Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.2% → 90.8%

Time: 12.8s

Alternatives: 9

Speedup: 18.6×

• 30.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 42:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 42.0)
   (* (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))) R)
   (* 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 <= 42.0) {
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	} 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 <= 42.0) {
		tmp = Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5)))) * R;
	} 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 <= 42.0:
		tmp = math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5)))) * R
	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 <= 42.0)
		tmp = Float64(hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))) * R);
	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 <= 42.0)
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	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, 42.0], N[(N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $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 < 42

   1. Initial program 67.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_1}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + {\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right), R\right) \]

      13. *-lowering-*.f64 86.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right)\right), R\right) \]

   5. Simplified 86.1%

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

   if 42 < phi2

   1. Initial program 62.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_2}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      12. --lowering--.f64 76.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   5. Simplified 76.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 42:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.8 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 1.26 \cdot 10^{+140}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)\right), \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 9.8e+30)
   (* (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))) R)
   (if (<= phi2 1.26e+140)
     (*
      R
      (sqrt
       (fma
        (- lambda1 lambda2)
        (* (- lambda1 lambda2) (+ 0.5 (* 0.5 (cos (+ phi2 phi1)))))
        (* (- phi1 phi2) (- phi1 phi2)))))
     (* R (- phi2 phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 9.8e+30) {
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	} else if (phi2 <= 1.26e+140) {
		tmp = R * sqrt(fma((lambda1 - lambda2), ((lambda1 - lambda2) * (0.5 + (0.5 * cos((phi2 + phi1))))), ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 9.8e+30)
		tmp = Float64(hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))) * R);
	elseif (phi2 <= 1.26e+140)
		tmp = Float64(R * sqrt(fma(Float64(lambda1 - lambda2), Float64(Float64(lambda1 - lambda2) * Float64(0.5 + Float64(0.5 * cos(Float64(phi2 + phi1))))), Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 9.8e+30], N[(N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 1.26e+140], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 9.79999999999999969e30

   1. Initial program 66.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_1}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + {\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right), R\right) \]

      13. *-lowering-*.f64 85.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right)\right), R\right) \]

   5. Simplified 85.1%

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

   if 9.79999999999999969e30 < phi2 < 1.26e140

   1. Initial program 81.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right), \color{blue}{R}\right) \]

   4. Applied egg-rr 81.1%

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

   if 1.26e140 < phi2

   1. Initial program 51.0%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_1}{\phi_2}\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R - \color{blue}{\frac{R \cdot \phi_1}{\phi_2}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_1}{\phi_2}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_1\right), \color{blue}{\phi_2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(\phi_1 \cdot R\right), \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 79.1%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right), \phi_2\right)\right)\right) \]

   5. Simplified 79.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto R \cdot \phi_2 - \color{blue}{R \cdot \phi_1} \]

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 - \phi_1\right)}\right) \]

      6. --lowering--.f64 88.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   8. Simplified 88.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.8 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 1.26 \cdot 10^{+140}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)\right), \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{+31}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.95 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)\right), \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.05e+31)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (if (<= phi2 1.95e+139)
     (*
      R
      (sqrt
       (fma
        (- lambda1 lambda2)
        (* (- lambda1 lambda2) (+ 0.5 (* 0.5 (cos (+ phi2 phi1)))))
        (* (- phi1 phi2) (- phi1 phi2)))))
     (* R (- phi2 phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.05e+31) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else if (phi2 <= 1.95e+139) {
		tmp = R * sqrt(fma((lambda1 - lambda2), ((lambda1 - lambda2) * (0.5 + (0.5 * cos((phi2 + phi1))))), ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.05e+31)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	elseif (phi2 <= 1.95e+139)
		tmp = Float64(R * sqrt(fma(Float64(lambda1 - lambda2), Float64(Float64(lambda1 - lambda2) * Float64(0.5 + Float64(0.5 * cos(Float64(phi2 + phi1))))), Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.05e+31], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.95e+139], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.04999999999999989e31

   1. Initial program 66.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_1}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + {\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right), R\right) \]

      13. *-lowering-*.f64 85.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right)\right), R\right) \]

   5. Simplified 85.1%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 79.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), R\right) \]

   8. Simplified 79.7%

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

   if 1.04999999999999989e31 < phi2 < 1.95000000000000003e139

   1. Initial program 81.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right), \color{blue}{R}\right) \]

   4. Applied egg-rr 81.1%

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

   if 1.95000000000000003e139 < phi2

   1. Initial program 51.0%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_1}{\phi_2}\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R - \color{blue}{\frac{R \cdot \phi_1}{\phi_2}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_1}{\phi_2}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_1\right), \color{blue}{\phi_2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(\phi_1 \cdot R\right), \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 79.1%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right), \phi_2\right)\right)\right) \]

   5. Simplified 79.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto R \cdot \phi_2 - \color{blue}{R \cdot \phi_1} \]

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 - \phi_1\right)}\right) \]

      6. --lowering--.f64 88.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   8. Simplified 88.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{+31}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.95 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\lambda_1 - \lambda_2, \left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)\right), \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4.8e+34)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (- phi2 phi1))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.8e+34) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.8e+34) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4.8e+34:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4.8e+34)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4.8e+34)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4.8e+34], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.79999999999999974e34

   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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_1}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + {\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right), R\right) \]

      13. *-lowering-*.f64 85.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right)\right), R\right) \]

   5. Simplified 85.2%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 79.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), R\right) \]

   8. Simplified 79.8%

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

   if 4.79999999999999974e34 < phi2

   1. Initial program 63.4%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_1}{\phi_2}\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R - \color{blue}{\frac{R \cdot \phi_1}{\phi_2}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_1}{\phi_2}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_1\right), \color{blue}{\phi_2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(\phi_1 \cdot R\right), \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 65.4%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right), \phi_2\right)\right)\right) \]

   5. Simplified 65.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto R \cdot \phi_2 - \color{blue}{R \cdot \phi_1} \]

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 - \phi_1\right)}\right) \]

      6. --lowering--.f64 70.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   8. Simplified 70.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-36}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

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 -2.7e-6)
   (* R (- phi2 phi1))
   (if (<= phi1 6.5e-36)
     (* R (* lambda1 (+ -1.0 (/ 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 <= -2.7e-6) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 6.5e-36) {
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)));
	} 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 (phi1 <= (-2.7d-6)) then
        tmp = r * (phi2 - phi1)
    else if (phi1 <= 6.5d-36) then
        tmp = r * (lambda1 * ((-1.0d0) + (lambda2 / lambda1)))
    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 (phi1 <= -2.7e-6) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 6.5e-36) {
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)));
	} 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 phi1 <= -2.7e-6:
		tmp = R * (phi2 - phi1)
	elif phi1 <= 6.5e-36:
		tmp = R * (lambda1 * (-1.0 + (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 <= -2.7e-6)
		tmp = Float64(R * Float64(phi2 - phi1));
	elseif (phi1 <= 6.5e-36)
		tmp = Float64(R * Float64(lambda1 * Float64(-1.0 + Float64(lambda2 / lambda1))));
	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 (phi1 <= -2.7e-6)
		tmp = R * (phi2 - phi1);
	elseif (phi1 <= 6.5e-36)
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)));
	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[phi1, -2.7e-6], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 6.5e-36], N[(R * N[(lambda1 * N[(-1.0 + N[(lambda2 / lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.69999999999999998e-6

   1. Initial program 65.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_1}{\phi_2}\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R - \color{blue}{\frac{R \cdot \phi_1}{\phi_2}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_1}{\phi_2}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_1\right), \color{blue}{\phi_2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(\phi_1 \cdot R\right), \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 67.1%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right), \phi_2\right)\right)\right) \]

   5. Simplified 67.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto R \cdot \phi_2 - \color{blue}{R \cdot \phi_1} \]

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 - \phi_1\right)}\right) \]

      6. --lowering--.f64 66.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   8. Simplified 66.1%

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

   if -2.69999999999999998e-6 < phi1 < 6.50000000000000012e-36

   1. Initial program 69.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_2}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      12. --lowering--.f64 98.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   5. Simplified 98.2%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 94.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), R\right) \]

   8. Simplified 94.1%

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

   9. Taylor expanded in lambda1 around -inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \left(1 + -1 \cdot \frac{\lambda_2}{\lambda_1}\right)\right)\right)}, R\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(-1 \cdot \lambda_1\right) \cdot \left(1 + -1 \cdot \frac{\lambda_2}{\lambda_1}\right)\right), R\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(-1 \cdot \lambda_1\right) \cdot \left(-1 \cdot \frac{\lambda_2}{\lambda_1} + 1\right)\right), R\right) \]

       3. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(-1 \cdot \lambda_1\right) \cdot \left(-1 \cdot \frac{\lambda_2}{\lambda_1}\right) + \left(-1 \cdot \lambda_1\right) \cdot 1\right), R\right) \]

       4. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(-1 \cdot \lambda_1\right) \cdot \left(-1 \cdot \frac{\lambda_2}{\lambda_1}\right) + -1 \cdot \lambda_1\right), R\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \left(-1 \cdot \frac{\lambda_2}{\lambda_1}\right) + -1 \cdot \lambda_1\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\lambda_1 \cdot \left(-1 \cdot \frac{\lambda_2}{\lambda_1}\right)\right)\right) + -1 \cdot \lambda_1\right), R\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\lambda_2}{\lambda_1}\right) \cdot \lambda_1\right)\right) + -1 \cdot \lambda_1\right), R\right) \]

       8. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\lambda_2}{\lambda_1}\right)\right) \cdot \lambda_1\right)\right) + -1 \cdot \lambda_1\right), R\right) \]

       9. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\lambda_2}{\lambda_1} \cdot \lambda_1\right)\right)\right)\right) + -1 \cdot \lambda_1\right), R\right) \]

       10. remove-double-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\lambda_2}{\lambda_1} \cdot \lambda_1 + -1 \cdot \lambda_1\right), R\right) \]

       11. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\lambda_1 \cdot \left(\frac{\lambda_2}{\lambda_1} + -1\right)\right), R\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\lambda_1 \cdot \left(\frac{\lambda_2}{\lambda_1} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), R\right) \]

       13. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\lambda_1 \cdot \left(\frac{\lambda_2}{\lambda_1} - 1\right)\right), R\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\lambda_1, \left(\frac{\lambda_2}{\lambda_1} - 1\right)\right), R\right) \]

       15. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\lambda_1, \left(\frac{\lambda_2}{\lambda_1} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), R\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\lambda_1, \left(\frac{\lambda_2}{\lambda_1} + -1\right)\right), R\right) \]

       17. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\lambda_1, \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right), R\right) \]

       18. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(-1, \left(\frac{\lambda_2}{\lambda_1}\right)\right)\right), R\right) \]

       19. /-lowering-/.f64 35.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\lambda_2, \lambda_1\right)\right)\right), R\right) \]

   11. Simplified 35.6%

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

   if 6.50000000000000012e-36 < phi1

   1. Initial program 61.2%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 20.3%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \color{blue}{R}\right) \]

   5. Simplified 20.3%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-36}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.9% accurate, 13.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.15 \cdot 10^{-139}:\\ \;\;\;\;0 - \lambda_1 \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.1 \cdot 10^{+105}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \end{array} \]

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 (<= lambda2 -1.15e-139)
   (- 0.0 (* lambda1 R))
   (if (<= lambda2 2.1e+105) (* R (- phi2 phi1)) (* lambda2 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 (lambda2 <= -1.15e-139) {
		tmp = 0.0 - (lambda1 * R);
	} else if (lambda2 <= 2.1e+105) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = lambda2 * 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 (lambda2 <= (-1.15d-139)) then
        tmp = 0.0d0 - (lambda1 * r)
    else if (lambda2 <= 2.1d+105) then
        tmp = r * (phi2 - phi1)
    else
        tmp = lambda2 * 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 (lambda2 <= -1.15e-139) {
		tmp = 0.0 - (lambda1 * R);
	} else if (lambda2 <= 2.1e+105) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = lambda2 * 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 lambda2 <= -1.15e-139:
		tmp = 0.0 - (lambda1 * R)
	elif lambda2 <= 2.1e+105:
		tmp = R * (phi2 - phi1)
	else:
		tmp = lambda2 * 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 (lambda2 <= -1.15e-139)
		tmp = Float64(0.0 - Float64(lambda1 * R));
	elseif (lambda2 <= 2.1e+105)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(lambda2 * 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 (lambda2 <= -1.15e-139)
		tmp = 0.0 - (lambda1 * R);
	elseif (lambda2 <= 2.1e+105)
		tmp = R * (phi2 - phi1);
	else
		tmp = lambda2 * 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[lambda2, -1.15e-139], N[(0.0 - N[(lambda1 * R), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 2.1e+105], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(lambda2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.15000000000000006e-139

   1. Initial program 63.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_2}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      12. --lowering--.f64 72.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   5. Simplified 72.3%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 70.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), R\right) \]

   8. Simplified 70.6%

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

   9. Taylor expanded in lambda1 around -inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \lambda_1\right)}, R\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right), R\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(0 - \lambda_1\right), R\right) \]

       3. --lowering--.f64 13.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \lambda_1\right), R\right) \]

   11. Simplified 13.1%

       \[\leadsto \color{blue}{\left(0 - \lambda_1\right)} \cdot R \]

   if -1.15000000000000006e-139 < lambda2 < 2.1000000000000001e105

   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 inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_1}{\phi_2}\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R - \color{blue}{\frac{R \cdot \phi_1}{\phi_2}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_1}{\phi_2}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_1\right), \color{blue}{\phi_2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(\phi_1 \cdot R\right), \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 36.3%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right), \phi_2\right)\right)\right) \]

   5. Simplified 36.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto R \cdot \phi_2 - \color{blue}{R \cdot \phi_1} \]

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 - \phi_1\right)}\right) \]

      6. --lowering--.f64 36.3%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   8. Simplified 36.3%

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

   if 2.1000000000000001e105 < lambda2

   1. Initial program 65.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_2}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      12. --lowering--.f64 79.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   5. Simplified 79.8%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 73.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), R\right) \]

   8. Simplified 73.7%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\lambda_2}, R\right) \]
   10. Step-by-step derivation

   11. Simplified 65.1%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.15 \cdot 10^{-139}:\\ \;\;\;\;0 - \lambda_1 \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.1 \cdot 10^{+105}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.4% accurate, 18.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.2 \cdot 10^{+106}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \end{array} \]

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 (<= lambda2 1.2e+106) (* R (- phi2 phi1)) (* lambda2 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 (lambda2 <= 1.2e+106) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = lambda2 * 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 (lambda2 <= 1.2d+106) then
        tmp = r * (phi2 - phi1)
    else
        tmp = lambda2 * 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 (lambda2 <= 1.2e+106) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = lambda2 * 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 lambda2 <= 1.2e+106:
		tmp = R * (phi2 - phi1)
	else:
		tmp = lambda2 * 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 (lambda2 <= 1.2e+106)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(lambda2 * 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 (lambda2 <= 1.2e+106)
		tmp = R * (phi2 - phi1);
	else
		tmp = lambda2 * 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[lambda2, 1.2e+106], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(lambda2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.2e106

   1. Initial program 66.2%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_1}{\phi_2}\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \left(R - \color{blue}{\frac{R \cdot \phi_1}{\phi_2}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_1}{\phi_2}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_1\right), \color{blue}{\phi_2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(\phi_1 \cdot R\right), \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 30.9%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right), \phi_2\right)\right)\right) \]

   5. Simplified 30.9%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto R \cdot \phi_2 - \color{blue}{R \cdot \phi_1} \]

      4. distribute-lft-out-- N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 - \phi_1\right)}\right) \]

      6. --lowering--.f64 31.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   8. Simplified 31.4%

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

   if 1.2e106 < lambda2

   1. Initial program 65.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_2}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      12. --lowering--.f64 79.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   5. Simplified 79.8%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 73.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), R\right) \]

   8. Simplified 73.7%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\lambda_2}, R\right) \]
   10. Step-by-step derivation

   11. Simplified 65.1%

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

Alternative 8: 37.4% accurate, 23.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.55 \cdot 10^{+32}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.55e+32) (* lambda2 R) (* phi2 R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.55e+32) {
		tmp = lambda2 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.55d+32) then
        tmp = lambda2 * r
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.55e+32) {
		tmp = lambda2 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.55e+32:
		tmp = lambda2 * R
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.55e+32)
		tmp = Float64(lambda2 * R);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.55e+32)
		tmp = lambda2 * R;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.55e+32], N[(lambda2 * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.54999999999999997e32

   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 phi1 around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}\right), \color{blue}{R}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_2}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      7. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

      8. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      12. --lowering--.f64 71.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   5. Simplified 71.9%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 69.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), R\right) \]

   8. Simplified 69.9%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\lambda_2}, R\right) \]
   10. Step-by-step derivation

   11. Simplified 19.6%

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

   if 1.54999999999999997e32 < phi2

   1. Initial program 63.4%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 68.9%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \color{blue}{R}\right) \]

   5. Simplified 68.9%

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

Alternative 9: 14.2% accurate, 46.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_2 \cdot R \end{array} \]

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 (* lambda2 R))

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in phi1 around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}\right), \color{blue}{R}\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\phi_2}^{2} + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}\right), R\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

   7. unswap-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]

   8. accelerator-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   10. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   12. --lowering--.f64 73.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

5. Simplified 73.0%

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

6. Taylor expanded in phi2 around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right), R\right) \]
7. Step-by-step derivation

   1. --lowering--.f64 71.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_2, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), R\right) \]

8. Simplified 71.2%

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

9. Taylor expanded in lambda2 around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\lambda_2}, R\right) \]
10. Step-by-step derivation

11. Simplified 17.9%

    \[\leadsto \color{blue}{\lambda_2} \cdot R \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024193 
(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))))))