Spherical law of cosines

Percentage Accurate: 73.4% → 93.8%

Time: 23.0s

Alternatives: 23

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Initial Program: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Alternative 1: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 1:\\ \;\;\;\;\cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))) 1.0)
     (*
      (acos
       (+
        t_0
        (*
         (cos phi1)
         (*
          (cos phi2)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2)))))))
      R)
     (* R (- lambda2 lambda1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if ((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))) <= 1.0) {
		tmp = acos((t_0 + (cos(phi1) * (cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))))) * R;
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    if ((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))) <= 1.0d0) then
        tmp = acos((t_0 + (cos(phi1) * (cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))))) * r
    else
        tmp = r * (lambda2 - lambda1)
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if ((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))) <= 1.0) {
		tmp = Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2))))))) * R;
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if (t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))) <= 1.0:
		tmp = math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2))))))) * R
	else:
		tmp = R * (lambda2 - lambda1)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))) <= 1.0)
		tmp = Float64(acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))))) * R);
	else
		tmp = Float64(R * Float64(lambda2 - lambda1));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if ((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))) <= 1.0)
		tmp = acos((t_0 + (cos(phi1) * (cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))))) * R;
	else
		tmp = R * (lambda2 - lambda1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))) < 1

   1. Initial program 69.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 93.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 93.1%

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

   5. Taylor expanded in phi1 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}\right)\right), R\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\cos \phi_1, \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\cos \phi_2, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      11. sin-lowering-sin.f64 93.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 93.1%

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

   if 1 < (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))

   1. Initial program 69.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 42.7%

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

   5. Simplified 42.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 24.6%

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

   8. Simplified 24.6%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. cos-diff N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. cos-diff N/A

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

      5. acos-cos-s N/A

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

      6. --lowering--.f64 5.7%

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

   10. Applied egg-rr 5.7%

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

4. Final simplification 93.1%

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

Alternative 2: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0 - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right), R, \frac{R \cdot \pi}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 9.8 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (-
           0.0
           (asin
            (+
             (* (sin phi1) (sin phi2))
             (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1)))))))
          R
          (/ (* R PI) 2.0))))
   (if (<= phi2 -2.9e-8)
     t_0
     (if (<= phi2 9.8e-11)
       (*
        R
        (acos
         (+
          (* (cos phi1) (* (sin lambda1) (sin lambda2)))
          (* (cos phi1) (* (cos lambda1) (cos lambda2))))))
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.0 - asin(((sin(phi1) * sin(phi2)) + (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1))))))), R, ((R * ((double) M_PI)) / 2.0));
	double tmp;
	if (phi2 <= -2.9e-8) {
		tmp = t_0;
	} else if (phi2 <= 9.8e-11) {
		tmp = R * acos(((cos(phi1) * (sin(lambda1) * sin(lambda2))) + (cos(phi1) * (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.0 - asin(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))))), R, Float64(Float64(R * pi) / 2.0))
	tmp = 0.0
	if (phi2 <= -2.9e-8)
		tmp = t_0;
	elseif (phi2 <= 9.8e-11)
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(sin(lambda1) * sin(lambda2))) + Float64(cos(phi1) * Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.0 - N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R + N[(N[(R * Pi), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.9e-8], t$95$0, If[LessEqual[phi2, 9.8e-11], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.9000000000000002e-8 or 9.7999999999999998e-11 < phi2

   1. Initial program 72.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right), \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 72.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right), R, \frac{R \cdot \pi}{2}\right)} \]

   if -2.9000000000000002e-8 < phi2 < 9.7999999999999998e-11

   1. Initial program 65.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 65.9%

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

   5. Simplified 65.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)\right)\right), R\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right), \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right), \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right), \cos \phi_1\right)\right)\right), R\right) \]

      17. cos-lowering-cos.f64 87.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(0 - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right), R, \frac{R \cdot \pi}{2}\right)\\ \mathbf{elif}\;\phi_2 \leq 9.8 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0 - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right), R, \frac{R \cdot \pi}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{R \cdot \pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -1.35e-8)
   (-
    (/ (* R PI) 2.0)
    (*
     R
     (asin
      (+
       (* (sin phi1) (sin phi2))
       (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1))))))))
   (if (<= phi2 8e-11)
     (*
      R
      (acos
       (+
        (* (cos phi1) (* (sin lambda1) (sin lambda2)))
        (* (cos phi1) (* (cos lambda1) (cos lambda2))))))
     (*
      R
      (acos
       (fma
        (sin phi2)
        (sin phi1)
        (* (cos phi2) (* (cos phi1) (cos (- lambda1 lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.35e-8) {
		tmp = ((R * ((double) M_PI)) / 2.0) - (R * asin(((sin(phi1) * sin(phi2)) + (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1)))))));
	} else if (phi2 <= 8e-11) {
		tmp = R * acos(((cos(phi1) * (sin(lambda1) * sin(lambda2))) + (cos(phi1) * (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos(fma(sin(phi2), sin(phi1), (cos(phi2) * (cos(phi1) * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.35e-8)
		tmp = Float64(Float64(Float64(R * pi) / 2.0) - Float64(R * asin(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))))));
	elseif (phi2 <= 8e-11)
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(sin(lambda1) * sin(lambda2))) + Float64(cos(phi1) * Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda1 - lambda2)))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -1.35e-8], N[(N[(N[(R * Pi), $MachinePrecision] / 2.0), $MachinePrecision] - N[(R * N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 8e-11], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.35000000000000001e-8

   1. Initial program 64.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right), \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 64.9%

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

   if -1.35000000000000001e-8 < phi2 < 7.99999999999999952e-11

   1. Initial program 65.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 65.9%

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

   5. Simplified 65.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \cos \phi_1\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)\right)\right), R\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right), \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right), \cos \phi_1\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right), \cos \phi_1\right)\right)\right), R\right) \]

      17. cos-lowering-cos.f64 87.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.3%

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

   if 7.99999999999999952e-11 < phi2

   1. Initial program 78.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_2 \cdot \sin \phi_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\sin \phi_2, \sin \phi_1, \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \sin \phi_1, \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\cos \phi_2, \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      13. --lowering--.f64 78.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 78.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{R \cdot \pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{R \cdot \pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -2.15e-8)
   (-
    (/ (* R PI) 2.0)
    (*
     R
     (asin
      (+
       (* (sin phi1) (sin phi2))
       (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1))))))))
   (if (<= phi2 6e-11)
     (*
      R
      (acos
       (*
        (cos phi1)
        (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))))
     (*
      R
      (acos
       (fma
        (sin phi2)
        (sin phi1)
        (* (cos phi2) (* (cos phi1) (cos (- lambda1 lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -2.15e-8) {
		tmp = ((R * ((double) M_PI)) / 2.0) - (R * asin(((sin(phi1) * sin(phi2)) + (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1)))))));
	} else if (phi2 <= 6e-11) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos(fma(sin(phi2), sin(phi1), (cos(phi2) * (cos(phi1) * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -2.15e-8)
		tmp = Float64(Float64(Float64(R * pi) / 2.0) - Float64(R * asin(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))))));
	elseif (phi2 <= 6e-11)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda1 - lambda2)))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -2.15e-8], N[(N[(N[(R * Pi), $MachinePrecision] / 2.0), $MachinePrecision] - N[(R * N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 6e-11], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.1500000000000001e-8

   1. Initial program 64.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right), \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 64.9%

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

   if -2.1500000000000001e-8 < phi2 < 6e-11

   1. Initial program 65.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 65.9%

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

   5. Simplified 65.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.3%

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

   if 6e-11 < phi2

   1. Initial program 78.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_2 \cdot \sin \phi_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\sin \phi_2, \sin \phi_1, \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \sin \phi_1, \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\cos \phi_2, \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      13. --lowering--.f64 78.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 78.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{R \cdot \pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{R \cdot \pi}{2} - R \cdot \sin^{-1} \left(t\_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -7.4e-9)
     (-
      (/ (* R PI) 2.0)
      (*
       R
       (asin (+ t_0 (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1))))))))
     (if (<= phi2 1.05e-10)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2))))))
       (*
        R
        (acos
         (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -7.4e-9) {
		tmp = ((R * ((double) M_PI)) / 2.0) - (R * asin((t_0 + (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1)))))));
	} else if (phi2 <= 1.05e-10) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi2 <= -7.4e-9) {
		tmp = ((R * Math.PI) / 2.0) - (R * Math.asin((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.cos((lambda2 - lambda1)))))));
	} else if (phi2 <= 1.05e-10) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi2 <= -7.4e-9:
		tmp = ((R * math.pi) / 2.0) - (R * math.asin((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.cos((lambda2 - lambda1)))))))
	elif phi2 <= 1.05e-10:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	else:
		tmp = R * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -7.4e-9)
		tmp = Float64(Float64(Float64(R * pi) / 2.0) - Float64(R * asin(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))))));
	elseif (phi2 <= 1.05e-10)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi2 <= -7.4e-9)
		tmp = ((R * pi) / 2.0) - (R * asin((t_0 + (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1)))))));
	elseif (phi2 <= 1.05e-10)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	else
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -7.4e-9], N[(N[(N[(R * Pi), $MachinePrecision] / 2.0), $MachinePrecision] - N[(R * N[ArcSin[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.05e-10], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -7.4e-9

   1. Initial program 64.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right), \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 64.9%

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

   if -7.4e-9 < phi2 < 1.05e-10

   1. Initial program 65.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 65.9%

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

   5. Simplified 65.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.3%

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

   if 1.05e-10 < phi2

   1. Initial program 78.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{R \cdot \pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{R}{\frac{2}{\pi}} - R \cdot \sin^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -2.5e-9)
     (-
      (/ R (/ 2.0 PI))
      (*
       R
       (asin (+ t_0 (* (cos phi1) (* (cos phi2) (cos (- lambda2 lambda1))))))))
     (if (<= phi2 1.05e-10)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2))))))
       (*
        R
        (acos
         (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -2.5e-9) {
		tmp = (R / (2.0 / ((double) M_PI))) - (R * asin((t_0 + (cos(phi1) * (cos(phi2) * cos((lambda2 - lambda1)))))));
	} else if (phi2 <= 1.05e-10) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi2 <= -2.5e-9) {
		tmp = (R / (2.0 / Math.PI)) - (R * Math.asin((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos((lambda2 - lambda1)))))));
	} else if (phi2 <= 1.05e-10) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi2 <= -2.5e-9:
		tmp = (R / (2.0 / math.pi)) - (R * math.asin((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos((lambda2 - lambda1)))))))
	elif phi2 <= 1.05e-10:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	else:
		tmp = R * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -2.5e-9)
		tmp = Float64(Float64(R / Float64(2.0 / pi)) - Float64(R * asin(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))))))));
	elseif (phi2 <= 1.05e-10)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi2 <= -2.5e-9)
		tmp = (R / (2.0 / pi)) - (R * asin((t_0 + (cos(phi1) * (cos(phi2) * cos((lambda2 - lambda1)))))));
	elseif (phi2 <= 1.05e-10)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	else
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.5e-9], N[(N[(R / N[(2.0 / Pi), $MachinePrecision]), $MachinePrecision] - N[(R * N[ArcSin[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.05e-10], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.5000000000000001e-9

   1. Initial program 64.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 99.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 99.0%

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

   6. Applied egg-rr 64.8%

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

   if -2.5000000000000001e-9 < phi2 < 1.05e-10

   1. Initial program 65.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 65.9%

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

   5. Simplified 65.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.3%

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

   if 1.05e-10 < phi2

   1. Initial program 78.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{R}{\frac{2}{\pi}} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 9.2 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          R
          (acos
           (+
            (* (sin phi1) (sin phi2))
            (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))))
   (if (<= phi2 -5.1e-8)
     t_0
     (if (<= phi2 9.2e-11)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2))))))
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi2 <= -5.1e-8) {
		tmp = t_0;
	} else if (phi2 <= 9.2e-11) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    if (phi2 <= (-5.1d-8)) then
        tmp = t_0
    else if (phi2 <= 9.2d-11) then
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	double tmp;
	if (phi2 <= -5.1e-8) {
		tmp = t_0;
	} else if (phi2 <= 9.2e-11) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	tmp = 0
	if phi2 <= -5.1e-8:
		tmp = t_0
	elif phi2 <= 9.2e-11:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	else:
		tmp = t_0
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))))
	tmp = 0.0
	if (phi2 <= -5.1e-8)
		tmp = t_0;
	elseif (phi2 <= 9.2e-11)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	tmp = 0.0;
	if (phi2 <= -5.1e-8)
		tmp = t_0;
	elseif (phi2 <= 9.2e-11)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -5.1e-8], t$95$0, If[LessEqual[phi2, 9.2e-11], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -5.10000000000000001e-8 or 9.20000000000000054e-11 < phi2

   1. Initial program 72.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   if -5.10000000000000001e-8 < phi2 < 9.20000000000000054e-11

   1. Initial program 65.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 65.9%

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

   5. Simplified 65.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.3%

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 9.2 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_1 \leq -1.18 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{elif}\;\phi_1 \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2)))))
   (if (<= phi1 -1.18e-20)
     (* R (acos (* (cos phi1) t_0)))
     (if (<= phi1 3.1e-5)
       (* R (acos (* (cos phi2) t_0)))
       (*
        R
        (acos
         (+
          (* (sin phi1) (sin phi2))
          (* (* (cos phi1) (cos phi2)) (cos lambda1)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2));
	double tmp;
	if (phi1 <= -1.18e-20) {
		tmp = R * acos((cos(phi1) * t_0));
	} else if (phi1 <= 3.1e-5) {
		tmp = R * acos((cos(phi2) * t_0));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda1))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))
    if (phi1 <= (-1.18d-20)) then
        tmp = r * acos((cos(phi1) * t_0))
    else if (phi1 <= 3.1d-5) then
        tmp = r * acos((cos(phi2) * t_0))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda1))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2));
	double tmp;
	if (phi1 <= -1.18e-20) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else if (phi1 <= 3.1e-5) {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda1))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2))
	tmp = 0
	if phi1 <= -1.18e-20:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	elif phi1 <= 3.1e-5:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda1))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))
	tmp = 0.0
	if (phi1 <= -1.18e-20)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	elseif (phi1 <= 3.1e-5)
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2));
	tmp = 0.0;
	if (phi1 <= -1.18e-20)
		tmp = R * acos((cos(phi1) * t_0));
	elseif (phi1 <= 3.1e-5)
		tmp = R * acos((cos(phi2) * t_0));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.18e-20], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.1e-5], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.1800000000000001e-20

   1. Initial program 76.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 55.2%

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

   5. Simplified 55.2%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 62.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 62.9%

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

   if -1.1800000000000001e-20 < phi1 < 3.10000000000000014e-5

   1. Initial program 63.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 88.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 88.1%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right), R\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 87.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 87.8%

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

   if 3.10000000000000014e-5 < phi1

   1. Initial program 73.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \lambda_1}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 52.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 52.6%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.18 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_1 \leq -1.18 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2)))))
   (if (<= phi1 -1.18e-20)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2));
	double tmp;
	if (phi1 <= -1.18e-20) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))
    if (phi1 <= (-1.18d-20)) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2));
	double tmp;
	if (phi1 <= -1.18e-20) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2))
	tmp = 0
	if phi1 <= -1.18e-20:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))
	tmp = 0.0
	if (phi1 <= -1.18e-20)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2));
	tmp = 0.0;
	if (phi1 <= -1.18e-20)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.18e-20], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.1800000000000001e-20

   1. Initial program 76.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 55.2%

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

   5. Simplified 55.2%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 62.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 62.9%

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

   if -1.1800000000000001e-20 < phi1

   1. Initial program 66.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 91.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right), R\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 66.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 66.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.18 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.05e-10)
   (*
    R
    (acos
     (*
      (cos phi1)
      (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))))
   (* R (acos (* (cos phi2) (cos (- lambda2 lambda1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.05e-10) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.05d-10) then
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    else
        tmp = r * acos((cos(phi2) * cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.05e-10) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.05e-10:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.05e-10)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.05e-10)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	else
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.05e-10], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.05e-10

   1. Initial program 65.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 51.3%

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

   5. Simplified 51.3%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 66.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 66.2%

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

   if 1.05e-10 < phi2

   1. Initial program 78.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

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

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

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

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

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

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

      12. --lowering--.f64 56.9%

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

   5. Simplified 56.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 2e-14)
     (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) t_0))))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 2e-14) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0)));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 2d-14) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0)))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 2e-14) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * t_0)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 2e-14:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * t_0)))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 2e-14)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 2e-14)
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0)));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 2e-14], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2e-14

   1. Initial program 65.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right), R\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 51.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right)\right), R\right) \]

   5. Simplified 51.1%

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

   if 2e-14 < phi2

   1. Initial program 78.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

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

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

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

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

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

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

      12. --lowering--.f64 56.9%

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

   5. Simplified 56.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -3.7e-7)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -3.7e-7) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi1 <= (-3.7d-7)) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -3.7e-7) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -3.7e-7:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -3.7e-7)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -3.7e-7)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.7e-7], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.70000000000000004e-7

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 54.5%

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

   5. Simplified 54.5%

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

   if -3.70000000000000004e-7 < phi1

   1. Initial program 66.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

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

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

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

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

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

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

      12. --lowering--.f64 49.2%

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

   5. Simplified 49.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.3e-6:
		tmp = R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda2)))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.30000000000000017e-6

   1. Initial program 65.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 51.5%

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

   5. Simplified 51.5%

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

   if 3.30000000000000017e-6 < phi2

   1. Initial program 78.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 60.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 60.9%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_2 \cdot \cos \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \cos \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \lambda_2\right)\right), R\right) \]

      4. cos-lowering-cos.f64 44.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right), R\right) \]

   8. Simplified 44.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1e-26)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi2) (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1e-26) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-1d-26)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi2) * cos(lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1e-26) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1e-26:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1e-26)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1e-26)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1e-26], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1e-26

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 55.8%

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

   5. Simplified 55.8%

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

   6. Taylor expanded in lambda2 around 0

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

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \cos \lambda_1\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \lambda_1\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \lambda_1\right)\right), R\right) \]

      4. cos-lowering-cos.f64 42.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right), R\right) \]

   8. Simplified 42.5%

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

   if -1e-26 < phi1

   1. Initial program 66.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 43.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 43.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_2 \cdot \cos \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \cos \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \lambda_2\right)\right), R\right) \]

      4. cos-lowering-cos.f64 32.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right), R\right) \]

   8. Simplified 32.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.3 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.3e+23)
   (* R (acos (* (cos phi2) (cos lambda1))))
   (* R (acos (* (cos phi1) (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.3e+23) {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-2.3d+23)) then
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.3e+23) {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.3e+23:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.3e+23)
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2.3e+23)
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.3e+23], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.3e23

   1. Initial program 55.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 99.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right), R\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 66.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 66.1%

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

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)}\right), R\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \phi_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \phi_2\right)\right), R\right) \]

      3. cos-lowering-cos.f64 39.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right)\right), R\right) \]

   10. Simplified 39.8%

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

   if -2.3e23 < lambda1

   1. Initial program 74.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 45.0%

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

   5. Simplified 45.0%

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

   6. Taylor expanded in lambda1 around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \cos \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \lambda_2\right)\right), R\right) \]

      4. cos-lowering-cos.f64 34.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right), R\right) \]

   8. Simplified 34.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.3 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.9e-6)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi2) (cos lambda1))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.9e-6) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-2.9d-6)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.9e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.9e-6:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.9e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.9e-6)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.9e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.9000000000000002e-6

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 54.5%

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

   5. Simplified 54.5%

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

   6. Taylor expanded in lambda2 around 0

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

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_1 \cdot \cos \lambda_1\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \lambda_1\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \lambda_1\right)\right), R\right) \]

      4. cos-lowering-cos.f64 43.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right), R\right) \]

   8. Simplified 43.3%

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

   if -2.9000000000000002e-6 < phi1

   1. Initial program 66.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 91.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 91.3%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right), R\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 66.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 66.2%

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

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)}\right), R\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \phi_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \phi_2\right)\right), R\right) \]

      3. cos-lowering-cos.f64 36.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right)\right), R\right) \]

   10. Simplified 36.6%

       \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.5e-5)
   (* R (acos (cos phi1)))
   (* R (acos (* (cos phi2) (cos lambda1))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e-5) {
		tmp = R * acos(cos(phi1));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-3.5d-5)) then
        tmp = r * acos(cos(phi1))
    else
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e-5) {
		tmp = R * Math.acos(Math.cos(phi1));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.5e-5:
		tmp = R * math.acos(math.cos(phi1))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.5e-5)
		tmp = Float64(R * acos(cos(phi1)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.5e-5)
		tmp = R * acos(cos(phi1));
	else
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.4999999999999997e-5

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 54.5%

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

   5. Simplified 54.5%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) + -1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \left(\mathsf{neg}\left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \lambda_2 \cdot \left(\mathsf{neg}\left(\sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      4. sin-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \lambda_2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin \lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      5. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\cos \lambda_1, \left(\lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \sin \lambda_1\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 34.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \mathsf{sin.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   8. Simplified 34.9%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \phi_1}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-lowering-cos.f64 32.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\phi_1\right)\right), R\right) \]

   11. Simplified 32.0%

       \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]

   if -3.4999999999999997e-5 < phi1

   1. Initial program 66.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 91.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 91.3%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right), R\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 66.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 66.2%

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

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)}\right), R\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \phi_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \phi_2\right)\right), R\right) \]

      3. cos-lowering-cos.f64 36.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right)\right), R\right) \]

   10. Simplified 36.6%

       \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 21.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.1 \cdot 10^{-237}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 1950:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -2.1e-237)
   (* R (acos (cos lambda1)))
   (if (<= lambda2 1950.0)
     (* R (acos (cos phi1)))
     (* R (acos (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -2.1e-237) {
		tmp = R * acos(cos(lambda1));
	} else if (lambda2 <= 1950.0) {
		tmp = R * acos(cos(phi1));
	} else {
		tmp = R * acos(cos(lambda2));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= (-2.1d-237)) then
        tmp = r * acos(cos(lambda1))
    else if (lambda2 <= 1950.0d0) then
        tmp = r * acos(cos(phi1))
    else
        tmp = r * acos(cos(lambda2))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -2.1e-237) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else if (lambda2 <= 1950.0) {
		tmp = R * Math.acos(Math.cos(phi1));
	} else {
		tmp = R * Math.acos(Math.cos(lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -2.1e-237:
		tmp = R * math.acos(math.cos(lambda1))
	elif lambda2 <= 1950.0:
		tmp = R * math.acos(math.cos(phi1))
	else:
		tmp = R * math.acos(math.cos(lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -2.1e-237)
		tmp = Float64(R * acos(cos(lambda1)));
	elseif (lambda2 <= 1950.0)
		tmp = Float64(R * acos(cos(phi1)));
	else
		tmp = Float64(R * acos(cos(lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -2.1e-237)
		tmp = R * acos(cos(lambda1));
	elseif (lambda2 <= 1950.0)
		tmp = R * acos(cos(phi1));
	else
		tmp = R * acos(cos(lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -2.1e-237], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 1950.0], N[(R * N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -2.1000000000000001e-237

   1. Initial program 68.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 44.8%

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

   5. Simplified 44.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 27.1%

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

   8. Simplified 27.1%

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

   9. Taylor expanded in lambda2 around 0

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

       1. cos-lowering-cos.f64 19.3%

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

   11. Simplified 19.3%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if -2.1000000000000001e-237 < lambda2 < 1950

   1. Initial program 83.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 43.6%

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

   5. Simplified 43.6%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) + -1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \left(\mathsf{neg}\left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \lambda_2 \cdot \left(\mathsf{neg}\left(\sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      4. sin-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \lambda_2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin \lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      5. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\cos \lambda_1, \left(\lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \sin \lambda_1\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 43.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \mathsf{sin.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   8. Simplified 43.6%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \phi_1}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-lowering-cos.f64 19.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\phi_1\right)\right), R\right) \]

   11. Simplified 19.3%

       \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]

   if 1950 < lambda2

   1. Initial program 53.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 26.6%

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

   8. Simplified 26.6%

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

   9. Taylor expanded in lambda1 around 0

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

       1. cos-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \lambda_2\right), R\right) \]

       2. cos-lowering-cos.f64 27.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right)\right), R\right) \]

   11. Simplified 27.0%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.1 \cdot 10^{-237}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 1950:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 33.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.5e-5)
   (* R (acos (cos phi1)))
   (* R (acos (cos (- lambda1 lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e-5) {
		tmp = R * acos(cos(phi1));
	} else {
		tmp = R * acos(cos((lambda1 - lambda2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-3.5d-5)) then
        tmp = r * acos(cos(phi1))
    else
        tmp = r * acos(cos((lambda1 - lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e-5) {
		tmp = R * Math.acos(Math.cos(phi1));
	} else {
		tmp = R * Math.acos(Math.cos((lambda1 - lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.5e-5:
		tmp = R * math.acos(math.cos(phi1))
	else:
		tmp = R * math.acos(math.cos((lambda1 - lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.5e-5)
		tmp = Float64(R * acos(cos(phi1)));
	else
		tmp = Float64(R * acos(cos(Float64(lambda1 - lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.5e-5)
		tmp = R * acos(cos(phi1));
	else
		tmp = R * acos(cos((lambda1 - lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.4999999999999997e-5

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 54.5%

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

   5. Simplified 54.5%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) + -1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \left(\mathsf{neg}\left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \lambda_2 \cdot \left(\mathsf{neg}\left(\sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      4. sin-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \lambda_2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin \lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      5. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\cos \lambda_1, \left(\lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \sin \lambda_1\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 34.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \mathsf{sin.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   8. Simplified 34.9%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \phi_1}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-lowering-cos.f64 32.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\phi_1\right)\right), R\right) \]

   11. Simplified 32.0%

       \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]

   if -3.4999999999999997e-5 < phi1

   1. Initial program 66.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 38.8%

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

   5. Simplified 38.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 27.9%

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

   8. Simplified 27.9%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 21.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.1e-6)
   (* R (acos (cos lambda1)))
   (* R (acos (cos lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.1e-6) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * acos(cos(lambda2));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 1.1d-6) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * acos(cos(lambda2))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.1e-6) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * Math.acos(Math.cos(lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.1e-6:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * math.acos(math.cos(lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.1e-6)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * acos(cos(lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.1e-6)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * acos(cos(lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.1e-6], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.1000000000000001e-6

   1. Initial program 75.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 44.4%

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

   5. Simplified 44.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 23.9%

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

   8. Simplified 23.9%

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

   9. Taylor expanded in lambda2 around 0

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

       1. cos-lowering-cos.f64 19.4%

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

   11. Simplified 19.4%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if 1.1000000000000001e-6 < lambda2

   1. Initial program 53.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 38.5%

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

   5. Simplified 38.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in lambda1 around 0

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

       1. cos-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \lambda_2\right), R\right) \]

       2. cos-lowering-cos.f64 26.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right)\right), R\right) \]

   11. Simplified 26.8%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 12.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2e-8) (* R (acos (cos lambda1))) (* R (- lambda2 lambda1))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2e-8) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-2d-8)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * (lambda2 - lambda1)
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2e-8) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2e-8:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * (lambda2 - lambda1)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2e-8)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * Float64(lambda2 - lambda1));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2e-8)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * (lambda2 - lambda1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2e-8], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2e-8

   1. Initial program 56.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 37.0%

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

   5. Simplified 37.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 27.6%

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

   8. Simplified 27.6%

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

   9. Taylor expanded in lambda2 around 0

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

       1. cos-lowering-cos.f64 27.7%

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

   11. Simplified 27.7%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if -2e-8 < lambda1

   1. Initial program 73.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

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

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 45.0%

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

   5. Simplified 45.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 23.5%

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

   8. Simplified 23.5%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. cos-diff N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. cos-diff N/A

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

      5. acos-cos-s N/A

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

      6. --lowering--.f64 5.8%

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

   10. Applied egg-rr 5.8%

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

4. Final simplification 12.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 5.7% accurate, 122.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \left(\lambda_2 - \lambda_1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   4. remove-double-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   5. mul-1-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. cos-neg N/A

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

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

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

   10. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

   11. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

   12. --lowering--.f64 42.7%

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

5. Simplified 42.7%

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

6. Taylor expanded in phi1 around 0

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

   1. sub-neg N/A

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

   2. remove-double-neg N/A

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

   3. distribute-neg-in N/A

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

   4. +-commutative N/A

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

   5. neg-mul-1 N/A

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

   6. cos-neg N/A

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

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

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

   8. neg-mul-1 N/A

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

   9. sub-neg N/A

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

   10. --lowering--.f64 24.6%

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

8. Simplified 24.6%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
9. Step-by-step derivation

   1. cos-diff N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. cos-diff N/A

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

   5. acos-cos-s N/A

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

   6. --lowering--.f64 5.7%

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

10. Applied egg-rr 5.7%

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

11. Final simplification 5.7%

    \[\leadsto R \cdot \left(\lambda_2 - \lambda_1\right) \]
12. Add Preprocessing

Alternative 23: 5.3% accurate, 204.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_1 \cdot R \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   4. remove-double-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   5. mul-1-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. cos-neg N/A

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

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

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

   10. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

   11. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

   12. --lowering--.f64 42.7%

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

5. Simplified 42.7%

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

6. Taylor expanded in phi1 around 0

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

   1. sub-neg N/A

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

   2. remove-double-neg N/A

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

   3. distribute-neg-in N/A

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

   4. +-commutative N/A

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

   5. neg-mul-1 N/A

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

   6. cos-neg N/A

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

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

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

   8. neg-mul-1 N/A

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

   9. sub-neg N/A

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

   10. --lowering--.f64 24.6%

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

8. Simplified 24.6%

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

9. Taylor expanded in lambda1 around inf

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

11. Simplified 4.4%

    \[\leadsto \color{blue}{\lambda_1} \cdot R \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))