Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.7% → 91.6%

Time: 12.1s

Alternatives: 13

Speedup: 5.9×

• 30.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.09999999999999992e-17

   1. Initial program 64.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 79.5

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

   5. Simplified 79.5%

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

   if 2.09999999999999992e-17 < phi2

   1. Initial program 53.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 84.2

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

   5. Simplified 84.2%

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

4. Final simplification 80.9%

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

Alternative 2: 85.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.2 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, \frac{\phi_1}{-\phi_2}, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 8.2e+14)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (* R (fma phi2 (/ phi1 (- phi2)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.2e+14) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else {
		tmp = R * fma(phi2, (phi1 / -phi2), phi2);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 8.2e+14)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * fma(phi2, Float64(phi1 / Float64(-phi2)), phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 8.2e+14], 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[(phi2 * N[(phi1 / (-phi2)), $MachinePrecision] + phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.2e14

   1. Initial program 64.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 78.7

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

   5. Simplified 78.7%

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

   if 8.2e14 < phi2

   1. Initial program 52.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 74.8

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

   5. Simplified 74.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.2 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, \frac{\phi_1}{-\phi_2}, \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.2 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_2, \frac{\phi_1}{-\phi_2}, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 8.2e+14)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (fma phi2 (/ phi1 (- phi2)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.2e+14) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * fma(phi2, (phi1 / -phi2), phi2);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 8.2e+14)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * fma(phi2, Float64(phi1 / Float64(-phi2)), phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 8.2e+14], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(phi1 / (-phi2)), $MachinePrecision] + phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.2e14

   1. Initial program 64.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 78.7

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

   5. Simplified 78.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 71.5

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

   8. Simplified 71.5%

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

   if 8.2e14 < phi2

   1. Initial program 52.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 74.8

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

   5. Simplified 74.8%

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

Alternative 4: 61.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.35 \cdot 10^{+117}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.86 \cdot 10^{-51}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq -5.1 \cdot 10^{-142}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.5, \frac{\phi_2 \cdot \phi_2}{\lambda_1}, \lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \lambda_1 - \lambda_2, \phi_2 \cdot \phi_2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.35e+117)
   (* R (fma phi1 (/ phi2 phi1) (- phi1)))
   (if (<= phi1 -1.86e-51)
     (* phi2 (- R (/ (* R phi1) phi2)))
     (if (<= phi1 -5.1e-142)
       (* R (fma -0.5 (/ (* phi2 phi2) lambda1) (- lambda2 lambda1)))
       (*
        R
        (sqrt (fma (- lambda1 lambda2) (- lambda1 lambda2) (* phi2 phi2))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.35e+117) {
		tmp = R * fma(phi1, (phi2 / phi1), -phi1);
	} else if (phi1 <= -1.86e-51) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else if (phi1 <= -5.1e-142) {
		tmp = R * fma(-0.5, ((phi2 * phi2) / lambda1), (lambda2 - lambda1));
	} else {
		tmp = R * sqrt(fma((lambda1 - lambda2), (lambda1 - lambda2), (phi2 * phi2)));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.35e+117)
		tmp = Float64(R * fma(phi1, Float64(phi2 / phi1), Float64(-phi1)));
	elseif (phi1 <= -1.86e-51)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	elseif (phi1 <= -5.1e-142)
		tmp = Float64(R * fma(-0.5, Float64(Float64(phi2 * phi2) / lambda1), Float64(lambda2 - lambda1)));
	else
		tmp = Float64(R * sqrt(fma(Float64(lambda1 - lambda2), Float64(lambda1 - lambda2), Float64(phi2 * phi2))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.35e+117], N[(R * N[(phi1 * N[(phi2 / phi1), $MachinePrecision] + (-phi1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -1.86e-51], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -5.1e-142], N[(R * N[(-0.5 * N[(N[(phi2 * phi2), $MachinePrecision] / lambda1), $MachinePrecision] + N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision] + N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -1.3500000000000001e117

   1. Initial program 46.3%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 86.1

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

   5. Simplified 86.1%

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

   if -1.3500000000000001e117 < phi1 < -1.8600000000000001e-51

   1. Initial program 55.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 55.8

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

   5. Simplified 55.8%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.9

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

   8. Simplified 42.9%

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

   if -1.8600000000000001e-51 < phi1 < -5.1000000000000001e-142

   1. Initial program 72.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 72.8

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

   5. Simplified 72.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.8

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

   8. Simplified 72.8%

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

   9. Taylor expanded in lambda1 around -inf

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

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

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

       8. remove-double-neg N/A

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

       9. *-rgt-identity N/A

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

       10. lower-fma.f64 N/A

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

   11. Simplified 58.8%

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

   12. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 58.8

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

   14. Simplified 58.8%

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

   if -5.1000000000000001e-142 < phi1

   1. Initial program 66.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 55.1

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

   8. Simplified 55.1%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.35 \cdot 10^{+117}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.86 \cdot 10^{-51}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq -5.1 \cdot 10^{-142}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.5, \frac{\phi_2 \cdot \phi_2}{\lambda_1}, \lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \lambda_1 - \lambda_2, \phi_2 \cdot \phi_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.35 \cdot 10^{+117}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.86 \cdot 10^{-51}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-187}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.5, \frac{\phi_2 \cdot \phi_2}{\lambda_1}, \lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.35e+117)
   (* R (fma phi1 (/ phi2 phi1) (- phi1)))
   (if (<= phi1 -1.86e-51)
     (* phi2 (- R (/ (* R phi1) phi2)))
     (if (<= phi1 2.2e-187)
       (* R (fma -0.5 (/ (* phi2 phi2) lambda1) (- lambda2 lambda1)))
       (* phi2 R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.35e+117) {
		tmp = R * fma(phi1, (phi2 / phi1), -phi1);
	} else if (phi1 <= -1.86e-51) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else if (phi1 <= 2.2e-187) {
		tmp = R * fma(-0.5, ((phi2 * phi2) / lambda1), (lambda2 - lambda1));
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.35e+117)
		tmp = Float64(R * fma(phi1, Float64(phi2 / phi1), Float64(-phi1)));
	elseif (phi1 <= -1.86e-51)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	elseif (phi1 <= 2.2e-187)
		tmp = Float64(R * fma(-0.5, Float64(Float64(phi2 * phi2) / lambda1), Float64(lambda2 - lambda1)));
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.35e+117], N[(R * N[(phi1 * N[(phi2 / phi1), $MachinePrecision] + (-phi1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -1.86e-51], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 2.2e-187], N[(R * N[(-0.5 * N[(N[(phi2 * phi2), $MachinePrecision] / lambda1), $MachinePrecision] + N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -1.3500000000000001e117

   1. Initial program 46.3%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 86.1

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

   5. Simplified 86.1%

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

   if -1.3500000000000001e117 < phi1 < -1.8600000000000001e-51

   1. Initial program 55.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 55.8

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

   5. Simplified 55.8%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.9

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

   8. Simplified 42.9%

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

   if -1.8600000000000001e-51 < phi1 < 2.20000000000000008e-187

   1. Initial program 71.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 69.7

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

   5. Simplified 69.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 69.7

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

   8. Simplified 69.7%

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

   9. Taylor expanded in lambda1 around -inf

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

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

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

       8. remove-double-neg N/A

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

       9. *-rgt-identity N/A

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

       10. lower-fma.f64 N/A

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

   11. Simplified 36.6%

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

   12. Taylor expanded in phi2 around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 36.8

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

   14. Simplified 36.8%

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

   if 2.20000000000000008e-187 < phi1

   1. Initial program 63.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 61.3

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

   5. Simplified 61.3%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 19.0

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

   8. Simplified 19.0%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.35 \cdot 10^{+117}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.86 \cdot 10^{-51}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-187}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.5, \frac{\phi_2 \cdot \phi_2}{\lambda_1}, \lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{+124}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5e+124)
   (* R (fma phi1 (/ phi2 phi1) (- phi1)))
   (*
    R
    (sqrt
     (+
      (* (- lambda1 lambda2) (- lambda1 lambda2))
      (* (- phi1 phi2) (- phi1 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 <= -5e+124) {
		tmp = R * fma(phi1, (phi2 / phi1), -phi1);
	} else {
		tmp = R * sqrt((((lambda1 - lambda2) * (lambda1 - lambda2)) + ((phi1 - phi2) * (phi1 - 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 <= -5e+124)
		tmp = Float64(R * fma(phi1, Float64(phi2 / phi1), Float64(-phi1)));
	else
		tmp = Float64(R * sqrt(Float64(Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5e+124], N[(R * N[(phi1 * N[(phi2 / phi1), $MachinePrecision] + (-phi1)), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.9999999999999996e124

   1. Initial program 45.3%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 85.9

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

   5. Simplified 85.9%

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

   if -4.9999999999999996e124 < phi1

   1. Initial program 65.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 63.9

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

   5. Simplified 63.9%

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

   6. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 63.2

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

   8. Simplified 63.2%

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

Alternative 7: 55.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.35 \cdot 10^{+117}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.86 \cdot 10^{-51}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 4.6 \cdot 10^{-183}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.35e+117)
   (* R (fma phi1 (/ phi2 phi1) (- phi1)))
   (if (<= phi1 -1.86e-51)
     (* phi2 (- R (/ (* R phi1) phi2)))
     (if (<= phi1 4.6e-183)
       (* lambda2 (- R (/ (* R lambda1) lambda2)))
       (* phi2 R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.35e+117) {
		tmp = R * fma(phi1, (phi2 / phi1), -phi1);
	} else if (phi1 <= -1.86e-51) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else if (phi1 <= 4.6e-183) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.35e+117)
		tmp = Float64(R * fma(phi1, Float64(phi2 / phi1), Float64(-phi1)));
	elseif (phi1 <= -1.86e-51)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	elseif (phi1 <= 4.6e-183)
		tmp = Float64(lambda2 * Float64(R - Float64(Float64(R * lambda1) / lambda2)));
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.35e+117], N[(R * N[(phi1 * N[(phi2 / phi1), $MachinePrecision] + (-phi1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -1.86e-51], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4.6e-183], N[(lambda2 * N[(R - N[(N[(R * lambda1), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -1.3500000000000001e117

   1. Initial program 46.3%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 86.1

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

   5. Simplified 86.1%

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

   if -1.3500000000000001e117 < phi1 < -1.8600000000000001e-51

   1. Initial program 55.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 55.8

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

   5. Simplified 55.8%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.9

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

   8. Simplified 42.9%

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

   if -1.8600000000000001e-51 < phi1 < 4.60000000000000032e-183

   1. Initial program 72.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 70.1

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

   8. Simplified 70.1%

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

   9. Taylor expanded in lambda2 around inf

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

       1. lower-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 26.5

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

   11. Simplified 26.5%

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

   if 4.60000000000000032e-183 < phi1

   1. Initial program 63.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 60.9

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

   5. Simplified 60.9%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 19.2

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

   8. Simplified 19.2%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.35 \cdot 10^{+117}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.86 \cdot 10^{-51}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 4.6 \cdot 10^{-183}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.05 \cdot 10^{-51}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 4.6 \cdot 10^{-183}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.05e-51)
   (* R (fma phi1 (/ phi2 phi1) (- phi1)))
   (if (<= phi1 4.6e-183)
     (* lambda2 (- R (/ (* R lambda1) lambda2)))
     (* phi2 R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.05e-51) {
		tmp = R * fma(phi1, (phi2 / phi1), -phi1);
	} else if (phi1 <= 4.6e-183) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.05e-51)
		tmp = Float64(R * fma(phi1, Float64(phi2 / phi1), Float64(-phi1)));
	elseif (phi1 <= 4.6e-183)
		tmp = Float64(lambda2 * Float64(R - Float64(Float64(R * lambda1) / lambda2)));
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.05e-51], N[(R * N[(phi1 * N[(phi2 / phi1), $MachinePrecision] + (-phi1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4.6e-183], N[(lambda2 * N[(R - N[(N[(R * lambda1), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.04999999999999987e-51

   1. Initial program 49.6%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 67.5

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

   5. Simplified 67.5%

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

   if -2.04999999999999987e-51 < phi1 < 4.60000000000000032e-183

   1. Initial program 72.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 70.1

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

   8. Simplified 70.1%

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

   9. Taylor expanded in lambda2 around inf

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

       1. lower-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 26.5

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

   11. Simplified 26.5%

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

   if 4.60000000000000032e-183 < phi1

   1. Initial program 63.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 60.9

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

   5. Simplified 60.9%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 19.2

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

   8. Simplified 19.2%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.05 \cdot 10^{-51}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 4.6 \cdot 10^{-183}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9.4 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 9.4e+190)
   (* R (fma phi1 (/ phi2 phi1) (- phi1)))
   (* R lambda2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 9.4e+190) {
		tmp = R * fma(phi1, (phi2 / phi1), -phi1);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 9.4e+190)
		tmp = Float64(R * fma(phi1, Float64(phi2 / phi1), Float64(-phi1)));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 9.4e+190], N[(R * N[(phi1 * N[(phi2 / phi1), $MachinePrecision] + (-phi1)), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 9.3999999999999996e190

   1. Initial program 62.1%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 34.8

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

   5. Simplified 34.8%

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

   if 9.3999999999999996e190 < lambda2

   1. Initial program 54.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 54.8

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

   5. Simplified 54.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 54.8

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

   8. Simplified 54.8%

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

   9. Taylor expanded in lambda2 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 65.8

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

   11. Simplified 65.8%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9.4 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1, \frac{\phi_2}{\phi_1}, -\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.2 \cdot 10^{-189}:\\ \;\;\;\;-R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 4.9 \cdot 10^{-116}:\\ \;\;\;\;-R \cdot \lambda_1\\ \mathbf{elif}\;\phi_2 \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 9.2e-189)
   (- (* R phi1))
   (if (<= phi2 4.9e-116)
     (- (* R lambda1))
     (if (<= phi2 3.3e+14) (* R lambda2) (* phi2 R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 9.2e-189) {
		tmp = -(R * phi1);
	} else if (phi2 <= 4.9e-116) {
		tmp = -(R * lambda1);
	} else if (phi2 <= 3.3e+14) {
		tmp = R * lambda2;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 9.2d-189) then
        tmp = -(r * phi1)
    else if (phi2 <= 4.9d-116) then
        tmp = -(r * lambda1)
    else if (phi2 <= 3.3d+14) then
        tmp = r * lambda2
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 9.2e-189) {
		tmp = -(R * phi1);
	} else if (phi2 <= 4.9e-116) {
		tmp = -(R * lambda1);
	} else if (phi2 <= 3.3e+14) {
		tmp = R * lambda2;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 9.2e-189:
		tmp = -(R * phi1)
	elif phi2 <= 4.9e-116:
		tmp = -(R * lambda1)
	elif phi2 <= 3.3e+14:
		tmp = R * lambda2
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 9.2e-189)
		tmp = Float64(-Float64(R * phi1));
	elseif (phi2 <= 4.9e-116)
		tmp = Float64(-Float64(R * lambda1));
	elseif (phi2 <= 3.3e+14)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 9.2e-189)
		tmp = -(R * phi1);
	elseif (phi2 <= 4.9e-116)
		tmp = -(R * lambda1);
	elseif (phi2 <= 3.3e+14)
		tmp = R * lambda2;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 9.2e-189], (-N[(R * phi1), $MachinePrecision]), If[LessEqual[phi2, 4.9e-116], (-N[(R * lambda1), $MachinePrecision]), If[LessEqual[phi2, 3.3e+14], N[(R * lambda2), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < 9.1999999999999993e-189

   1. Initial program 63.2%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 21.7

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

   5. Simplified 21.7%

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

   if 9.1999999999999993e-189 < phi2 < 4.89999999999999977e-116

   1. Initial program 73.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 53.1

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

   8. Simplified 53.1%

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

   9. Taylor expanded in lambda1 around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

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

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

       4. mul-1-neg N/A

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

       5. lower-*.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 16.4

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

   11. Simplified 16.4%

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

   if 4.89999999999999977e-116 < phi2 < 3.3e14

   1. Initial program 67.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 60.8

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

   5. Simplified 60.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 45.5

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

   8. Simplified 45.5%

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

   9. Taylor expanded in lambda2 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 20.3

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

   11. Simplified 20.3%

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

   if 3.3e14 < phi2

   1. Initial program 52.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 50.2

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

   5. Simplified 50.2%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 67.7

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

   8. Simplified 67.7%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.2 \cdot 10^{-189}:\\ \;\;\;\;-R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 4.9 \cdot 10^{-116}:\\ \;\;\;\;-R \cdot \lambda_1\\ \mathbf{elif}\;\phi_2 \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.8% accurate, 15.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6 \cdot 10^{-130}:\\ \;\;\;\;-R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 6e-130)
   (- (* R phi1))
   (if (<= phi2 3.3e+14) (* R lambda2) (* phi2 R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6e-130) {
		tmp = -(R * phi1);
	} else if (phi2 <= 3.3e+14) {
		tmp = R * lambda2;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 6d-130) then
        tmp = -(r * phi1)
    else if (phi2 <= 3.3d+14) then
        tmp = r * lambda2
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6e-130) {
		tmp = -(R * phi1);
	} else if (phi2 <= 3.3e+14) {
		tmp = R * lambda2;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 6e-130:
		tmp = -(R * phi1)
	elif phi2 <= 3.3e+14:
		tmp = R * lambda2
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 6e-130)
		tmp = Float64(-Float64(R * phi1));
	elseif (phi2 <= 3.3e+14)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 6e-130)
		tmp = -(R * phi1);
	elseif (phi2 <= 3.3e+14)
		tmp = R * lambda2;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 6e-130], (-N[(R * phi1), $MachinePrecision]), If[LessEqual[phi2, 3.3e+14], N[(R * lambda2), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 5.99999999999999972e-130

   1. Initial program 63.9%

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

   3. Taylor expanded in phi1 around -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 22.2

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

   5. Simplified 22.2%

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

   if 5.99999999999999972e-130 < phi2 < 3.3e14

   1. Initial program 68.1%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 48.4

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

   8. Simplified 48.4%

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

   9. Taylor expanded in lambda2 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 18.2

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

   11. Simplified 18.2%

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

   if 3.3e14 < phi2

   1. Initial program 52.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 50.2

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

   5. Simplified 50.2%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 67.7

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

   8. Simplified 67.7%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6 \cdot 10^{-130}:\\ \;\;\;\;-R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.3e+14) (* R lambda2) (* phi2 R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.3e+14) {
		tmp = R * lambda2;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 3.3d+14) then
        tmp = r * lambda2
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.3e+14) {
		tmp = R * lambda2;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.3e+14:
		tmp = R * lambda2
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.3e+14)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.3e+14)
		tmp = R * lambda2;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.3e+14], N[(R * lambda2), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.3e14

   1. Initial program 64.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 63.2

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

   5. Simplified 63.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 48.8

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

   8. Simplified 48.8%

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

   9. Taylor expanded in lambda2 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 13.7

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

   11. Simplified 13.7%

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

   if 3.3e14 < phi2

   1. Initial program 52.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 50.2

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

   5. Simplified 50.2%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 67.7

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

   8. Simplified 67.7%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.0% accurate, 46.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in phi2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 59.9

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

5. Simplified 59.9%

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

6. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 19.6

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

8. Simplified 19.6%

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

9. Final simplification 19.6%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))