2.2e+150) {
		tmp = R * sqrt(fma((phi1 - phi2), (phi1 - phi2), (((lambda1 - lambda2) * (lambda1 - lambda2)) * (0.5 + (0.5 * cos((phi2 + phi1)))))));
	} else {
		tmp = R * fma(phi2, -(phi1 / phi2), phi2);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.65)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	elseif (phi2 <= 2.2e+150)
		tmp = Float64(R * sqrt(fma(Float64(phi1 - phi2), Float64(phi1 - phi2), Float64(Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)) * Float64(0.5 + Float64(0.5 * cos(Float64(phi2 + phi1))))))));
	else
		tmp = Float64(R * fma(phi2, Float64(-Float64(phi1 / phi2)), phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.65], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.2e+150], N[(R * N[Sqrt[N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision] + N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * (-N[(phi1 / phi2), $MachinePrecision]) + phi2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 0.650000000000000022

   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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 72.4

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

   8. Simplified 72.4%

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

   if 0.650000000000000022 < phi2 < 2.19999999999999999e150

   1. Initial program 58.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

   4. Applied egg-rr 58.9%

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

   if 2.19999999999999999e150 < phi2

   1. Initial program 48.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 94.9

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

   5. Simplified 94.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.65:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{+150}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\phi_1 - \phi_2, \phi_1 - \phi_2, \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, -\frac{\phi_1}{\phi_2}, \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, -\frac{\phi_1}{\phi_2}, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.95e+15)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (fma phi2 (- (/ phi1 phi2)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.95e+15) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * fma(phi2, -(phi1 / phi2), phi2);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.95e+15)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * fma(phi2, Float64(-Float64(phi1 / phi2)), phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.95e+15], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * (-N[(phi1 / phi2), $MachinePrecision]) + phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.95e15

   1. Initial program 64.4%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 78.7

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

   5. Simplified 78.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 71.5

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

   8. Simplified 71.5%

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

   if 1.95e15 < phi2

   1. Initial program 52.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 74.8

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

   5. Simplified 74.8%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, -\frac{\phi_1}{\phi_2}, \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{-51}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-183}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.7e-51)
   (* R (fma phi1 (/ phi2 phi1) (- phi1)))
   (if (<= phi1 6.5e-183)
     (* lambda2 (- R (/ (* R 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 <= -1.7e-51) {
		tmp = R * fma(phi1, (phi2 / phi1), -phi1);
	} else if (phi1 <= 6.5e-183) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} 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 (phi1 <= -1.7e-51)
		tmp = Float64(R * fma(phi1, Float64(phi2 / phi1), Float64(-phi1)));
	elseif (phi1 <= 6.5e-183)
		tmp = Float64(lambda2 * Float64(R - Float64(Float64(R * lambda1) / lambda2)));
	else
		tmp = Float64(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, -1.7e-51], N[(R * N[(phi1 * N[(phi2 / phi1), $MachinePrecision] + (-phi1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 6.5e-183], N[(lambda2 * N[(R - N[(N[(R * lambda1), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.70000000000000001e-51

   1. Initial program 49.6%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 67.5

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

   5. Simplified 67.5%

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

   if -1.70000000000000001e-51 < phi1 < 6.50000000000000014e-183

   1. Initial program 72.2%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 60.1

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

   5. Simplified 60.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 60.1

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

   8. Simplified 60.1%

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

   9. Taylor expanded in lambda2 around inf

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

       1. lower-*.f64 N/A

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

       2. mul-1-neg N/A

          \[\leadsto \lambda_2 \cdot \left(R + \color{blue}{\left(\mathsf{neg}\left(\frac{R \cdot \lambda_1}{\lambda_2}\right)\right)}\right) \]

       3. unsub-neg N/A

          \[\leadsto \lambda_2 \cdot \color{blue}{\left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \lambda_2 \cdot \color{blue}{\left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \lambda_2 \cdot \left(R - \color{blue}{\frac{R \cdot \lambda_1}{\lambda_2}}\right) \]

       6. *-commutative N/A

          \[\leadsto \lambda_2 \cdot \left(R - \frac{\color{blue}{\lambda_1 \cdot R}}{\lambda_2}\right) \]

       7. lower-*.f64 26.5

          \[\leadsto \lambda_2 \cdot \left(R - \frac{\color{blue}{\lambda_1 \cdot R}}{\lambda_2}\right) \]

   11. Simplified 26.5%

       \[\leadsto \color{blue}{\lambda_2 \cdot \left(R - \frac{\lambda_1 \cdot R}{\lambda_2}\right)} \]

   if 6.50000000000000014e-183 < phi1

   1. Initial program 63.3%

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

   3. Taylor expanded in phi1 around -inf

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

   5. Simplified 10.3%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 19.2

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

   8. Simplified 19.2%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{-51}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-183}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.05 \cdot 10^{+191}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

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 (<= lambda2 1.05e+191)
   (* R (fma phi1 (/ phi2 phi1) (- phi1)))
   (* R lambda2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.05e+191) {
		tmp = R * fma(phi1, (phi2 / phi1), -phi1);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.05e+191)
		tmp = Float64(R * fma(phi1, Float64(phi2 / phi1), Float64(-phi1)));
	else
		tmp = Float64(R * lambda2);
	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, 1.05e+191], N[(R * N[(phi1 * N[(phi2 / phi1), $MachinePrecision] + (-phi1)), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.05e191

   1. Initial program 62.1%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 34.8

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

   5. Simplified 34.8%

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

   if 1.05e191 < lambda2

   1. Initial program 54.8%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 65.8

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

   8. Simplified 65.8%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

       2. lower-*.f64 65.8

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   11. Simplified 65.8%

       \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.05 \cdot 10^{+191}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.9 \cdot 10^{-189}:\\ \;\;\;\;-R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 1.25 \cdot 10^{-115}:\\ \;\;\;\;-R \cdot \lambda_1\\ \mathbf{elif}\;\phi_2 \leq 4.9 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 6.9e-189)
   (- (* R phi1))
   (if (<= phi2 1.25e-115)
     (- (* R lambda1))
     (if (<= phi2 4.9e+14) (* R 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 <= 6.9e-189) {
		tmp = -(R * phi1);
	} else if (phi2 <= 1.25e-115) {
		tmp = -(R * lambda1);
	} else if (phi2 <= 4.9e+14) {
		tmp = R * lambda2;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 6.9d-189) then
        tmp = -(r * phi1)
    else if (phi2 <= 1.25d-115) then
        tmp = -(r * lambda1)
    else if (phi2 <= 4.9d+14) then
        tmp = r * lambda2
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.9e-189) {
		tmp = -(R * phi1);
	} else if (phi2 <= 1.25e-115) {
		tmp = -(R * lambda1);
	} else if (phi2 <= 4.9e+14) {
		tmp = R * lambda2;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 6.9e-189:
		tmp = -(R * phi1)
	elif phi2 <= 1.25e-115:
		tmp = -(R * lambda1)
	elif phi2 <= 4.9e+14:
		tmp = R * lambda2
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 6.9e-189)
		tmp = Float64(-Float64(R * phi1));
	elseif (phi2 <= 1.25e-115)
		tmp = Float64(-Float64(R * lambda1));
	elseif (phi2 <= 4.9e+14)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 6.9e-189)
		tmp = -(R * phi1);
	elseif (phi2 <= 1.25e-115)
		tmp = -(R * lambda1);
	elseif (phi2 <= 4.9e+14)
		tmp = R * lambda2;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 6.9e-189], (-N[(R * phi1), $MachinePrecision]), If[LessEqual[phi2, 1.25e-115], (-N[(R * lambda1), $MachinePrecision]), If[LessEqual[phi2, 4.9e+14], N[(R * lambda2), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < 6.9000000000000003e-189

   1. Initial program 63.2%

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

   3. Taylor expanded in phi1 around -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 21.7

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

   5. Simplified 21.7%

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

   if 6.9000000000000003e-189 < phi2 < 1.2500000000000001e-115

   1. Initial program 73.2%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 93.3

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

   5. Simplified 93.3%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 87.9

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

   8. Simplified 87.9%

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

   9. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \lambda_1\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto R \cdot \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} \]

       2. lower-neg.f64 16.4

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

   11. Simplified 16.4%

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

   if 1.2500000000000001e-115 < phi2 < 4.9e14

   1. Initial program 67.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 61.3

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

   8. Simplified 61.3%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

       2. lower-*.f64 20.3

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   11. Simplified 20.3%

       \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   if 4.9e14 < phi2

   1. Initial program 52.6%

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

   3. Taylor expanded in phi1 around -inf

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

   5. Simplified 48.5%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 67.7

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

   8. Simplified 67.7%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.9 \cdot 10^{-189}:\\ \;\;\;\;-R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 1.25 \cdot 10^{-115}:\\ \;\;\;\;-R \cdot \lambda_1\\ \mathbf{elif}\;\phi_2 \leq 4.9 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.2% accurate, 15.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-115}:\\ \;\;\;\;-R \cdot \lambda_1\\ \mathbf{elif}\;\phi_2 \leq 4.9 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

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 1.25e-115)
   (- (* R lambda1))
   (if (<= phi2 4.9e+14) (* R 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 <= 1.25e-115) {
		tmp = -(R * lambda1);
	} else if (phi2 <= 4.9e+14) {
		tmp = R * lambda2;
	} 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 <= 1.25d-115) then
        tmp = -(r * lambda1)
    else if (phi2 <= 4.9d+14) then
        tmp = r * lambda2
    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 <= 1.25e-115) {
		tmp = -(R * lambda1);
	} else if (phi2 <= 4.9e+14) {
		tmp = R * lambda2;
	} 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 <= 1.25e-115:
		tmp = -(R * lambda1)
	elif phi2 <= 4.9e+14:
		tmp = R * lambda2
	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 <= 1.25e-115)
		tmp = Float64(-Float64(R * lambda1));
	elseif (phi2 <= 4.9e+14)
		tmp = Float64(R * lambda2);
	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 <= 1.25e-115)
		tmp = -(R * lambda1);
	elseif (phi2 <= 4.9e+14)
		tmp = R * lambda2;
	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, 1.25e-115], (-N[(R * lambda1), $MachinePrecision]), If[LessEqual[phi2, 4.9e+14], N[(R * lambda2), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.2500000000000001e-115

   1. Initial program 64.0%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 79.0

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

   5. Simplified 79.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 72.8

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

   8. Simplified 72.8%

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

   9. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \lambda_1\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto R \cdot \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} \]

       2. lower-neg.f64 17.4

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

   11. Simplified 17.4%

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

   if 1.2500000000000001e-115 < phi2 < 4.9e14

   1. Initial program 67.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 61.3

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

   8. Simplified 61.3%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

       2. lower-*.f64 20.3

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   11. Simplified 20.3%

       \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   if 4.9e14 < phi2

   1. Initial program 52.6%

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

   3. Taylor expanded in phi1 around -inf

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

   5. Simplified 48.5%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 67.7

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

   8. Simplified 67.7%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-115}:\\ \;\;\;\;-R \cdot \lambda_1\\ \mathbf{elif}\;\phi_2 \leq 4.9 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.2% accurate, 23.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.9 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

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 4.9e+14) (* R 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 <= 4.9e+14) {
		tmp = R * lambda2;
	} 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 <= 4.9d+14) then
        tmp = r * lambda2
    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 <= 4.9e+14) {
		tmp = R * lambda2;
	} 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 <= 4.9e+14:
		tmp = R * lambda2
	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 <= 4.9e+14)
		tmp = Float64(R * lambda2);
	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 <= 4.9e+14)
		tmp = R * lambda2;
	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, 4.9e+14], N[(R * lambda2), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.9e14

   1. Initial program 64.4%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 78.7

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

   5. Simplified 78.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 71.5

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

   8. Simplified 71.5%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

       2. lower-*.f64 13.7

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   11. Simplified 13.7%

       \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   if 4.9e14 < phi2

   1. Initial program 52.6%

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

   3. Taylor expanded in phi1 around -inf

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

   5. Simplified 48.5%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 67.7

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

   8. Simplified 67.7%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.9 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.0% 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 61.5%

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

3. Taylor expanded in phi1 around -inf

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

5. Simplified 32.8%

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

6. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 19.6

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

8. Simplified 19.6%

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

9. Final simplification 19.6%

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

Reproduce

herbie shell --seed 2024215 
(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))))))