Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.2% → 90.9%

Time: 13.5s

Alternatives: 12

Speedup: 6.2×

• 30.3% 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: 59.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.6e-5)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.6e-5) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.6e-5) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.6e-5:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.6e-5)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.6e-5)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.6e-5], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.59999999999999993e-5

   1. Initial program 61.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 88.1

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

   5. Applied rewrites 88.1%

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

   if -1.59999999999999993e-5 < phi1

   1. Initial program 61.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.55 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

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 -2.55e-13)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (* 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 <= -2.55e-13) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} 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 <= -2.55e-13) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5))));
	} 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 <= -2.55e-13:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5))))
	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 <= -2.55e-13)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.55e-13)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	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, -2.55e-13], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.55e-13

   1. Initial program 61.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -2.55e-13 < phi1

   1. Initial program 61.4%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 71.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.55 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\phi_1 - \phi_2, \phi_1 - \phi_2, \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

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 -3.5e+110)
   (fma phi2 R (- (* phi1 R)))
   (if (<= phi1 -1.6e-5)
     (*
      R
      (sqrt
       (fma
        (- phi1 phi2)
        (- phi1 phi2)
        (*
         (* (- lambda1 lambda2) (- lambda1 lambda2))
         (+ 0.5 (* 0.5 (cos (+ phi1 phi2))))))))
     (* 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 <= -3.5e+110) {
		tmp = fma(phi2, R, -(phi1 * R));
	} else if (phi1 <= -1.6e-5) {
		tmp = R * sqrt(fma((phi1 - phi2), (phi1 - phi2), (((lambda1 - lambda2) * (lambda1 - lambda2)) * (0.5 + (0.5 * cos((phi1 + phi2)))))));
	} else {
		tmp = R * 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 <= -3.5e+110)
		tmp = fma(phi2, R, Float64(-Float64(phi1 * R)));
	elseif (phi1 <= -1.6e-5)
		tmp = Float64(R * sqrt(fma(Float64(phi1 - phi2), Float64(phi1 - phi2), Float64(Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)) * Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + phi2))))))));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.5e+110], N[(phi2 * R + (-N[(phi1 * R), $MachinePrecision])), $MachinePrecision], If[LessEqual[phi1, -1.6e-5], N[(R * N[Sqrt[N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision] + N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.4999999999999999e110

   1. Initial program 52.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 69.1%

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

   10. Applied rewrites 71.2%

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

   if -3.4999999999999999e110 < phi1 < -1.59999999999999993e-5

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 77.2

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-cos.f64 N/A

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

   4. Applied rewrites 77.2%

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

   if -1.59999999999999993e-5 < phi1

   1. Initial program 61.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 71.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\phi_1 - \phi_2, \phi_1 - \phi_2, \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq -8.8 \cdot 10^{-263}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \lambda_1 - \lambda_2, \phi_2 \cdot \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, -\lambda_2\right)\\ \end{array} \end{array} \]

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 -2.7e+17)
   (fma phi2 R (- (* phi1 R)))
   (if (<= phi1 -8.8e-263)
     (* R (sqrt (fma (- lambda1 lambda2) (- lambda1 lambda2) (* phi2 phi2))))
     (* R (hypot phi2 (- 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 <= -2.7e+17) {
		tmp = fma(phi2, R, -(phi1 * R));
	} else if (phi1 <= -8.8e-263) {
		tmp = R * sqrt(fma((lambda1 - lambda2), (lambda1 - lambda2), (phi2 * phi2)));
	} else {
		tmp = R * hypot(phi2, -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 <= -2.7e+17)
		tmp = fma(phi2, R, Float64(-Float64(phi1 * R)));
	elseif (phi1 <= -8.8e-263)
		tmp = Float64(R * sqrt(fma(Float64(lambda1 - lambda2), Float64(lambda1 - lambda2), Float64(phi2 * phi2))));
	else
		tmp = Float64(R * hypot(phi2, Float64(-lambda2)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.7e+17], N[(phi2 * R + (-N[(phi1 * R), $MachinePrecision])), $MachinePrecision], If[LessEqual[phi1, -8.8e-263], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision] + N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + (-lambda2) ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.7e17

   1. Initial program 59.4%

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

   3. Taylor expanded in phi1 around -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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 66.9%

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

   10. Applied rewrites 68.5%

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

   if -2.7e17 < phi1 < -8.8000000000000001e-263

   1. Initial program 65.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 84.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.5

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

   10. Applied rewrites 61.5%

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

   if -8.8000000000000001e-263 < phi1

   1. Initial program 60.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 64.5%

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

   9. Taylor expanded in lambda1 around 0

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

   11. Applied rewrites 52.7%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq -8.8 \cdot 10^{-263}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \lambda_1 - \lambda_2, \phi_2 \cdot \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, -\lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0136:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.0136)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0136) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0136) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.0136:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.0136)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.0136)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.0136], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0135999999999999992

   1. Initial program 60.1%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 72.2%

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

   if -0.0135999999999999992 < phi1

   1. Initial program 62.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 71.5%

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

Alternative 6: 80.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 22:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 22.0)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (- phi2 phi1))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 22.0) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 <= 22.0) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 <= 22.0:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 22.0)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	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 <= 22.0)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 - phi1);
	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, 22.0], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 22

   1. Initial program 60.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 76.2%

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

   if 22 < phi2

   1. Initial program 64.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 68.9%

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

Alternative 7: 62.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.7 \cdot 10^{+119}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \lambda_1 - \lambda_2, \phi_2 \cdot \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -8e-223)
   (fma phi2 R (- (* phi1 R)))
   (if (<= phi2 2.7e+119)
     (* R (sqrt (fma (- lambda1 lambda2) (- lambda1 lambda2) (* phi2 phi2))))
     (* R (- phi2 phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -8e-223) {
		tmp = fma(phi2, R, -(phi1 * R));
	} else if (phi2 <= 2.7e+119) {
		tmp = R * sqrt(fma((lambda1 - lambda2), (lambda1 - lambda2), (phi2 * phi2)));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -8e-223)
		tmp = fma(phi2, R, Float64(-Float64(phi1 * R)));
	elseif (phi2 <= 2.7e+119)
		tmp = Float64(R * sqrt(fma(Float64(lambda1 - lambda2), Float64(lambda1 - lambda2), Float64(phi2 * phi2))));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -8e-223], N[(phi2 * R + (-N[(phi1 * R), $MachinePrecision])), $MachinePrecision], If[LessEqual[phi2, 2.7e+119], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision] + N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -7.9999999999999998e-223

   1. Initial program 56.0%

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

   3. Taylor expanded in phi1 around -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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 26.3%

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

   10. Applied rewrites 27.4%

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

   if -7.9999999999999998e-223 < phi2 < 2.6999999999999998e119

   1. Initial program 66.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 63.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.4

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

   10. Applied rewrites 51.2%

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

   if 2.6999999999999998e119 < phi2

   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 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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 77.1%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.7 \cdot 10^{+119}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \lambda_1 - \lambda_2, \phi_2 \cdot \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.3 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 5.7 \cdot 10^{-85}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.125, \lambda_2 \cdot \left(\phi_2 \cdot \phi_2\right), \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -3.3e-223)
   (fma phi2 R (- (* phi1 R)))
   (if (<= phi2 5.7e-85)
     (* R (fma -0.125 (* lambda2 (* phi2 phi2)) lambda2))
     (if (<= phi2 3.6e+84)
       (* phi1 (- (/ (* R phi2) phi1) R))
       (* R (- phi2 phi1))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -3.3e-223) {
		tmp = fma(phi2, R, -(phi1 * R));
	} else if (phi2 <= 5.7e-85) {
		tmp = R * fma(-0.125, (lambda2 * (phi2 * phi2)), lambda2);
	} else if (phi2 <= 3.6e+84) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -3.3e-223)
		tmp = fma(phi2, R, Float64(-Float64(phi1 * R)));
	elseif (phi2 <= 5.7e-85)
		tmp = Float64(R * fma(-0.125, Float64(lambda2 * Float64(phi2 * phi2)), lambda2));
	elseif (phi2 <= 3.6e+84)
		tmp = Float64(phi1 * Float64(Float64(Float64(R * phi2) / phi1) - R));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -3.3e-223], N[(phi2 * R + (-N[(phi1 * R), $MachinePrecision])), $MachinePrecision], If[LessEqual[phi2, 5.7e-85], N[(R * N[(-0.125 * N[(lambda2 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision] + lambda2), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.6e+84], N[(phi1 * N[(N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -3.29999999999999994e-223

   1. Initial program 56.0%

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

   3. Taylor expanded in phi1 around -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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 26.3%

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

   10. Applied rewrites 27.4%

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

   if -3.29999999999999994e-223 < phi2 < 5.69999999999999979e-85

   1. Initial program 59.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in lambda2 around inf

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

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(\lambda_2 + \frac{-1}{8} \cdot \color{blue}{\left(\lambda_2 \cdot {\phi_2}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 19.7%

       \[\leadsto R \cdot \mathsf{fma}\left(-0.125, \lambda_2 \cdot \color{blue}{\left(\phi_2 \cdot \phi_2\right)}, \lambda_2\right) \]

   if 5.69999999999999979e-85 < phi2 < 3.5999999999999999e84

   1. Initial program 83.7%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in phi1 around inf

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

   8. Applied rewrites 63.9%

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

   if 3.5999999999999999e84 < phi2

   1. Initial program 56.1%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 71.3%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.3 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 5.7 \cdot 10^{-85}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.125, \lambda_2 \cdot \left(\phi_2 \cdot \phi_2\right), \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.3 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-85}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.125, \lambda_2 \cdot \left(\phi_2 \cdot \phi_2\right), \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -3.3e-223)
   (fma phi2 R (- (* phi1 R)))
   (if (<= phi2 7.5e-85)
     (* R (fma -0.125 (* lambda2 (* phi2 phi2)) lambda2))
     (* R (- phi2 phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -3.3e-223) {
		tmp = fma(phi2, R, -(phi1 * R));
	} else if (phi2 <= 7.5e-85) {
		tmp = R * fma(-0.125, (lambda2 * (phi2 * phi2)), lambda2);
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -3.3e-223)
		tmp = fma(phi2, R, Float64(-Float64(phi1 * R)));
	elseif (phi2 <= 7.5e-85)
		tmp = Float64(R * fma(-0.125, Float64(lambda2 * Float64(phi2 * phi2)), lambda2));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -3.3e-223], N[(phi2 * R + (-N[(phi1 * R), $MachinePrecision])), $MachinePrecision], If[LessEqual[phi2, 7.5e-85], N[(R * N[(-0.125 * N[(lambda2 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision] + lambda2), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.29999999999999994e-223

   1. Initial program 56.0%

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

   3. Taylor expanded in phi1 around -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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 26.3%

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

   10. Applied rewrites 27.4%

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

   if -3.29999999999999994e-223 < phi2 < 7.5000000000000003e-85

   1. Initial program 59.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in lambda2 around inf

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

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(\lambda_2 + \frac{-1}{8} \cdot \color{blue}{\left(\lambda_2 \cdot {\phi_2}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 19.7%

       \[\leadsto R \cdot \mathsf{fma}\left(-0.125, \lambda_2 \cdot \color{blue}{\left(\phi_2 \cdot \phi_2\right)}, \lambda_2\right) \]

   if 7.5000000000000003e-85 < phi2

   1. Initial program 69.4%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 64.3%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.3 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, -\phi_1 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-85}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.125, \lambda_2 \cdot \left(\phi_2 \cdot \phi_2\right), \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.3% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

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 -1.6e-5) (* 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 <= -1.6e-5) {
		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 <= (-1.6d-5)) 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 <= -1.6e-5) {
		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 <= -1.6e-5:
		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 <= -1.6e-5)
		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 <= -1.6e-5)
		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, -1.6e-5], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.59999999999999993e-5

   1. Initial program 61.2%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if -1.59999999999999993e-5 < phi1

   1. Initial program 61.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.3% accurate, 31.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in phi1 around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 33.8

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

5. Applied rewrites 33.8%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 34.9%

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

Alternative 12: 31.5% 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 61.5%

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

3. Taylor expanded in phi2 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 20.3

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

5. Applied rewrites 20.3%

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

6. Final simplification 20.3%

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

Reproduce

herbie shell --seed 2024221 
(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))))))