Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.8% → 91.5%

Time: 10.6s

Alternatives: 16

Speedup: 1.3×

• 30.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.25 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.7 \cdot 10^{+93}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot -0.5\right), \left(\lambda_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \sin \left(\phi_2 \cdot -0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.25e-17)
   (* R (hypot (* (cos (* -0.5 phi1)) (- lambda1 lambda2)) phi1))
   (if (<= phi2 2.7e+93)
     (* R (hypot (* (cos (* -0.5 phi2)) (- lambda1 lambda2)) phi2))
     (*
      R
      (hypot
       (fma
        (* lambda2 (cos (* 0.5 phi1)))
        (cos (* phi2 -0.5))
        (* (* lambda2 (sin (* 0.5 phi1))) (sin (* phi2 -0.5))))
       (- phi1 phi2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.25e-17) {
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else if (phi2 <= 2.7e+93) {
		tmp = R * hypot((cos((-0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	} else {
		tmp = R * hypot(fma((lambda2 * cos((0.5 * phi1))), cos((phi2 * -0.5)), ((lambda2 * sin((0.5 * phi1))) * sin((phi2 * -0.5)))), (phi1 - phi2));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.25e-17)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	elseif (phi2 <= 2.7e+93)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi2)) * Float64(lambda1 - lambda2)), phi2));
	else
		tmp = Float64(R * hypot(fma(Float64(lambda2 * cos(Float64(0.5 * phi1))), cos(Float64(phi2 * -0.5)), Float64(Float64(lambda2 * sin(Float64(0.5 * phi1))) * sin(Float64(phi2 * -0.5)))), Float64(phi1 - phi2)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.25e-17], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.7e+93], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda2 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] + N[(N[(lambda2 * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.24999999999999989e-17

   1. Initial program 55.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if 2.24999999999999989e-17 < phi2 < 2.6999999999999999e93

   1. Initial program 72.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if 2.6999999999999999e93 < phi2

   1. Initial program 39.7%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 86.5

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

   5. Applied rewrites 86.5%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 92.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \lambda_2, \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right) \cdot \lambda_2\right), \color{blue}{\phi_1} - \phi_2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.9%

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

Alternative 2: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq 2.25 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.7 \cdot 10^{+93}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(t\_0 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(t\_0, \cos \left(0.5 \cdot \phi_1\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 phi2))))
   (if (<= phi2 2.25e-17)
     (* R (hypot (* (cos (* -0.5 phi1)) (- lambda1 lambda2)) phi1))
     (if (<= phi2 2.7e+93)
       (* R (hypot (* t_0 (- lambda1 lambda2)) phi2))
       (*
        R
        (hypot
         (*
          (fma
           t_0
           (cos (* 0.5 phi1))
           (* (sin (* -0.5 phi2)) (sin (* 0.5 phi1))))
          lambda2)
         (- phi1 phi2)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((-0.5 * phi2));
	double tmp;
	if (phi2 <= 2.25e-17) {
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else if (phi2 <= 2.7e+93) {
		tmp = R * hypot((t_0 * (lambda1 - lambda2)), phi2);
	} else {
		tmp = R * hypot((fma(t_0, cos((0.5 * phi1)), (sin((-0.5 * phi2)) * sin((0.5 * phi1)))) * lambda2), (phi1 - phi2));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(-0.5 * phi2))
	tmp = 0.0
	if (phi2 <= 2.25e-17)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	elseif (phi2 <= 2.7e+93)
		tmp = Float64(R * hypot(Float64(t_0 * Float64(lambda1 - lambda2)), phi2));
	else
		tmp = Float64(R * hypot(Float64(fma(t_0, cos(Float64(0.5 * phi1)), Float64(sin(Float64(-0.5 * phi2)) * sin(Float64(0.5 * phi1)))) * lambda2), Float64(phi1 - phi2)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 2.25e-17], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.7e+93], N[(R * N[Sqrt[N[(t$95$0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(t$95$0 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.24999999999999989e-17

   1. Initial program 55.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if 2.24999999999999989e-17 < phi2 < 2.6999999999999999e93

   1. Initial program 72.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if 2.6999999999999999e93 < phi2

   1. Initial program 39.7%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 86.5

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

   5. Applied rewrites 86.5%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.8%

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

Alternative 3: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{-16}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \lambda_2, \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -5.8 \cdot 10^{-54}:\\ \;\;\;\;\left(R - R \cdot \frac{\phi_1}{\phi_2}\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 1.12 \cdot 10^{-241}:\\ \;\;\;\;R \cdot \left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \mathsf{fma}\left(-1, \lambda_1, \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.8e-16)
   (* R (hypot (* (cos (* -0.5 phi1)) lambda2) phi1))
   (if (<= phi1 -5.8e-54)
     (* (- R (* R (/ phi1 phi2))) phi2)
     (if (<= phi1 1.12e-241)
       (* R (* (cos (* -0.5 (+ phi2 phi1))) (fma -1.0 lambda1 lambda2)))
       (* R (hypot (* (cos (* -0.5 phi2)) lambda2) phi2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.8e-16) {
		tmp = R * hypot((cos((-0.5 * phi1)) * lambda2), phi1);
	} else if (phi1 <= -5.8e-54) {
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	} else if (phi1 <= 1.12e-241) {
		tmp = R * (cos((-0.5 * (phi2 + phi1))) * fma(-1.0, lambda1, lambda2));
	} else {
		tmp = R * hypot((cos((-0.5 * phi2)) * lambda2), phi2);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.8e-16)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi1)) * lambda2), phi1));
	elseif (phi1 <= -5.8e-54)
		tmp = Float64(Float64(R - Float64(R * Float64(phi1 / phi2))) * phi2);
	elseif (phi1 <= 1.12e-241)
		tmp = Float64(R * Float64(cos(Float64(-0.5 * Float64(phi2 + phi1))) * fma(-1.0, lambda1, lambda2)));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi2)) * lambda2), phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.8e-16], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -5.8e-54], N[(N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision], If[LessEqual[phi1, 1.12e-241], N[(R * N[(N[Cos[N[(-0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * lambda1 + lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -1.79999999999999991e-16

   1. Initial program 52.6%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 84.3

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 72.8%

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

   if -1.79999999999999991e-16 < phi1 < -5.80000000000000029e-54

   1. Initial program 59.4%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -5.80000000000000029e-54 < phi1 < 1.11999999999999993e-241

   1. Initial program 57.8%

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

   3. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 35.2%

      \[\leadsto R \cdot \left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \lambda_1, \lambda_2\right)}\right) \]

   if 1.11999999999999993e-241 < phi1

   1. Initial program 52.7%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 46.0%

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

Alternative 4: 90.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{-27}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.3e-27)
   (* R (hypot (* (cos (* -0.5 phi1)) (- lambda1 lambda2)) phi1))
   (* R (hypot (* (cos (* -0.5 phi2)) (- lambda1 lambda2)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.3e-27) {
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * hypot((cos((-0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.3e-27) {
		tmp = R * Math.hypot((Math.cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * Math.hypot((Math.cos((-0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.3e-27:
		tmp = R * math.hypot((math.cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1)
	else:
		tmp = R * math.hypot((math.cos((-0.5 * phi2)) * (lambda1 - lambda2)), phi2)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.3e-27)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi2)) * Float64(lambda1 - lambda2)), phi2));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.3e-27)
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	else
		tmp = R * hypot((cos((-0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.3e-27], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.30000000000000009e-27

   1. Initial program 54.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -1.30000000000000009e-27 < phi1

   1. Initial program 54.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 78.0

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

   5. Applied rewrites 78.0%

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

Alternative 5: 84.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.6 \cdot 10^{+105}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 4.6e+105)
   (* R (hypot (* (cos (* -0.5 phi1)) (- lambda1 lambda2)) phi1))
   (* 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 <= 4.6e+105) {
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} 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 <= 4.6e+105) {
		tmp = R * Math.hypot((Math.cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} 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 <= 4.6e+105:
		tmp = R * math.hypot((math.cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1)
	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 <= 4.6e+105)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	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 <= 4.6e+105)
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	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, 4.6e+105], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.5999999999999996e105

   1. Initial program 57.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if 4.5999999999999996e105 < phi2

   1. Initial program 38.7%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 70.5%

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

Alternative 6: 64.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.82 \cdot 10^{-153}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \lambda_2, \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{+92}:\\ \;\;\;\;R \cdot \left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \mathsf{fma}\left(-1, \lambda_1, \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 1.82e-153)
   (* R (hypot (* (cos (* -0.5 phi1)) lambda2) phi1))
   (if (<= phi2 3.8e+92)
     (* R (* (cos (* -0.5 (+ phi2 phi1))) (fma -1.0 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 <= 1.82e-153) {
		tmp = R * hypot((cos((-0.5 * phi1)) * lambda2), phi1);
	} else if (phi2 <= 3.8e+92) {
		tmp = R * (cos((-0.5 * (phi2 + phi1))) * fma(-1.0, 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 <= 1.82e-153)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi1)) * lambda2), phi1));
	elseif (phi2 <= 3.8e+92)
		tmp = Float64(R * Float64(cos(Float64(-0.5 * Float64(phi2 + phi1))) * fma(-1.0, lambda1, lambda2)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.82e-153], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.8e+92], N[(R * N[(N[Cos[N[(-0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * lambda1 + lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.81999999999999993e-153

   1. Initial program 56.3%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 57.8%

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

   if 1.81999999999999993e-153 < phi2 < 3.8e92

   1. Initial program 59.6%

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

   3. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 25.2%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 33.2%

      \[\leadsto R \cdot \left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \lambda_1, \lambda_2\right)}\right) \]

   if 3.8e92 < phi2

   1. Initial program 39.7%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 67.3%

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

Alternative 7: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.8 \cdot 10^{-54}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 1.2 \cdot 10^{-235}:\\ \;\;\;\;R \cdot \left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \mathsf{fma}\left(-1, \lambda_1, \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - \frac{\phi_2 \cdot R}{\phi_1}\right)\\ \end{array} \end{array} \]

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 (<= phi1 -5.8e-54)
   (* R (- phi2 phi1))
   (if (<= phi1 1.2e-235)
     (* R (* (cos (* -0.5 (+ phi2 phi1))) (fma -1.0 lambda1 lambda2)))
     (* (- phi1) (- R (/ (* phi2 R) 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 (phi1 <= -5.8e-54) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 1.2e-235) {
		tmp = R * (cos((-0.5 * (phi2 + phi1))) * fma(-1.0, lambda1, lambda2));
	} else {
		tmp = -phi1 * (R - ((phi2 * R) / 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 (phi1 <= -5.8e-54)
		tmp = Float64(R * Float64(phi2 - phi1));
	elseif (phi1 <= 1.2e-235)
		tmp = Float64(R * Float64(cos(Float64(-0.5 * Float64(phi2 + phi1))) * fma(-1.0, lambda1, lambda2)));
	else
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(Float64(phi2 * R) / 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[phi1, -5.8e-54], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.2e-235], N[(R * N[(N[Cos[N[(-0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * lambda1 + lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-phi1) * N[(R - N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.80000000000000029e-54

   1. Initial program 53.2%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 69.2%

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

   if -5.80000000000000029e-54 < phi1 < 1.20000000000000005e-235

   1. Initial program 57.1%

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

   3. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 34.7%

      \[\leadsto R \cdot \left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \lambda_1, \lambda_2\right)}\right) \]

   if 1.20000000000000005e-235 < phi1

   1. Initial program 53.1%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 76.7

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

   5. Applied rewrites 76.7%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.5%

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

   8. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 11.5

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

   10. Applied rewrites 11.5%

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

Alternative 8: 51.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -4.2 \cdot 10^{+103}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot \left(t\_0 \cdot R\right)\\ \mathbf{elif}\;\lambda_1 \leq 9 \cdot 10^{+102}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - \frac{\phi_2 \cdot R}{\phi_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot t\_0\\ \end{array} \end{array} \]

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
 (let* ((t_0 (cos (* -0.5 (+ phi2 phi1)))))
   (if (<= lambda1 -4.2e+103)
     (* (- lambda1) (* t_0 R))
     (if (<= lambda1 9e+102)
       (* (- phi1) (- R (/ (* phi2 R) phi1)))
       (* (* R lambda2) t_0)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((-0.5 * (phi2 + phi1)));
	double tmp;
	if (lambda1 <= -4.2e+103) {
		tmp = -lambda1 * (t_0 * R);
	} else if (lambda1 <= 9e+102) {
		tmp = -phi1 * (R - ((phi2 * R) / phi1));
	} else {
		tmp = (R * lambda2) * t_0;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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
    real(8) :: tmp
    t_0 = cos(((-0.5d0) * (phi2 + phi1)))
    if (lambda1 <= (-4.2d+103)) then
        tmp = -lambda1 * (t_0 * r)
    else if (lambda1 <= 9d+102) then
        tmp = -phi1 * (r - ((phi2 * r) / phi1))
    else
        tmp = (r * lambda2) * t_0
    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 t_0 = Math.cos((-0.5 * (phi2 + phi1)));
	double tmp;
	if (lambda1 <= -4.2e+103) {
		tmp = -lambda1 * (t_0 * R);
	} else if (lambda1 <= 9e+102) {
		tmp = -phi1 * (R - ((phi2 * R) / phi1));
	} else {
		tmp = (R * lambda2) * t_0;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((-0.5 * (phi2 + phi1)))
	tmp = 0
	if lambda1 <= -4.2e+103:
		tmp = -lambda1 * (t_0 * R)
	elif lambda1 <= 9e+102:
		tmp = -phi1 * (R - ((phi2 * R) / phi1))
	else:
		tmp = (R * lambda2) * t_0
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(-0.5 * Float64(phi2 + phi1)))
	tmp = 0.0
	if (lambda1 <= -4.2e+103)
		tmp = Float64(Float64(-lambda1) * Float64(t_0 * R));
	elseif (lambda1 <= 9e+102)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(Float64(phi2 * R) / phi1)));
	else
		tmp = Float64(Float64(R * lambda2) * t_0);
	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)
	t_0 = cos((-0.5 * (phi2 + phi1)));
	tmp = 0.0;
	if (lambda1 <= -4.2e+103)
		tmp = -lambda1 * (t_0 * R);
	elseif (lambda1 <= 9e+102)
		tmp = -phi1 * (R - ((phi2 * R) / phi1));
	else
		tmp = (R * lambda2) * t_0;
	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_] := Block[{t$95$0 = N[Cos[N[(-0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -4.2e+103], N[((-lambda1) * N[(t$95$0 * R), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 9e+102], N[((-phi1) * N[(R - N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * lambda2), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -4.2000000000000003e103

   1. Initial program 48.9%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower-cos.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   8. Applied rewrites 58.2%

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

   9. Taylor expanded in lambda1 around inf

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

   11. Applied rewrites 55.9%

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

   if -4.2000000000000003e103 < lambda1 < 9.00000000000000042e102

   1. Initial program 58.0%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 92.9

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

   5. Applied rewrites 92.9%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 32.8

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

   10. Applied rewrites 32.8%

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

   if 9.00000000000000042e102 < lambda1

   1. Initial program 48.8%

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

   3. Taylor expanded in lambda2 around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lower-cos.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 15.4

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

   5. Applied rewrites 15.4%

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

Alternative 9: 54.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.3 \cdot 10^{+174}:\\ \;\;\;\;\left(R \cdot \lambda_1\right) \cdot \cos \left(\mathsf{fma}\left(-0.5, \phi_2 + \phi_1, \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 7.5 \cdot 10^{+101}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \end{array} \end{array} \]

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 (<= lambda1 -3.3e+174)
   (* (* R lambda1) (cos (fma -0.5 (+ phi2 phi1) (PI))))
   (if (<= lambda1 7.5e+101)
     (* R (- phi2 phi1))
     (* (* R lambda2) (cos (* -0.5 (+ phi2 phi1)))))))

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.3000000000000001e174

   1. Initial program 55.3%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. cos-neg-rev N/A

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

      9. cos-+PI-rev N/A

         \[\leadsto \left(R \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \mathsf{PI}\left(\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \left(R \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) + \mathsf{PI}\left(\right)\right)} \]

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

          \[\leadsto \left(R \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\phi_1 + \phi_2\right)} + \mathsf{PI}\left(\right)\right) \]

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-PI.f64 47.5

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

   5. Applied rewrites 47.5%

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

   if -3.3000000000000001e174 < lambda1 < 7.4999999999999995e101

   1. Initial program 55.8%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 28.0

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

   5. Applied rewrites 28.0%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 30.3%

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

   if 7.4999999999999995e101 < lambda1

   1. Initial program 48.8%

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

   3. Taylor expanded in lambda2 around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lower-cos.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 15.4

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

   5. Applied rewrites 15.4%

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

Alternative 10: 56.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.5 \cdot 10^{+228}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - \frac{\phi_2 \cdot R}{\phi_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \end{array} \end{array} \]

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 (<= lambda2 3.5e+228)
   (* (- phi1) (- R (/ (* phi2 R) phi1)))
   (* (* R lambda2) (cos (* -0.5 (+ 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 (lambda2 <= 3.5e+228) {
		tmp = -phi1 * (R - ((phi2 * R) / phi1));
	} else {
		tmp = (R * lambda2) * cos((-0.5 * (phi2 + phi1)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 <= 3.5d+228) then
        tmp = -phi1 * (r - ((phi2 * r) / phi1))
    else
        tmp = (r * lambda2) * cos(((-0.5d0) * (phi2 + phi1)))
    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 <= 3.5e+228) {
		tmp = -phi1 * (R - ((phi2 * R) / phi1));
	} else {
		tmp = (R * lambda2) * Math.cos((-0.5 * (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 lambda2 <= 3.5e+228:
		tmp = -phi1 * (R - ((phi2 * R) / phi1))
	else:
		tmp = (R * lambda2) * math.cos((-0.5 * (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 (lambda2 <= 3.5e+228)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(Float64(phi2 * R) / phi1)));
	else
		tmp = Float64(Float64(R * lambda2) * cos(Float64(-0.5 * 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 (lambda2 <= 3.5e+228)
		tmp = -phi1 * (R - ((phi2 * R) / phi1));
	else
		tmp = (R * lambda2) * cos((-0.5 * (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[lambda2, 3.5e+228], N[((-phi1) * N[(R - N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * lambda2), $MachinePrecision] * N[Cos[N[(-0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.5000000000000002e228

   1. Initial program 56.5%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 77.6

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

   5. Applied rewrites 77.6%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.7%

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

   8. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 28.8

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

   10. Applied rewrites 28.8%

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

   if 3.5000000000000002e228 < lambda2

   1. Initial program 26.1%

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

   3. Taylor expanded in lambda2 around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lower-cos.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 67.9

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

   5. Applied rewrites 67.9%

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

Alternative 11: 58.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+198}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - \frac{\phi_2 \cdot R}{\phi_1}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 (<= (- lambda1 lambda2) -5e+198)
   (* (- phi1) (- R (/ (* phi2 R) phi1)))
   (* 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 ((lambda1 - lambda2) <= -5e+198) {
		tmp = -phi1 * (R - ((phi2 * R) / phi1));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 ((lambda1 - lambda2) <= (-5d+198)) then
        tmp = -phi1 * (r - ((phi2 * r) / phi1))
    else
        tmp = r * (phi2 - phi1)
    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 ((lambda1 - lambda2) <= -5e+198) {
		tmp = -phi1 * (R - ((phi2 * R) / phi1));
	} 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 (lambda1 - lambda2) <= -5e+198:
		tmp = -phi1 * (R - ((phi2 * R) / phi1))
	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 (Float64(lambda1 - lambda2) <= -5e+198)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(Float64(phi2 * R) / phi1)));
	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 ((lambda1 - lambda2) <= -5e+198)
		tmp = -phi1 * (R - ((phi2 * R) / phi1));
	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[N[(lambda1 - lambda2), $MachinePrecision], -5e+198], N[((-phi1) * N[(R - N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -5.00000000000000049e198

   1. Initial program 41.2%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 58.7

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

   5. Applied rewrites 58.7%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(-0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.9%

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

   8. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 18.0

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

   10. Applied rewrites 18.0%

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

   if -5.00000000000000049e198 < (-.f64 lambda1 lambda2)

   1. Initial program 57.6%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 27.7

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

   5. Applied rewrites 27.7%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 29.6%

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

Alternative 12: 58.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{+54}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - R \cdot \frac{\phi_2}{\phi_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R - R \cdot \frac{\phi_1}{\phi_2}\right) \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -6.5e+54)
   (* (- phi1) (- R (* R (/ phi2 phi1))))
   (* (- R (* R (/ phi1 phi2))) phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.5e+54) {
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	} else {
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 <= (-6.5d+54)) then
        tmp = -phi1 * (r - (r * (phi2 / phi1)))
    else
        tmp = (r - (r * (phi1 / phi2))) * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.5e+54) {
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	} else {
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -6.5e+54:
		tmp = -phi1 * (R - (R * (phi2 / phi1)))
	else:
		tmp = (R - (R * (phi1 / phi2))) * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -6.5e+54)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(R * Float64(phi2 / phi1))));
	else
		tmp = Float64(Float64(R - Float64(R * Float64(phi1 / phi2))) * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -6.5e+54)
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	else
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -6.5e+54], N[((-phi1) * N[(R - N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.5e54

   1. Initial program 56.5%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if -6.5e54 < phi1

   1. Initial program 53.9%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 14.1

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

   5. Applied rewrites 14.1%

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

Alternative 13: 58.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{+54}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R - R \cdot \frac{\phi_1}{\phi_2}\right) \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -6.5e+54)
   (* R (- phi2 phi1))
   (* (- R (* R (/ phi1 phi2))) phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.5e+54) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 <= (-6.5d+54)) then
        tmp = r * (phi2 - phi1)
    else
        tmp = (r - (r * (phi1 / phi2))) * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.5e+54) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -6.5e+54:
		tmp = R * (phi2 - phi1)
	else:
		tmp = (R - (R * (phi1 / phi2))) * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -6.5e+54)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(Float64(R - Float64(R * Float64(phi1 / phi2))) * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -6.5e+54)
		tmp = R * (phi2 - phi1);
	else
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -6.5e+54], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.5e54

   1. Initial program 56.5%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 84.6%

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

   if -6.5e54 < phi1

   1. Initial program 53.9%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 14.1

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

   5. Applied rewrites 14.1%

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

Alternative 14: 52.0% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.6 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -9.6e-28) (* R (- phi1)) (* R phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.6e-28) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 <= (-9.6d-28)) then
        tmp = r * -phi1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.6e-28) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -9.6e-28:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -9.6e-28)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -9.6e-28)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -9.6e-28], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.6000000000000008e-28

   1. Initial program 54.0%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if -9.6000000000000008e-28 < phi1

   1. Initial program 54.5%

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

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 12.3

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

   5. Applied rewrites 12.3%

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

Alternative 15: 57.7% accurate, 31.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * (phi2 - phi1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in phi1 around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 25.1

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

5. Applied rewrites 25.1%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 26.6%

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

Alternative 16: 30.5% accurate, 46.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * phi2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 11.7

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

5. Applied rewrites 11.7%

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

Reproduce

herbie shell --seed 2024359 
(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))))))