Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.2% → 95.8%

Time: 14.7s

Alternatives: 13

Speedup: N/A×

• 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: 60.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 95.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 56.9%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

4. Applied rewrites 94.9%

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

Alternative 2: 92.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1000:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(\frac{\phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1000.0)
   (* (hypot (- phi1 phi2) (* (cos (/ phi1 2.0)) (- lambda1 lambda2))) R)
   (*
    (hypot
     (* phi2 (- (/ phi1 phi2) 1.0))
     (* (cos (/ (+ phi2 phi1) 2.0)) (- lambda1 lambda2)))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1000.0) {
		tmp = hypot((phi1 - phi2), (cos((phi1 / 2.0)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1000.0) {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((phi1 / 2.0)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi2 * ((phi1 / phi2) - 1.0)), (Math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1000.0:
		tmp = math.hypot((phi1 - phi2), (math.cos((phi1 / 2.0)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi2 * ((phi1 / phi2) - 1.0)), (math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1000.0)
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(phi1 / 2.0)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi2 * Float64(Float64(phi1 / phi2) - 1.0)), Float64(cos(Float64(Float64(phi2 + phi1) / 2.0)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1000.0)
		tmp = hypot((phi1 - phi2), (cos((phi1 / 2.0)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1000.0], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi2 * N[(N[(phi1 / phi2), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1e3

   1. Initial program 46.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in phi1 around inf

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

   7. Applied rewrites 88.5%

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

   if -1e3 < phi1

   1. Initial program 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 93.0

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

   7. Applied rewrites 93.0%

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.2 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, 1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -9.2e+177)
   (* (hypot (- phi1 phi2) (* 1.0 (- lambda1 lambda2))) R)
   (*
    (hypot
     (* phi2 (- (/ phi1 phi2) 1.0))
     (* (cos (/ (+ phi2 phi1) 2.0)) (- lambda1 lambda2)))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.2e+177) {
		tmp = hypot((phi1 - phi2), (1.0 * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.2e+177) {
		tmp = Math.hypot((phi1 - phi2), (1.0 * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi2 * ((phi1 / phi2) - 1.0)), (Math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -9.2e+177:
		tmp = math.hypot((phi1 - phi2), (1.0 * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi2 * ((phi1 / phi2) - 1.0)), (math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -9.2e+177)
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(1.0 * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi2 * Float64(Float64(phi1 / phi2) - 1.0)), Float64(cos(Float64(Float64(phi2 + phi1) / 2.0)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -9.2e+177)
		tmp = hypot((phi1 - phi2), (1.0 * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -9.2e+177], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi2 * N[(N[(phi1 / phi2), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.1999999999999996e177

   1. Initial program 43.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 90.2%

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

   5. Taylor expanded in phi1 around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, 1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 86.6%

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

   if -9.1999999999999996e177 < phi1

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 91.3

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

   7. Applied rewrites 91.3%

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.55 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.55e+228)
   (* (fma (/ (* phi1 R) phi2) -1.0 R) phi2)
   (*
    (hypot
     (* phi2 (- (/ phi1 phi2) 1.0))
     (* (cos (/ (+ phi2 phi1) 2.0)) (- lambda1 lambda2)))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.55e+228) {
		tmp = fma(((phi1 * R) / phi2), -1.0, R) * phi2;
	} else {
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.55e+228)
		tmp = Float64(fma(Float64(Float64(phi1 * R) / phi2), -1.0, R) * phi2);
	else
		tmp = Float64(hypot(Float64(phi2 * Float64(Float64(phi1 / phi2) - 1.0)), Float64(cos(Float64(Float64(phi2 + phi1) / 2.0)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.55e+228], N[(N[(N[(N[(phi1 * R), $MachinePrecision] / phi2), $MachinePrecision] * -1.0 + R), $MachinePrecision] * phi2), $MachinePrecision], N[(N[Sqrt[N[(phi2 * N[(N[(phi1 / phi2), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.5499999999999999e228

   1. Initial program 44.0%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if -1.5499999999999999e228 < phi1

   1. Initial program 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 40.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_1 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ t_2 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;R \cdot \sqrt{{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + t\_2}\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, R, \frac{\left(R \cdot \lambda_2\right) \cdot t\_1}{-\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \sqrt{t\_0 \cdot t\_0 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
        (t_1 (cos (* 0.5 (+ phi2 phi1))))
        (t_2 (* (- phi1 phi2) (- phi1 phi2))))
   (if (<= phi2 -4.2e-87)
     (* (fma (/ (* phi1 R) phi2) -1.0 R) phi2)
     (if (<= phi2 -5e-213)
       (*
        R
        (sqrt (+ (pow (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) 2.0) t_2)))
       (if (<= phi2 8.2e-28)
         (* (fma t_1 R (/ (* (* R lambda2) t_1) (- lambda1))) (- lambda1))
         (if (<= phi2 1.8e+135)
           (* R (sqrt (+ (* t_0 t_0) t_2)))
           (* (- phi1) (fma (/ (* phi2 R) phi1) -1.0 R))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double t_1 = cos((0.5 * (phi2 + phi1)));
	double t_2 = (phi1 - phi2) * (phi1 - phi2);
	double tmp;
	if (phi2 <= -4.2e-87) {
		tmp = fma(((phi1 * R) / phi2), -1.0, R) * phi2;
	} else if (phi2 <= -5e-213) {
		tmp = R * sqrt((pow((cos((0.5 * phi1)) * (lambda1 - lambda2)), 2.0) + t_2));
	} else if (phi2 <= 8.2e-28) {
		tmp = fma(t_1, R, (((R * lambda2) * t_1) / -lambda1)) * -lambda1;
	} else if (phi2 <= 1.8e+135) {
		tmp = R * sqrt(((t_0 * t_0) + t_2));
	} else {
		tmp = -phi1 * fma(((phi2 * R) / phi1), -1.0, R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	t_1 = cos(Float64(0.5 * Float64(phi2 + phi1)))
	t_2 = Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2))
	tmp = 0.0
	if (phi2 <= -4.2e-87)
		tmp = Float64(fma(Float64(Float64(phi1 * R) / phi2), -1.0, R) * phi2);
	elseif (phi2 <= -5e-213)
		tmp = Float64(R * sqrt(Float64((Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)) ^ 2.0) + t_2)));
	elseif (phi2 <= 8.2e-28)
		tmp = Float64(fma(t_1, R, Float64(Float64(Float64(R * lambda2) * t_1) / Float64(-lambda1))) * Float64(-lambda1));
	elseif (phi2 <= 1.8e+135)
		tmp = Float64(R * sqrt(Float64(Float64(t_0 * t_0) + t_2)));
	else
		tmp = Float64(Float64(-phi1) * fma(Float64(Float64(phi2 * R) / phi1), -1.0, R));
	end
	return tmp
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]}, Block[{t$95$1 = N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.2e-87], N[(N[(N[(N[(phi1 * R), $MachinePrecision] / phi2), $MachinePrecision] * -1.0 + R), $MachinePrecision] * phi2), $MachinePrecision], If[LessEqual[phi2, -5e-213], N[(R * N[Sqrt[N[(N[Power[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 8.2e-28], N[(N[(t$95$1 * R + N[(N[(N[(R * lambda2), $MachinePrecision] * t$95$1), $MachinePrecision] / (-lambda1)), $MachinePrecision]), $MachinePrecision] * (-lambda1)), $MachinePrecision], If[LessEqual[phi2, 1.8e+135], N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-phi1) * N[(N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision] * -1.0 + R), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if phi2 < -4.20000000000000014e-87

   1. Initial program 49.5%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 22.5

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

   5. Applied rewrites 22.5%

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

   if -4.20000000000000014e-87 < phi2 < -4.99999999999999977e-213

   1. Initial program 71.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if -4.99999999999999977e-213 < phi2 < 8.2000000000000005e-28

   1. Initial program 59.1%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   if 8.2000000000000005e-28 < phi2 < 1.7999999999999999e135

   1. Initial program 59.9%

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

   if 1.7999999999999999e135 < phi2

   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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;R \cdot \sqrt{{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}{-\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \sqrt{{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ t_1 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, R, \frac{\left(R \cdot \lambda_2\right) \cdot t\_1}{-\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{+135}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          R
          (sqrt
           (+
            (pow (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) 2.0)
            (* (- phi1 phi2) (- phi1 phi2))))))
        (t_1 (cos (* 0.5 (+ phi2 phi1)))))
   (if (<= phi2 -4.2e-87)
     (* (fma (/ (* phi1 R) phi2) -1.0 R) phi2)
     (if (<= phi2 -5e-213)
       t_0
       (if (<= phi2 8.2e-28)
         (* (fma t_1 R (/ (* (* R lambda2) t_1) (- lambda1))) (- lambda1))
         (if (<= phi2 1.8e+135)
           t_0
           (* (- phi1) (fma (/ (* phi2 R) phi1) -1.0 R))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * sqrt((pow((cos((0.5 * phi1)) * (lambda1 - lambda2)), 2.0) + ((phi1 - phi2) * (phi1 - phi2))));
	double t_1 = cos((0.5 * (phi2 + phi1)));
	double tmp;
	if (phi2 <= -4.2e-87) {
		tmp = fma(((phi1 * R) / phi2), -1.0, R) * phi2;
	} else if (phi2 <= -5e-213) {
		tmp = t_0;
	} else if (phi2 <= 8.2e-28) {
		tmp = fma(t_1, R, (((R * lambda2) * t_1) / -lambda1)) * -lambda1;
	} else if (phi2 <= 1.8e+135) {
		tmp = t_0;
	} else {
		tmp = -phi1 * fma(((phi2 * R) / phi1), -1.0, R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * sqrt(Float64((Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)) ^ 2.0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
	t_1 = cos(Float64(0.5 * Float64(phi2 + phi1)))
	tmp = 0.0
	if (phi2 <= -4.2e-87)
		tmp = Float64(fma(Float64(Float64(phi1 * R) / phi2), -1.0, R) * phi2);
	elseif (phi2 <= -5e-213)
		tmp = t_0;
	elseif (phi2 <= 8.2e-28)
		tmp = Float64(fma(t_1, R, Float64(Float64(Float64(R * lambda2) * t_1) / Float64(-lambda1))) * Float64(-lambda1));
	elseif (phi2 <= 1.8e+135)
		tmp = t_0;
	else
		tmp = Float64(Float64(-phi1) * fma(Float64(Float64(phi2 * R) / phi1), -1.0, R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[Sqrt[N[(N[Power[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -4.2e-87], N[(N[(N[(N[(phi1 * R), $MachinePrecision] / phi2), $MachinePrecision] * -1.0 + R), $MachinePrecision] * phi2), $MachinePrecision], If[LessEqual[phi2, -5e-213], t$95$0, If[LessEqual[phi2, 8.2e-28], N[(N[(t$95$1 * R + N[(N[(N[(R * lambda2), $MachinePrecision] * t$95$1), $MachinePrecision] / (-lambda1)), $MachinePrecision]), $MachinePrecision] * (-lambda1)), $MachinePrecision], If[LessEqual[phi2, 1.8e+135], t$95$0, N[((-phi1) * N[(N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision] * -1.0 + R), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -4.20000000000000014e-87

   1. Initial program 49.5%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 22.5

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

   5. Applied rewrites 22.5%

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

   if -4.20000000000000014e-87 < phi2 < -4.99999999999999977e-213 or 8.2000000000000005e-28 < phi2 < 1.7999999999999999e135

   1. Initial program 64.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if -4.99999999999999977e-213 < phi2 < 8.2000000000000005e-28

   1. Initial program 59.1%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   if 1.7999999999999999e135 < phi2

   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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;R \cdot \sqrt{{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}{-\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \sqrt{{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{if}\;\phi_2 \leq -2.2 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq 2.1 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, R, \frac{\left(R \cdot \lambda_2\right) \cdot t\_0}{-\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\phi_2, \phi_2, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 (+ phi2 phi1)))))
   (if (<= phi2 -2.2e-211)
     (* (fma (/ (* phi1 R) phi2) -1.0 R) phi2)
     (if (<= phi2 2.1e-20)
       (* (fma t_0 R (/ (* (* R lambda2) t_0) (- lambda1))) (- lambda1))
       (if (<= phi2 1.8e+135)
         (*
          R
          (sqrt
           (fma
            phi2
            phi2
            (pow (* (cos (* 0.5 phi2)) (- lambda1 lambda2)) 2.0))))
         (* (- phi1) (fma (/ (* phi2 R) phi1) -1.0 R)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * (phi2 + phi1)));
	double tmp;
	if (phi2 <= -2.2e-211) {
		tmp = fma(((phi1 * R) / phi2), -1.0, R) * phi2;
	} else if (phi2 <= 2.1e-20) {
		tmp = fma(t_0, R, (((R * lambda2) * t_0) / -lambda1)) * -lambda1;
	} else if (phi2 <= 1.8e+135) {
		tmp = R * sqrt(fma(phi2, phi2, pow((cos((0.5 * phi2)) * (lambda1 - lambda2)), 2.0)));
	} else {
		tmp = -phi1 * fma(((phi2 * R) / phi1), -1.0, R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * Float64(phi2 + phi1)))
	tmp = 0.0
	if (phi2 <= -2.2e-211)
		tmp = Float64(fma(Float64(Float64(phi1 * R) / phi2), -1.0, R) * phi2);
	elseif (phi2 <= 2.1e-20)
		tmp = Float64(fma(t_0, R, Float64(Float64(Float64(R * lambda2) * t_0) / Float64(-lambda1))) * Float64(-lambda1));
	elseif (phi2 <= 1.8e+135)
		tmp = Float64(R * sqrt(fma(phi2, phi2, (Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2)) ^ 2.0))));
	else
		tmp = Float64(Float64(-phi1) * fma(Float64(Float64(phi2 * R) / phi1), -1.0, R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -2.2e-211], N[(N[(N[(N[(phi1 * R), $MachinePrecision] / phi2), $MachinePrecision] * -1.0 + R), $MachinePrecision] * phi2), $MachinePrecision], If[LessEqual[phi2, 2.1e-20], N[(N[(t$95$0 * R + N[(N[(N[(R * lambda2), $MachinePrecision] * t$95$0), $MachinePrecision] / (-lambda1)), $MachinePrecision]), $MachinePrecision] * (-lambda1)), $MachinePrecision], If[LessEqual[phi2, 1.8e+135], N[(R * N[Sqrt[N[(phi2 * phi2 + N[Power[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-phi1) * N[(N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision] * -1.0 + R), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -2.19999999999999998e-211

   1. Initial program 54.8%

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

   3. Taylor expanded in phi2 around 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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 22.7

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

   5. Applied rewrites 22.7%

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

   if -2.19999999999999998e-211 < phi2 < 2.0999999999999999e-20

   1. Initial program 59.5%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 37.2%

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

   if 2.0999999999999999e-20 < phi2 < 1.7999999999999999e135

   1. Initial program 60.3%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 55.0

         \[\leadsto R \cdot \sqrt{\mathsf{fma}\left(\phi_2, \phi_2, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)} \]

   5. Applied rewrites 55.0%

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

   if 1.7999999999999999e135 < phi2

   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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.2 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq 2.1 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}{-\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\phi_2, \phi_2, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.2 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq 24500000:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -2.2e-211)
   (* (fma (/ (* phi1 R) phi2) -1.0 R) phi2)
   (if (<= phi2 24500000.0)
     (*
      (+
       (* (cos (* 0.5 (+ phi2 phi1))) (- R))
       (/
        (*
         (* R lambda2)
         (- (cos (* 0.5 phi1)) (* 0.5 (* phi2 (sin (* 0.5 phi1))))))
        lambda1))
      lambda1)
     (* (- phi1) (fma (/ (* phi2 R) phi1) -1.0 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -2.2e-211) {
		tmp = fma(((phi1 * R) / phi2), -1.0, R) * phi2;
	} else if (phi2 <= 24500000.0) {
		tmp = ((cos((0.5 * (phi2 + phi1))) * -R) + (((R * lambda2) * (cos((0.5 * phi1)) - (0.5 * (phi2 * sin((0.5 * phi1)))))) / lambda1)) * lambda1;
	} else {
		tmp = -phi1 * fma(((phi2 * R) / phi1), -1.0, R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -2.2e-211)
		tmp = Float64(fma(Float64(Float64(phi1 * R) / phi2), -1.0, R) * phi2);
	elseif (phi2 <= 24500000.0)
		tmp = Float64(Float64(Float64(cos(Float64(0.5 * Float64(phi2 + phi1))) * Float64(-R)) + Float64(Float64(Float64(R * lambda2) * Float64(cos(Float64(0.5 * phi1)) - Float64(0.5 * Float64(phi2 * sin(Float64(0.5 * phi1)))))) / lambda1)) * lambda1);
	else
		tmp = Float64(Float64(-phi1) * fma(Float64(Float64(phi2 * R) / phi1), -1.0, R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -2.2e-211], N[(N[(N[(N[(phi1 * R), $MachinePrecision] / phi2), $MachinePrecision] * -1.0 + R), $MachinePrecision] * phi2), $MachinePrecision], If[LessEqual[phi2, 24500000.0], N[(N[(N[(N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision] + N[(N[(N[(R * lambda2), $MachinePrecision] * N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] - N[(0.5 * N[(phi2 * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision], N[((-phi1) * N[(N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision] * -1.0 + R), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.19999999999999998e-211

   1. Initial program 54.8%

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

   3. Taylor expanded in phi2 around 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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 22.7

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

   5. Applied rewrites 22.7%

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

   if -2.19999999999999998e-211 < phi2 < 2.45e7

   1. Initial program 61.0%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 36.3%

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

   6. Taylor expanded in phi2 around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 36.3

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

   8. Applied rewrites 36.3%

      \[\leadsto -\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 36.3

         \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   10. Applied rewrites 36.3%

       \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   if 2.45e7 < phi2

   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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.2 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq 24500000:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.2 \cdot 10^{-211} \lor \neg \left(\phi_2 \leq 24500000\right):\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -2.2e-211) (not (<= phi2 24500000.0)))
   (* (- phi1) (fma (/ (* phi2 R) phi1) -1.0 R))
   (*
    (+
     (* (cos (* 0.5 (+ phi2 phi1))) (- R))
     (/
      (*
       (* R lambda2)
       (- (cos (* 0.5 phi1)) (* 0.5 (* phi2 (sin (* 0.5 phi1))))))
      lambda1))
    lambda1)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -2.2e-211) || !(phi2 <= 24500000.0)) {
		tmp = -phi1 * fma(((phi2 * R) / phi1), -1.0, R);
	} else {
		tmp = ((cos((0.5 * (phi2 + phi1))) * -R) + (((R * lambda2) * (cos((0.5 * phi1)) - (0.5 * (phi2 * sin((0.5 * phi1)))))) / lambda1)) * lambda1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -2.2e-211) || !(phi2 <= 24500000.0))
		tmp = Float64(Float64(-phi1) * fma(Float64(Float64(phi2 * R) / phi1), -1.0, R));
	else
		tmp = Float64(Float64(Float64(cos(Float64(0.5 * Float64(phi2 + phi1))) * Float64(-R)) + Float64(Float64(Float64(R * lambda2) * Float64(cos(Float64(0.5 * phi1)) - Float64(0.5 * Float64(phi2 * sin(Float64(0.5 * phi1)))))) / lambda1)) * lambda1);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -2.2e-211], N[Not[LessEqual[phi2, 24500000.0]], $MachinePrecision]], N[((-phi1) * N[(N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision] * -1.0 + R), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision] + N[(N[(N[(R * lambda2), $MachinePrecision] * N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] - N[(0.5 * N[(phi2 * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.19999999999999998e-211 or 2.45e7 < phi2

   1. Initial program 54.6%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 41.5

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

   5. Applied rewrites 41.5%

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

   if -2.19999999999999998e-211 < phi2 < 2.45e7

   1. Initial program 61.0%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 36.3%

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

   6. Taylor expanded in phi2 around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 36.3

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

   8. Applied rewrites 36.3%

      \[\leadsto -\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 36.3

         \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   10. Applied rewrites 36.3%

       \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.2 \cdot 10^{-211} \lor \neg \left(\phi_2 \leq 24500000\right):\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq 5.8 \cdot 10^{+31}:\\ \;\;\;\;\left(t\_0 \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, R, \frac{\left(R \cdot \lambda_2\right) \cdot t\_0}{-\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 (+ phi2 phi1)))))
   (if (<= lambda2 5.8e+31)
     (*
      (+
       (* t_0 (- R))
       (/
        (*
         (* R lambda2)
         (- (cos (* 0.5 phi1)) (* 0.5 (* phi2 (sin (* 0.5 phi1))))))
        lambda1))
      lambda1)
     (* (fma t_0 R (/ (* (* R lambda2) t_0) (- lambda1))) (- lambda1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * (phi2 + phi1)));
	double tmp;
	if (lambda2 <= 5.8e+31) {
		tmp = ((t_0 * -R) + (((R * lambda2) * (cos((0.5 * phi1)) - (0.5 * (phi2 * sin((0.5 * phi1)))))) / lambda1)) * lambda1;
	} else {
		tmp = fma(t_0, R, (((R * lambda2) * t_0) / -lambda1)) * -lambda1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * Float64(phi2 + phi1)))
	tmp = 0.0
	if (lambda2 <= 5.8e+31)
		tmp = Float64(Float64(Float64(t_0 * Float64(-R)) + Float64(Float64(Float64(R * lambda2) * Float64(cos(Float64(0.5 * phi1)) - Float64(0.5 * Float64(phi2 * sin(Float64(0.5 * phi1)))))) / lambda1)) * lambda1);
	else
		tmp = Float64(fma(t_0, R, Float64(Float64(Float64(R * lambda2) * t_0) / Float64(-lambda1))) * Float64(-lambda1));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, 5.8e+31], N[(N[(N[(t$95$0 * (-R)), $MachinePrecision] + N[(N[(N[(R * lambda2), $MachinePrecision] * N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] - N[(0.5 * N[(phi2 * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision], N[(N[(t$95$0 * R + N[(N[(N[(R * lambda2), $MachinePrecision] * t$95$0), $MachinePrecision] / (-lambda1)), $MachinePrecision]), $MachinePrecision] * (-lambda1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.8000000000000001e31

   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 lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 22.4%

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

   6. Taylor expanded in phi2 around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 26.5

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

   8. Applied rewrites 26.5%

      \[\leadsto -\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 26.5

         \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   10. Applied rewrites 26.5%

       \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   if 5.8000000000000001e31 < lambda2

   1. Initial program 47.8%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 41.7%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.8 \cdot 10^{+31}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}{-\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ t_2 := \sin \left(0.5 \cdot \phi_2\right)\\ \mathbf{if}\;\phi_1 \leq 2.1 \cdot 10^{-199}:\\ \;\;\;\;\left(t\_1 \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, R, \frac{\left(\left(-R\right) \cdot \lambda_2\right) \cdot \left(t\_0 + \phi_1 \cdot \mathsf{fma}\left(\phi_1, \mathsf{fma}\left(-0.125, t\_0, 0.020833333333333332 \cdot \left(\phi_1 \cdot t\_2\right)\right), -0.5 \cdot t\_2\right)\right)}{\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi2)))
        (t_1 (cos (* 0.5 (+ phi2 phi1))))
        (t_2 (sin (* 0.5 phi2))))
   (if (<= phi1 2.1e-199)
     (*
      (+
       (* t_1 (- R))
       (/
        (*
         (* R lambda2)
         (- (cos (* 0.5 phi1)) (* 0.5 (* phi2 (sin (* 0.5 phi1))))))
        lambda1))
      lambda1)
     (*
      (fma
       t_1
       R
       (/
        (*
         (* (- R) lambda2)
         (+
          t_0
          (*
           phi1
           (fma
            phi1
            (fma -0.125 t_0 (* 0.020833333333333332 (* phi1 t_2)))
            (* -0.5 t_2)))))
        lambda1))
      (- lambda1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi2));
	double t_1 = cos((0.5 * (phi2 + phi1)));
	double t_2 = sin((0.5 * phi2));
	double tmp;
	if (phi1 <= 2.1e-199) {
		tmp = ((t_1 * -R) + (((R * lambda2) * (cos((0.5 * phi1)) - (0.5 * (phi2 * sin((0.5 * phi1)))))) / lambda1)) * lambda1;
	} else {
		tmp = fma(t_1, R, (((-R * lambda2) * (t_0 + (phi1 * fma(phi1, fma(-0.125, t_0, (0.020833333333333332 * (phi1 * t_2))), (-0.5 * t_2))))) / lambda1)) * -lambda1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi2))
	t_1 = cos(Float64(0.5 * Float64(phi2 + phi1)))
	t_2 = sin(Float64(0.5 * phi2))
	tmp = 0.0
	if (phi1 <= 2.1e-199)
		tmp = Float64(Float64(Float64(t_1 * Float64(-R)) + Float64(Float64(Float64(R * lambda2) * Float64(cos(Float64(0.5 * phi1)) - Float64(0.5 * Float64(phi2 * sin(Float64(0.5 * phi1)))))) / lambda1)) * lambda1);
	else
		tmp = Float64(fma(t_1, R, Float64(Float64(Float64(Float64(-R) * lambda2) * Float64(t_0 + Float64(phi1 * fma(phi1, fma(-0.125, t_0, Float64(0.020833333333333332 * Float64(phi1 * t_2))), Float64(-0.5 * t_2))))) / lambda1)) * Float64(-lambda1));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, 2.1e-199], N[(N[(N[(t$95$1 * (-R)), $MachinePrecision] + N[(N[(N[(R * lambda2), $MachinePrecision] * N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] - N[(0.5 * N[(phi2 * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision], N[(N[(t$95$1 * R + N[(N[(N[((-R) * lambda2), $MachinePrecision] * N[(t$95$0 + N[(phi1 * N[(phi1 * N[(-0.125 * t$95$0 + N[(0.020833333333333332 * N[(phi1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision] * (-lambda1)), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 2.10000000000000002e-199

   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 -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in phi2 around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 30.9

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

   8. Applied rewrites 30.9%

      \[\leadsto -\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 30.9

         \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   10. Applied rewrites 30.9%

       \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   if 2.10000000000000002e-199 < phi1

   1. Initial program 57.5%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 22.1%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto -\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\phi_1 \cdot \left(\frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{1}{48} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\phi_1 \cdot \left(\frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{1}{48} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

      2. lift-cos.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\phi_1 \cdot \left(\frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{1}{48} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

      3. lift-*.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\phi_1 \cdot \left(\frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{1}{48} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

      4. lower-*.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\phi_1 \cdot \left(\frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{1}{48} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

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

         \[\leadsto -\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\phi_1 \cdot \left(\frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{1}{48} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

      6. lower-fma.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \mathsf{fma}\left(\phi_1, \frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{1}{48} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   8. Applied rewrites 30.2%

      \[\leadsto -\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) + \phi_1 \cdot \mathsf{fma}\left(\phi_1, \mathsf{fma}\left(-0.125, \cos \left(0.5 \cdot \phi_2\right), 0.020833333333333332 \cdot \left(\phi_1 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right), -0.5 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 2.1 \cdot 10^{-199}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{\left(\left(-R\right) \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) + \phi_1 \cdot \mathsf{fma}\left(\phi_1, \mathsf{fma}\left(-0.125, \cos \left(0.5 \cdot \phi_2\right), 0.020833333333333332 \cdot \left(\phi_1 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right), -0.5 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right)}{\lambda_1}\right) \cdot \left(-\lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ \mathbf{if}\;\lambda_2 \leq 5.8 \cdot 10^{+31}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot \mathsf{fma}\left(-1, \frac{t\_0}{\lambda_1}, \frac{t\_0}{\lambda_2}\right)\right) \cdot \left(-\lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (cos (* 0.5 (+ phi1 phi2))))))
   (if (<= lambda2 5.8e+31)
     (*
      (+
       (* (cos (* 0.5 (+ phi2 phi1))) (- R))
       (/
        (*
         (* R lambda2)
         (- (cos (* 0.5 phi1)) (* 0.5 (* phi2 (sin (* 0.5 phi1))))))
        lambda1))
      lambda1)
     (* (* lambda2 (fma -1.0 (/ t_0 lambda1) (/ t_0 lambda2))) (- lambda1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * cos((0.5 * (phi1 + phi2)));
	double tmp;
	if (lambda2 <= 5.8e+31) {
		tmp = ((cos((0.5 * (phi2 + phi1))) * -R) + (((R * lambda2) * (cos((0.5 * phi1)) - (0.5 * (phi2 * sin((0.5 * phi1)))))) / lambda1)) * lambda1;
	} else {
		tmp = (lambda2 * fma(-1.0, (t_0 / lambda1), (t_0 / lambda2))) * -lambda1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * cos(Float64(0.5 * Float64(phi1 + phi2))))
	tmp = 0.0
	if (lambda2 <= 5.8e+31)
		tmp = Float64(Float64(Float64(cos(Float64(0.5 * Float64(phi2 + phi1))) * Float64(-R)) + Float64(Float64(Float64(R * lambda2) * Float64(cos(Float64(0.5 * phi1)) - Float64(0.5 * Float64(phi2 * sin(Float64(0.5 * phi1)))))) / lambda1)) * lambda1);
	else
		tmp = Float64(Float64(lambda2 * fma(-1.0, Float64(t_0 / lambda1), Float64(t_0 / lambda2))) * Float64(-lambda1));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[Cos[N[(0.5 * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, 5.8e+31], N[(N[(N[(N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision] + N[(N[(N[(R * lambda2), $MachinePrecision] * N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] - N[(0.5 * N[(phi2 * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision] * lambda1), $MachinePrecision], N[(N[(lambda2 * N[(-1.0 * N[(t$95$0 / lambda1), $MachinePrecision] + N[(t$95$0 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-lambda1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.8000000000000001e31

   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 lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 22.4%

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

   6. Taylor expanded in phi2 around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 26.5

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

   8. Applied rewrites 26.5%

      \[\leadsto -\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), R, \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 26.5

         \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   10. Applied rewrites 26.5%

       \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R + \frac{-\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1 \]

   if 5.8000000000000001e31 < lambda2

   1. Initial program 47.8%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 38.2%

      \[\leadsto -\left(\lambda_2 \cdot \mathsf{fma}\left(-1, \frac{R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}{\lambda_1}, \frac{R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}{\lambda_2}\right)\right) \cdot \lambda_1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.8 \cdot 10^{+31}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(-R\right) + \frac{\left(R \cdot \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) - 0.5 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\lambda_1}\right) \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 \cdot \mathsf{fma}\left(-1, \frac{R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}{\lambda_1}, \frac{R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}{\lambda_2}\right)\right) \cdot \left(-\lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\\ \left(\lambda_2 \cdot \mathsf{fma}\left(-1, \frac{t\_0}{\lambda_1}, \frac{t\_0}{\lambda_2}\right)\right) \cdot \left(-\lambda_1\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (cos (* 0.5 (+ phi1 phi2))))))
   (* (* lambda2 (fma -1.0 (/ t_0 lambda1) (/ t_0 lambda2))) (- lambda1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * cos((0.5 * (phi1 + phi2)));
	return (lambda2 * fma(-1.0, (t_0 / lambda1), (t_0 / lambda2))) * -lambda1;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * cos(Float64(0.5 * Float64(phi1 + phi2))))
	return Float64(Float64(lambda2 * fma(-1.0, Float64(t_0 / lambda1), Float64(t_0 / lambda2))) * Float64(-lambda1))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[Cos[N[(0.5 * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(lambda2 * N[(-1.0 * N[(t$95$0 / lambda1), $MachinePrecision] + N[(t$95$0 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-lambda1)), $MachinePrecision]]

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in lambda1 around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 26.7%

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

6. Taylor expanded in lambda2 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 N/A

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

8. Applied rewrites 25.0%

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

9. Final simplification 25.0%

   \[\leadsto \left(\lambda_2 \cdot \mathsf{fma}\left(-1, \frac{R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}{\lambda_1}, \frac{R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}{\lambda_2}\right)\right) \cdot \left(-\lambda_1\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))))))