Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.8% → 94.8%

Time: 9.7s

Alternatives: 12

Speedup: 2.5×

• 30.2% 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: 59.8% 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: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(-0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

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 9.2e-39)
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1))
   (* R (hypot (- phi1 phi2) (* (cos (* -0.5 phi2)) (- lambda1 lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 9.2e-39) {
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * hypot((phi1 - phi2), (cos((-0.5 * phi2)) * (lambda1 - lambda2)));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 9.2e-39) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * Math.hypot((phi1 - phi2), (Math.cos((-0.5 * phi2)) * (lambda1 - lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 9.2e-39:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1)
	else:
		tmp = R * math.hypot((phi1 - phi2), (math.cos((-0.5 * phi2)) * (lambda1 - lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 9.2e-39)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	else
		tmp = Float64(R * hypot(Float64(phi1 - phi2), Float64(cos(Float64(-0.5 * phi2)) * Float64(lambda1 - lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 9.2e-39)
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	else
		tmp = R * hypot((phi1 - phi2), (cos((-0.5 * phi2)) * (lambda1 - lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 9.2e-39], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 9.20000000000000033e-39

   1. Initial program 55.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if 9.20000000000000033e-39 < phi2

   1. Initial program 55.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 87.1

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

   7. Applied rewrites 87.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \color{blue}{\cos \left(-0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \end{array} \end{array} \]

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 (<= phi1 -1.6e-6)
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1))
   (* R (hypot (* (cos (* 0.5 phi2)) (- lambda1 lambda2)) 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 (phi1 <= -1.6e-6) {
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.6e-6) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * Math.hypot((Math.cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.6e-6:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1)
	else:
		tmp = R * math.hypot((math.cos((0.5 * phi2)) * (lambda1 - lambda2)), 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 (phi1 <= -1.6e-6)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2)), phi2));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.6e-6)
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	else
		tmp = R * hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.6e-6], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.5999999999999999e-6

   1. Initial program 45.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if -1.5999999999999999e-6 < phi1

   1. Initial program 58.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 79.0

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

   5. Applied rewrites 79.0%

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

Alternative 3: 89.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 9.2e-39)
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1))
   (* R (hypot (- phi1 phi2) (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 9.2e-39) {
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * hypot((phi1 - phi2), (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 9.2e-39) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * Math.hypot((phi1 - phi2), (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 9.2e-39:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1)
	else:
		tmp = R * math.hypot((phi1 - phi2), (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 9.2e-39)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	else
		tmp = Float64(R * hypot(Float64(phi1 - phi2), Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 9.2e-39)
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	else
		tmp = R * hypot((phi1 - phi2), (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 9.2e-39], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 9.20000000000000033e-39

   1. Initial program 55.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if 9.20000000000000033e-39 < phi2

   1. Initial program 55.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 66.2

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

   7. Applied rewrites 66.2%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.8%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.6 \cdot 10^{+46}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, -\lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(-\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4.6e+46)
   (* R (hypot (- phi1 phi2) (- lambda2)))
   (* R (hypot (- phi2) (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.6e+46) {
		tmp = R * hypot((phi1 - phi2), -lambda2);
	} else {
		tmp = R * hypot(-phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.6e+46) {
		tmp = R * Math.hypot((phi1 - phi2), -lambda2);
	} else {
		tmp = R * Math.hypot(-phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4.6e+46:
		tmp = R * math.hypot((phi1 - phi2), -lambda2)
	else:
		tmp = R * math.hypot(-phi2, (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4.6e+46)
		tmp = Float64(R * hypot(Float64(phi1 - phi2), Float64(-lambda2)));
	else
		tmp = Float64(R * hypot(Float64(-phi2), Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4.6e+46)
		tmp = R * hypot((phi1 - phi2), -lambda2);
	else
		tmp = R * hypot(-phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4.6e+46], N[(R * N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + (-lambda2) ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[(-phi2) ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.6000000000000001e46

   1. Initial program 42.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]

   11. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, -1 \cdot \lambda_2\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.6%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, -\lambda_2\right) \]

   if -4.6000000000000001e46 < phi1

   1. Initial program 58.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 79.5

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

   7. Applied rewrites 79.5%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 84.3%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]

   11. Taylor expanded in phi1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 69.3

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

   13. Applied rewrites 69.3%

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

Alternative 5: 79.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{+47}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(-1 + \frac{\phi_2}{\phi_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(-\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.3e+47)
   (* R (* phi1 (+ -1.0 (/ phi2 phi1))))
   (* R (hypot (- phi2) (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.3e+47) {
		tmp = R * (phi1 * (-1.0 + (phi2 / phi1)));
	} else {
		tmp = R * hypot(-phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.3e+47) {
		tmp = R * (phi1 * (-1.0 + (phi2 / phi1)));
	} else {
		tmp = R * Math.hypot(-phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.3e+47:
		tmp = R * (phi1 * (-1.0 + (phi2 / phi1)))
	else:
		tmp = R * math.hypot(-phi2, (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.3e+47)
		tmp = Float64(R * Float64(phi1 * Float64(-1.0 + Float64(phi2 / phi1))));
	else
		tmp = Float64(R * hypot(Float64(-phi2), Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.3e+47)
		tmp = R * (phi1 * (-1.0 + (phi2 / phi1)));
	else
		tmp = R * hypot(-phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.3e+47], N[(R * N[(phi1 * N[(-1.0 + N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[(-phi2) ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.2999999999999999e47

   1. Initial program 42.9%

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

   3. Taylor expanded in phi1 around -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. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if -3.2999999999999999e47 < phi1

   1. Initial program 58.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 79.5

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

   7. Applied rewrites 79.5%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 84.3%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]

   11. Taylor expanded in phi1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 69.3

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

   13. Applied rewrites 69.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{+47}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(-1 + \frac{\phi_2}{\phi_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(-\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. pow-to-exp N/A

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

   8. exp-lft-sqr N/A

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

   9. lower-hypot.f64 N/A

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

4. Applied rewrites 50.6%

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

5. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 78.7

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

7. Applied rewrites 78.7%

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

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
9. Step-by-step derivation

10. Applied rewrites 83.2%

    \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
11. Add Preprocessing

Alternative 7: 58.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{+47}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - R \cdot \frac{\phi_2}{\phi_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.4e+47)
   (* (- phi1) (- R (* R (/ phi2 phi1))))
   (* (fma R (/ (- phi1) phi2) R) phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.4e+47) {
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	} else {
		tmp = fma(R, (-phi1 / phi2), R) * phi2;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.4e+47)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(R * Float64(phi2 / phi1))));
	else
		tmp = Float64(fma(R, Float64(Float64(-phi1) / phi2), R) * phi2);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.4e+47], N[((-phi1) * N[(R - N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * N[((-phi1) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.39999999999999994e47

   1. Initial program 42.9%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 63.6

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

   5. Applied rewrites 63.6%

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

   if -1.39999999999999994e47 < phi1

   1. Initial program 58.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 21.1

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

   5. Applied rewrites 21.1%

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

Alternative 8: 58.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{+47}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(-1 + \frac{\phi_2}{\phi_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.4e+47)
   (* R (* phi1 (+ -1.0 (/ phi2 phi1))))
   (* (fma R (/ (- phi1) phi2) R) phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.4e+47) {
		tmp = R * (phi1 * (-1.0 + (phi2 / phi1)));
	} else {
		tmp = fma(R, (-phi1 / phi2), R) * phi2;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.4e+47)
		tmp = Float64(R * Float64(phi1 * Float64(-1.0 + Float64(phi2 / phi1))));
	else
		tmp = Float64(fma(R, Float64(Float64(-phi1) / phi2), R) * phi2);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.4e+47], N[(R * N[(phi1 * N[(-1.0 + N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * N[((-phi1) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.39999999999999994e47

   1. Initial program 42.9%

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

   3. Taylor expanded in phi1 around -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. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if -1.39999999999999994e47 < phi1

   1. Initial program 58.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 21.1

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

   5. Applied rewrites 21.1%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{+47}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(-1 + \frac{\phi_2}{\phi_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{+187}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.5e+187) (* R (- phi1)) (* (fma R (/ (- phi1) phi2) R) phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.5e+187) {
		tmp = R * -phi1;
	} else {
		tmp = fma(R, (-phi1 / phi2), R) * phi2;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.5e+187)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(fma(R, Float64(Float64(-phi1) / phi2), R) * phi2);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.5e+187], N[(R * (-phi1)), $MachinePrecision], N[(N[(R * N[((-phi1) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.5000000000000001e187

   1. Initial program 39.5%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   if -2.5000000000000001e187 < phi1

   1. Initial program 56.9%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 23.1

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

   5. Applied rewrites 23.1%

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

Alternative 10: 56.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{+204}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R - \frac{\phi_1 \cdot R}{\phi_2}\right) \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1e+204) (* R (- phi1)) (* (- R (/ (* phi1 R) 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 (phi1 <= -1e+204) {
		tmp = R * -phi1;
	} else {
		tmp = (R - ((phi1 * R) / phi2)) * phi2;
	}
	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 (phi1 <= (-1d+204)) then
        tmp = r * -phi1
    else
        tmp = (r - ((phi1 * r) / phi2)) * phi2
    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 (phi1 <= -1e+204) {
		tmp = R * -phi1;
	} else {
		tmp = (R - ((phi1 * R) / phi2)) * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1e+204:
		tmp = R * -phi1
	else:
		tmp = (R - ((phi1 * R) / 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 (phi1 <= -1e+204)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(Float64(R - Float64(Float64(phi1 * R) / phi2)) * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1e+204)
		tmp = R * -phi1;
	else
		tmp = (R - ((phi1 * R) / phi2)) * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1e+204], N[(R * (-phi1)), $MachinePrecision], N[(N[(R - N[(N[(phi1 * R), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.99999999999999989e203

   1. Initial program 39.5%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   if -9.99999999999999989e203 < phi1

   1. Initial program 56.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 23.6

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

   8. Applied rewrites 23.6%

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

Alternative 11: 52.3% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{+41}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4.2e+41) (* R (- phi1)) (* R phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.2e+41) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	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 (phi1 <= (-4.2d+41)) then
        tmp = r * -phi1
    else
        tmp = r * phi2
    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 (phi1 <= -4.2e+41) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4.2e+41:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4.2e+41)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4.2e+41)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4.2e+41], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.1999999999999999e41

   1. Initial program 42.9%

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

   3. Taylor expanded in phi1 around -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 60.9

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

   5. Applied rewrites 60.9%

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

   if -4.1999999999999999e41 < phi1

   1. Initial program 58.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 19.9

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

   5. Applied rewrites 19.9%

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

Alternative 12: 31.7% accurate, 46.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 17.5

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

5. Applied rewrites 17.5%

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

Reproduce

herbie shell --seed 2024309 
(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))))))