Spherical law of cosines

Percentage Accurate: 73.4% → 96.5%

Time: 24.5s

Alternatives: 28

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: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0.0004:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_1 \cdot \left(t\_1 \cdot \cos \lambda_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (acos (+ t_0 (* t_1 (cos (- lambda1 lambda2))))) 0.0004)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (+
        t_0
        (+
         (* (cos phi1) (* (cos phi2) (* (sin lambda1) (sin lambda2))))
         (* (cos lambda1) (* t_1 (cos lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0004) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos((t_0 + ((cos(phi1) * (cos(phi2) * (sin(lambda1) * sin(lambda2)))) + (cos(lambda1) * (t_1 * cos(lambda2))))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    t_1 = cos(phi1) * cos(phi2)
    if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0004d0) then
        tmp = (lambda2 - lambda1) * r
    else
        tmp = r * acos((t_0 + ((cos(phi1) * (cos(phi2) * (sin(lambda1) * sin(lambda2)))) + (cos(lambda1) * (t_1 * cos(lambda2))))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 t_1 = Math.cos(phi1) * Math.cos(phi2);
	double tmp;
	if (Math.acos((t_0 + (t_1 * Math.cos((lambda1 - lambda2))))) <= 0.0004) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi1) * (Math.cos(phi2) * (Math.sin(lambda1) * Math.sin(lambda2)))) + (Math.cos(lambda1) * (t_1 * Math.cos(lambda2))))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[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)
	t_1 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if math.acos((t_0 + (t_1 * math.cos((lambda1 - lambda2))))) <= 0.0004:
		tmp = (lambda2 - lambda1) * R
	else:
		tmp = R * math.acos((t_0 + ((math.cos(phi1) * (math.cos(phi2) * (math.sin(lambda1) * math.sin(lambda2)))) + (math.cos(lambda1) * (t_1 * math.cos(lambda2))))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (acos(Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2))))) <= 0.0004)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * Float64(cos(phi2) * Float64(sin(lambda1) * sin(lambda2)))) + Float64(cos(lambda1) * Float64(t_1 * cos(lambda2)))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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);
	t_1 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0004)
		tmp = (lambda2 - lambda1) * R;
	else
		tmp = R * acos((t_0 + ((cos(phi1) * (cos(phi2) * (sin(lambda1) * sin(lambda2)))) + (cos(lambda1) * (t_1 * 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.
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]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0004], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[(t$95$1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 4.00000000000000019e-4

   1. Initial program 8.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 8.3%

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

   5. Simplified 8.3%

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

   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 8.3%

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

   8. Simplified 8.3%

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

      1. cos-diff N/A

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

      2. *-commutative N/A

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

      3. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

      9. sin-lowering-sin.f64 8.3%

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

   10. Applied egg-rr 8.3%

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

       1. *-commutative N/A

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

       2. cos-diff N/A

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

       3. acos-cos-s N/A

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

       4. --lowering--.f64 47.8%

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

   12. Applied egg-rr 47.8%

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

   if 4.00000000000000019e-4 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 77.5%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. distribute-rgt-in N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      19. cos-lowering-cos.f64 98.9%

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

   6. Applied egg-rr 98.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0.0004:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0.0004:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + t\_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (acos (+ t_0 (* t_1 (cos (- lambda1 lambda2))))) 0.0004)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (+
        t_0
        (*
         t_1
         (+
          (* (sin lambda1) (sin lambda2))
          (* (cos lambda1) (cos lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0004) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos((t_0 + (t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    t_1 = cos(phi1) * cos(phi2)
    if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0004d0) then
        tmp = (lambda2 - lambda1) * r
    else
        tmp = r * acos((t_0 + (t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 t_1 = Math.cos(phi1) * Math.cos(phi2);
	double tmp;
	if (Math.acos((t_0 + (t_1 * Math.cos((lambda1 - lambda2))))) <= 0.0004) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * Math.acos((t_0 + (t_1 * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2))))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[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)
	t_1 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if math.acos((t_0 + (t_1 * math.cos((lambda1 - lambda2))))) <= 0.0004:
		tmp = (lambda2 - lambda1) * R
	else:
		tmp = R * math.acos((t_0 + (t_1 * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2))))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (acos(Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2))))) <= 0.0004)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(t_1 * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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);
	t_1 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0004)
		tmp = (lambda2 - lambda1) * R;
	else
		tmp = R * acos((t_0 + (t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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.
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]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0004], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(t$95$1 * 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]]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 4.00000000000000019e-4

   1. Initial program 8.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 8.3%

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

   5. Simplified 8.3%

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

   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 8.3%

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

   8. Simplified 8.3%

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

      1. cos-diff N/A

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

      2. *-commutative N/A

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

      3. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

      9. sin-lowering-sin.f64 8.3%

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

   10. Applied egg-rr 8.3%

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

       1. *-commutative N/A

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

       2. cos-diff N/A

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

       3. acos-cos-s N/A

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

       4. --lowering--.f64 47.8%

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

   12. Applied egg-rr 47.8%

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

   if 4.00000000000000019e-4 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 77.5%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 98.9%

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

   4. Applied egg-rr 98.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0.0004:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.004:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right) \cdot \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -0.004)
     (*
      R
      (-
       (/ PI 2.0)
       (asin (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1)))))))
     (if (<= phi2 1.65e-6)
       (*
        R
        (acos
         (+
          t_0
          (*
           (+ (* -0.5 (* phi2 phi2)) 1.0)
           (*
            (cos phi1)
            (+
             (* (sin lambda1) (sin lambda2))
             (* (cos lambda1) (cos lambda2))))))))
       (*
        R
        (acos
         (fma (* (cos phi2) (cos (- lambda1 lambda2))) (cos phi1) t_0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= -0.004) {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1))))));
	} else if (phi2 <= 1.65e-6) {
		tmp = R * acos((t_0 + (((-0.5 * (phi2 * phi2)) + 1.0) * (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))));
	} else {
		tmp = R * acos(fma((cos(phi2) * cos((lambda1 - lambda2))), cos(phi1), t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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 <= -0.004)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda2 - lambda1)))))));
	elseif (phi2 <= 1.65e-6)
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(Float64(-0.5 * Float64(phi2 * phi2)) + 1.0) * Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))))));
	else
		tmp = Float64(R * acos(fma(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), cos(phi1), t_0)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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, -0.004], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.65e-6], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 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]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0040000000000000001

   1. Initial program 77.7%

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

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.5%

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

      1. acos-asin N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   6. Applied egg-rr 77.7%

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

   if -0.0040000000000000001 < phi2 < 1.65000000000000008e-6

   1. Initial program 69.6%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 89.2%

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \color{blue}{\left(\frac{-1}{2} \cdot \left({\phi_2}^{2} \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right)\right), R\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \left(\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right) \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \left(\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right) \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right), \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot {\phi_2}^{2}\right), 1\right), \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({\phi_2}^{2}\right)\right), 1\right), \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\phi_2 \cdot \phi_2\right)\right), 1\right), \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\phi_2, \phi_2\right)\right), 1\right), \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\phi_2, \phi_2\right)\right), 1\right), \mathsf{*.f64}\left(\cos \phi_1, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\phi_2, \phi_2\right)\right), 1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\phi_2, \phi_2\right)\right), 1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 88.8%

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

   if 1.65000000000000008e-6 < phi2

   1. Initial program 78.4%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. sin-lowering-sin.f64 78.4%

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.004:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right) \cdot \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.003:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right) \cdot \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -0.003)
     (*
      R
      (-
       (/ PI 2.0)
       (asin (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1)))))))
     (if (<= phi2 1.65e-6)
       (*
        R
        (acos
         (+
          (*
           (+ (* -0.5 (* phi2 phi2)) 1.0)
           (*
            (cos phi1)
            (+
             (* (sin lambda1) (sin lambda2))
             (* (cos lambda1) (cos lambda2)))))
          (* (sin phi1) phi2))))
       (*
        R
        (acos
         (fma (* (cos phi2) (cos (- lambda1 lambda2))) (cos phi1) t_0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= -0.003) {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1))))));
	} else if (phi2 <= 1.65e-6) {
		tmp = R * acos(((((-0.5 * (phi2 * phi2)) + 1.0) * (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(fma((cos(phi2) * cos((lambda1 - lambda2))), cos(phi1), t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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 <= -0.003)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda2 - lambda1)))))));
	elseif (phi2 <= 1.65e-6)
		tmp = Float64(R * acos(Float64(Float64(Float64(Float64(-0.5 * Float64(phi2 * phi2)) + 1.0) * Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(fma(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), cos(phi1), t_0)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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, -0.003], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.65e-6], N[(R * N[ArcCos[N[(N[(N[(N[(-0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 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] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0030000000000000001

   1. Initial program 77.7%

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

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.5%

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

      1. acos-asin N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   6. Applied egg-rr 77.7%

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

   if -0.0030000000000000001 < phi2 < 1.65000000000000008e-6

   1. Initial program 69.6%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 89.2%

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\phi_2 \cdot \left(\sin \phi_1 + \frac{-1}{2} \cdot \left(\phi_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right), R\right) \]

   7. Simplified 88.7%

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

   if 1.65000000000000008e-6 < phi2

   1. Initial program 78.4%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. sin-lowering-sin.f64 78.4%

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.003:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right) \cdot \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -6.5e-5)
     (*
      R
      (-
       (/ PI 2.0)
       (asin (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1)))))))
     (if (<= phi2 1.35e-6)
       (*
        R
        (acos
         (+
          (* (sin phi1) phi2)
          (*
           (cos phi1)
           (+
            (* (sin lambda1) (sin lambda2))
            (* (cos lambda1) (cos lambda2)))))))
       (*
        R
        (acos
         (fma (* (cos phi2) (cos (- lambda1 lambda2))) (cos phi1) t_0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= -6.5e-5) {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1))))));
	} else if (phi2 <= 1.35e-6) {
		tmp = R * acos(((sin(phi1) * phi2) + (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = R * acos(fma((cos(phi2) * cos((lambda1 - lambda2))), cos(phi1), t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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 <= -6.5e-5)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda2 - lambda1)))))));
	elseif (phi2 <= 1.35e-6)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * phi2) + Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))))));
	else
		tmp = Float64(R * acos(fma(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), cos(phi1), t_0)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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, -6.5e-5], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.35e-6], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision] + 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]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.49999999999999943e-5

   1. Initial program 77.0%

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

      1. cos-diff N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.1%

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

      1. acos-asin N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   6. Applied egg-rr 77.1%

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

   if -6.49999999999999943e-5 < phi2 < 1.34999999999999999e-6

   1. Initial program 69.7%

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

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 89.1%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

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

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

   7. Simplified 89.1%

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

   if 1.34999999999999999e-6 < phi2

   1. Initial program 78.4%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. sin-lowering-sin.f64 78.4%

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.3 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -1.06e-13)
   (*
    R
    (acos
     (fma
      (sin phi2)
      (sin phi1)
      (* (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1))))))
   (if (<= phi2 1.3e-9)
     (*
      R
      (acos
       (*
        (cos phi1)
        (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
     (*
      R
      (acos
       (fma
        (* (cos phi2) (cos (- lambda1 lambda2)))
        (cos phi1)
        (* (sin phi1) (sin phi2))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= -1.06e-13) {
		tmp = R * acos(fma(sin(phi2), sin(phi1), ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1)))));
	} else if (phi2 <= 1.3e-9) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(fma((cos(phi2) * cos((lambda1 - lambda2))), cos(phi1), (sin(phi1) * sin(phi2))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.06e-13)
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda2 - lambda1))))));
	elseif (phi2 <= 1.3e-9)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(fma(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), cos(phi1), Float64(sin(phi1) * sin(phi2)))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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, -1.06e-13], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.3e-9], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.06e-13

   1. Initial program 76.4%

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

      1. cos-diff N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.2%

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

      1. associate-+l+ N/A

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

      2. *-commutative N/A

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

      3. fma-define N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. cos-diff N/A

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

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

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

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

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

      10. 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. Applied egg-rr 76.5%

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

   if -1.06e-13 < phi2 < 1.3000000000000001e-9

   1. Initial program 69.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 69.7%

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

   5. Simplified 69.7%

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

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

      3. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

      9. sin-lowering-sin.f64 88.5%

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

   7. Applied egg-rr 88.5%

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

   if 1.3000000000000001e-9 < phi2

   1. Initial program 78.4%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. sin-lowering-sin.f64 78.4%

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.3 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -1.06e-13)
   (*
    R
    (acos
     (fma
      (sin phi2)
      (sin phi1)
      (* (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1))))))
   (if (<= phi2 2.9e-8)
     (*
      R
      (acos
       (*
        (cos phi1)
        (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin 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);
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 <= -1.06e-13) {
		tmp = R * acos(fma(sin(phi2), sin(phi1), ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1)))));
	} else if (phi2 <= 2.9e-8) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(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])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.06e-13)
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda2 - lambda1))))));
	elseif (phi2 <= 2.9e-8)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(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.
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, -1.06e-13], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.9e-8], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[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 < -1.06e-13

   1. Initial program 76.4%

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

      1. cos-diff N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.2%

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

      1. associate-+l+ N/A

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

      2. *-commutative N/A

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

      3. fma-define N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. cos-diff N/A

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

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

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

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

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

      10. 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. Applied egg-rr 76.5%

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

   if -1.06e-13 < phi2 < 2.9000000000000002e-8

   1. Initial program 69.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 69.7%

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

   5. Simplified 69.7%

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

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

      3. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

      9. sin-lowering-sin.f64 88.5%

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

   7. Applied egg-rr 88.5%

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

   if 2.9000000000000002e-8 < phi2

   1. Initial program 78.4%

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

      1. *-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. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. --lowering--.f64 N/A

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

      14. cos-lowering-cos.f64 78.4%

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -1.06e-13)
   (*
    R
    (-
     (/ PI 2.0)
     (asin
      (+
       (* (sin phi1) (sin phi2))
       (* (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1)))))))
   (if (<= phi2 1.1e-8)
     (*
      R
      (acos
       (*
        (cos phi1)
        (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin 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);
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 <= -1.06e-13) {
		tmp = R * ((((double) M_PI) / 2.0) - asin(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1))))));
	} else if (phi2 <= 1.1e-8) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(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])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.06e-13)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda2 - lambda1)))))));
	elseif (phi2 <= 1.1e-8)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(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.
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, -1.06e-13], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[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[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.1e-8], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[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 < -1.06e-13

   1. Initial program 76.4%

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

      1. cos-diff N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.2%

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

      1. acos-asin N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   6. Applied egg-rr 76.5%

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

   if -1.06e-13 < phi2 < 1.0999999999999999e-8

   1. Initial program 69.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 69.7%

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

   5. Simplified 69.7%

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

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

      3. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

      9. sin-lowering-sin.f64 88.5%

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

   7. Applied egg-rr 88.5%

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

   if 1.0999999999999999e-8 < phi2

   1. Initial program 78.4%

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

      1. *-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. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. --lowering--.f64 N/A

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

      14. cos-lowering-cos.f64 78.4%

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -1.06e-13)
     (*
      R
      (-
       (/ PI 2.0)
       (asin (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1)))))))
     (if (<= phi2 2.4e-8)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (*
        R
        (acos
         (+ t_0 (* (cos phi2) (* (cos phi1) (cos (- lambda1 lambda2)))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= -1.06e-13) {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1))))));
	} else if (phi2 <= 2.4e-8) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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 <= -1.06e-13)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda2 - lambda1)))))));
	elseif (phi2 <= 2.4e-8)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + 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.
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, -1.06e-13], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.4e-8], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + 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 < -1.06e-13

   1. Initial program 76.4%

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

      1. cos-diff N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.2%

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

      1. acos-asin N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   6. Applied egg-rr 76.5%

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

   if -1.06e-13 < phi2 < 2.39999999999999998e-8

   1. Initial program 69.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 69.7%

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

   5. Simplified 69.7%

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

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

      3. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

      9. sin-lowering-sin.f64 88.5%

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

   7. Applied egg-rr 88.5%

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

   if 2.39999999999999998e-8 < phi2

   1. Initial program 78.4%

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

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

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -1.06e-13)
     (*
      R
      (-
       (/ PI 2.0)
       (asin (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1)))))))
     (if (<= phi2 1e-9)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2))))))
       (*
        R
        (acos
         (+ t_0 (* (cos phi2) (* (cos phi1) (cos (- lambda1 lambda2)))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= -1.06e-13) {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1))))));
	} else if (phi2 <= 1e-9) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= -1.06e-13) {
		tmp = R * ((Math.PI / 2.0) - Math.asin((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda2 - lambda1))))));
	} else if (phi2 <= 1e-9) {
		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(phi2) * (Math.cos(phi1) * Math.cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[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 <= -1.06e-13:
		tmp = R * ((math.pi / 2.0) - math.asin((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda2 - lambda1))))))
	elif phi2 <= 1e-9:
		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(phi2) * (math.cos(phi1) * math.cos((lambda1 - lambda2))))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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 <= -1.06e-13)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda2 - lambda1)))))));
	elseif (phi2 <= 1e-9)
		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(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda1 - lambda2)))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 <= -1.06e-13)
		tmp = R * ((pi / 2.0) - asin((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1))))));
	elseif (phi2 <= 1e-9)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	else
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * 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.
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, -1.06e-13], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1e-9], 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[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 < -1.06e-13

   1. Initial program 76.4%

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

      1. cos-diff N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.2%

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

      1. acos-asin N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \sin^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   6. Applied egg-rr 76.5%

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

   if -1.06e-13 < phi2 < 1.00000000000000006e-9

   1. Initial program 69.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 69.7%

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

   5. Simplified 69.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   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. *-lowering-*.f64 N/A

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

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

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

      11. cos-lowering-cos.f64 88.5%

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

   7. Applied egg-rr 88.5%

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

   if 1.00000000000000006e-9 < phi2

   1. Initial program 78.4%

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

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

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-115}:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -1.45e-115)
     (* (acos (+ t_1 (* (* (cos phi1) (cos phi2)) t_0))) R)
     (if (<= phi2 4e-10)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2))))))
       (* R (acos (+ t_1 (* (cos phi2) (* (cos phi1) t_0)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -1.45e-115) {
		tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0))) * R;
	} else if (phi2 <= 4e-10) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = sin(phi1) * sin(phi2)
    if (phi2 <= (-1.45d-115)) then
        tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0))) * r
    else if (phi2 <= 4d-10) then
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    else
        tmp = r * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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((lambda1 - lambda2));
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi2 <= -1.45e-115) {
		tmp = Math.acos((t_1 + ((Math.cos(phi1) * Math.cos(phi2)) * t_0))) * R;
	} else if (phi2 <= 4e-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_1 + (Math.cos(phi2) * (Math.cos(phi1) * t_0))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi2 <= -1.45e-115:
		tmp = math.acos((t_1 + ((math.cos(phi1) * math.cos(phi2)) * t_0))) * R
	elif phi2 <= 4e-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_1 + (math.cos(phi2) * (math.cos(phi1) * t_0))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -1.45e-115)
		tmp = Float64(acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))) * R);
	elseif (phi2 <= 4e-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_1 + Float64(cos(phi2) * Float64(cos(phi1) * t_0)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi2 <= -1.45e-115)
		tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0))) * R;
	elseif (phi2 <= 4e-10)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	else
		tmp = R * acos((t_1 + (cos(phi2) * (cos(phi1) * 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.
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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.45e-115], N[(N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 4e-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$1 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.4499999999999999e-115

   1. Initial program 77.7%

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

   if -1.4499999999999999e-115 < phi2 < 4.00000000000000015e-10

   1. Initial program 67.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 67.2%

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

   5. Simplified 67.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   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. *-lowering-*.f64 N/A

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

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

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

      11. cos-lowering-cos.f64 87.4%

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

   7. Applied egg-rr 87.4%

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

   if 4.00000000000000015e-10 < phi2

   1. Initial program 78.4%

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

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

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-115}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 phi2) (* (cos phi1) (cos (- lambda1 lambda2)))))))))
   (if (<= phi2 -1.45e-115)
     t_0
     (if (<= phi2 1.25e-8)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2))))))
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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(phi2) * (cos(phi1) * cos((lambda1 - lambda2))))));
	double tmp;
	if (phi2 <= -1.45e-115) {
		tmp = t_0;
	} else if (phi2 <= 1.25e-8) {
		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.
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(phi2) * (cos(phi1) * cos((lambda1 - lambda2))))))
    if (phi2 <= (-1.45d-115)) then
        tmp = t_0
    else if (phi2 <= 1.25d-8) 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;
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(phi2) * (Math.cos(phi1) * Math.cos((lambda1 - lambda2))))));
	double tmp;
	if (phi2 <= -1.45e-115) {
		tmp = t_0;
	} else if (phi2 <= 1.25e-8) {
		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])
[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(phi2) * (math.cos(phi1) * math.cos((lambda1 - lambda2))))))
	tmp = 0
	if phi2 <= -1.45e-115:
		tmp = t_0
	elif phi2 <= 1.25e-8:
		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])
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(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda1 - lambda2)))))))
	tmp = 0.0
	if (phi2 <= -1.45e-115)
		tmp = t_0;
	elseif (phi2 <= 1.25e-8)
		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])){:}
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(phi2) * (cos(phi1) * cos((lambda1 - lambda2))))));
	tmp = 0.0;
	if (phi2 <= -1.45e-115)
		tmp = t_0;
	elseif (phi2 <= 1.25e-8)
		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.
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[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.45e-115], t$95$0, If[LessEqual[phi2, 1.25e-8], 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 < -1.4499999999999999e-115 or 1.2499999999999999e-8 < 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. Step-by-step derivation

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

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

   4. Applied egg-rr 78.0%

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

   if -1.4499999999999999e-115 < phi2 < 1.2499999999999999e-8

   1. Initial program 67.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 67.2%

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

   5. Simplified 67.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   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. *-lowering-*.f64 N/A

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

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

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

      11. cos-lowering-cos.f64 87.4%

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

   7. Applied egg-rr 87.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-115}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (<= lambda1 -3.1e-6)
     (* R (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda1))))))
     (if (<= lambda1 1e-6)
       (* R (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos lambda2)))))
       (*
        R
        (acos
         (*
          (cos phi2)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 (lambda1 <= -3.1e-6) {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	} else if (lambda1 <= 1e-6) {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	} else {
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (lambda1 <= (-3.1d-6)) then
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))))
    else if (lambda1 <= 1d-6) then
        tmp = r * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))))
    else
        tmp = r * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 (lambda1 <= -3.1e-6) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda1)))));
	} else if (lambda1 <= 1e-6) {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[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 lambda1 <= -3.1e-6:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda1)))))
	elif lambda1 <= 1e-6:
		tmp = R * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	else:
		tmp = R * math.acos((math.cos(phi2) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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 (lambda1 <= -3.1e-6)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda1))))));
	elseif (lambda1 <= 1e-6)
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 (lambda1 <= -3.1e-6)
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	elseif (lambda1 <= 1e-6)
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	else
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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.
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[lambda1, -3.1e-6], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 1e-6], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[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]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.1e-6

   1. Initial program 52.6%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. cos-lowering-cos.f64 52.6%

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

   7. Simplified 52.6%

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

   if -3.1e-6 < lambda1 < 9.99999999999999955e-7

   1. Initial program 88.8%

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

   3. Taylor expanded in 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 88.8%

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

   5. Simplified 88.8%

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

   if 9.99999999999999955e-7 < lambda1

   1. Initial program 60.9%

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

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. cos-lowering-cos.f64 51.3%

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

   7. Simplified 51.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -6.6 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (<= lambda1 -6.6e-6)
     (* R (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda1))))))
     (if (<= lambda1 2.7e-5)
       (* R (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda2))))))
       (*
        R
        (acos
         (*
          (cos phi2)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 (lambda1 <= -6.6e-6) {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	} else if (lambda1 <= 2.7e-5) {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (lambda1 <= (-6.6d-6)) then
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))))
    else if (lambda1 <= 2.7d-5) then
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))))
    else
        tmp = r * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 (lambda1 <= -6.6e-6) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda1)))));
	} else if (lambda1 <= 2.7e-5) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[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 lambda1 <= -6.6e-6:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda1)))))
	elif lambda1 <= 2.7e-5:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi2) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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 (lambda1 <= -6.6e-6)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda1))))));
	elseif (lambda1 <= 2.7e-5)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 (lambda1 <= -6.6e-6)
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	elseif (lambda1 <= 2.7e-5)
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	else
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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.
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[lambda1, -6.6e-6], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 2.7e-5], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[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]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -6.60000000000000034e-6

   1. Initial program 52.6%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. cos-lowering-cos.f64 52.6%

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

   7. Simplified 52.6%

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

   if -6.60000000000000034e-6 < lambda1 < 2.6999999999999999e-5

   1. Initial program 88.8%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. cos-neg N/A

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

      11. cos-lowering-cos.f64 88.8%

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

   5. Simplified 88.8%

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

   if 2.6999999999999999e-5 < lambda1

   1. Initial program 60.9%

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

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. cos-lowering-cos.f64 51.3%

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

   7. Simplified 51.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.6 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.058:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -6e-11)
     (* R (acos (* (cos phi1) t_0)))
     (if (<= phi1 0.058)
       (* 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);
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 <= -6e-11) {
		tmp = R * acos((cos(phi1) * t_0));
	} else if (phi1 <= 0.058) {
		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.
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 <= (-6d-11)) then
        tmp = r * acos((cos(phi1) * t_0))
    else if (phi1 <= 0.058d0) 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;
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 <= -6e-11) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else if (phi1 <= 0.058) {
		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])
[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 <= -6e-11:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	elif phi1 <= 0.058:
		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])
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 <= -6e-11)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	elseif (phi1 <= 0.058)
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda1))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 <= -6e-11)
		tmp = R * acos((cos(phi1) * t_0));
	elseif (phi1 <= 0.058)
		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.
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, -6e-11], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.058], 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[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -6e-11

   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 56.5%

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

   5. Simplified 56.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   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. *-lowering-*.f64 N/A

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

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

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

      11. cos-lowering-cos.f64 70.6%

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

   7. Applied egg-rr 70.6%

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

   if -6e-11 < phi1 < 0.0580000000000000029

   1. Initial program 66.0%

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

      1. cos-diff N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 87.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. cos-lowering-cos.f64 86.9%

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

   7. Simplified 86.9%

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

   if 0.0580000000000000029 < phi1

   1. Initial program 82.0%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. cos-lowering-cos.f64 57.8%

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

   7. Simplified 57.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.058:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (<= phi2 7e-5)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 (phi2 <= 7e-5) {
		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.
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 (phi2 <= 7d-5) 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;
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 (phi2 <= 7e-5) {
		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])
[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 phi2 <= 7e-5:
		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])
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 (phi2 <= 7e-5)
		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])){:}
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 (phi2 <= 7e-5)
		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.
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[phi2, 7e-5], 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 phi2 < 6.9999999999999994e-5

   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. 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 54.2%

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

   5. Simplified 54.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   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. *-lowering-*.f64 N/A

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

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

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

      11. cos-lowering-cos.f64 67.6%

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

   7. Applied egg-rr 67.6%

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

   if 6.9999999999999994e-5 < 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. 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-lft-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. cos-lowering-cos.f64 52.9%

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

   7. Simplified 52.9%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{\frac{\sin \phi_1 \cdot \sin \phi_2}{2}}{\frac{\phi_1}{\cos \left(\lambda_1 - \lambda_2\right) \cdot 2} \cdot \tan \phi_2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 7e-5)
   (*
    R
    (acos
     (*
      (cos phi1)
      (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))))
   (*
    R
    (acos
     (/
      (/ (* (sin phi1) (sin phi2)) 2.0)
      (* (/ phi1 (* (cos (- lambda1 lambda2)) 2.0)) (tan phi2)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 7e-5) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos((((sin(phi1) * sin(phi2)) / 2.0) / ((phi1 / (cos((lambda1 - lambda2)) * 2.0)) * tan(phi2))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= 7d-5) then
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    else
        tmp = r * acos((((sin(phi1) * sin(phi2)) / 2.0d0) / ((phi1 / (cos((lambda1 - lambda2)) * 2.0d0)) * tan(phi2))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= 7e-5) {
		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.sin(phi1) * Math.sin(phi2)) / 2.0) / ((phi1 / (Math.cos((lambda1 - lambda2)) * 2.0)) * Math.tan(phi2))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 7e-5:
		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.sin(phi1) * math.sin(phi2)) / 2.0) / ((phi1 / (math.cos((lambda1 - lambda2)) * 2.0)) * math.tan(phi2))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 7e-5)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(Float64(sin(phi1) * sin(phi2)) / 2.0) / Float64(Float64(phi1 / Float64(cos(Float64(lambda1 - lambda2)) * 2.0)) * tan(phi2)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 <= 7e-5)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	else
		tmp = R * acos((((sin(phi1) * sin(phi2)) / 2.0) / ((phi1 / (cos((lambda1 - lambda2)) * 2.0)) * tan(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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, 7e-5], 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[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] / N[(N[(phi1 / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Tan[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.9999999999999994e-5

   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. 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 54.2%

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

   5. Simplified 54.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   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. *-lowering-*.f64 N/A

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

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

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

      11. cos-lowering-cos.f64 67.6%

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

   7. Applied egg-rr 67.6%

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

   if 6.9999999999999994e-5 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. cos-mult N/A

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

      3. associate-*l/ N/A

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

      4. sin-mult N/A

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

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}{2} + \frac{1}{\frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}}\right)\right), R\right) \]

      6. frac-add N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)} + 2 \cdot 1}{2 \cdot \frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}}\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)} + 2 \cdot 1\right), \left(2 \cdot \frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}\right)\right)\right), R\right) \]

   4. Applied egg-rr 40.3%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}{\phi_1 \cdot \sin \phi_2}\right)}, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\lambda_1 - \lambda_2\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      9. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \left(\cos \phi_2 + \cos \phi_2\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \left(2 \cdot \cos \phi_2\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(2, \cos \phi_2\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\phi_2\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\phi_2\right)\right)\right), \mathsf{*.f64}\left(\phi_1, \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      14. sin-lowering-sin.f64 41.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\phi_2\right)\right)\right), \mathsf{*.f64}\left(\phi_1, \mathsf{sin.f64}\left(\phi_2\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 41.6%

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

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

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\frac{\frac{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(2 \cdot \cos \phi_2\right)}{\phi_1 \cdot \sin \phi_2}}{2 \cdot \frac{1}{\sin \phi_1 \cdot \sin \phi_2}}\right), \color{blue}{R}\right) \]

   9. Applied egg-rr 41.7%

      \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\frac{\sin \phi_1 \cdot \sin \phi_2}{2}}{\frac{\phi_1}{\cos \left(\lambda_1 - \lambda_2\right) \cdot 2} \cdot \tan \phi_2}\right) \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{\frac{\sin \phi_1 \cdot \sin \phi_2}{2}}{\frac{\phi_1}{\cos \left(\lambda_1 - \lambda_2\right) \cdot 2} \cdot \tan \phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.9 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{\frac{\sin \phi_1 \cdot \sin \phi_2}{2}}{\frac{\phi_1}{\cos \left(\lambda_1 - \lambda_2\right) \cdot 2} \cdot \tan \phi_2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4.9e-6)
   (* R (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
   (*
    R
    (acos
     (/
      (/ (* (sin phi1) (sin phi2)) 2.0)
      (* (/ phi1 (* (cos (- lambda1 lambda2)) 2.0)) (tan phi2)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.9e-6) {
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R * acos((((sin(phi1) * sin(phi2)) / 2.0) / ((phi1 / (cos((lambda1 - lambda2)) * 2.0)) * tan(phi2))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= 4.9d-6) then
        tmp = r * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r * acos((((sin(phi1) * sin(phi2)) / 2.0d0) / ((phi1 / (cos((lambda1 - lambda2)) * 2.0d0)) * tan(phi2))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.9e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R * Math.acos((((Math.sin(phi1) * Math.sin(phi2)) / 2.0) / ((phi1 / (Math.cos((lambda1 - lambda2)) * 2.0)) * Math.tan(phi2))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4.9e-6:
		tmp = R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R * math.acos((((math.sin(phi1) * math.sin(phi2)) / 2.0) / ((phi1 / (math.cos((lambda1 - lambda2)) * 2.0)) * math.tan(phi2))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4.9e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(Float64(sin(phi1) * sin(phi2)) / 2.0) / Float64(Float64(phi1 / Float64(cos(Float64(lambda1 - lambda2)) * 2.0)) * tan(phi2)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4.9e-6)
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R * acos((((sin(phi1) * sin(phi2)) / 2.0) / ((phi1 / (cos((lambda1 - lambda2)) * 2.0)) * tan(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4.9e-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[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] / N[(N[(phi1 / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Tan[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.89999999999999967e-6

   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. 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 54.2%

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

   5. Simplified 54.2%

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

   if 4.89999999999999967e-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. Step-by-step derivation

      1. +-commutative N/A

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

      2. cos-mult N/A

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

      3. associate-*l/ N/A

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

      4. sin-mult N/A

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

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}{2} + \frac{1}{\frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}}\right)\right), R\right) \]

      6. frac-add N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)} + 2 \cdot 1}{2 \cdot \frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}}\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)} + 2 \cdot 1\right), \left(2 \cdot \frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}\right)\right)\right), R\right) \]

   4. Applied egg-rr 40.3%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}{\phi_1 \cdot \sin \phi_2}\right)}, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\lambda_1 - \lambda_2\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \left(\cos \phi_2 + \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      9. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \left(\cos \phi_2 + \cos \phi_2\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \left(2 \cdot \cos \phi_2\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(2, \cos \phi_2\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\phi_2\right)\right)\right), \left(\phi_1 \cdot \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\phi_2\right)\right)\right), \mathsf{*.f64}\left(\phi_1, \sin \phi_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

      14. sin-lowering-sin.f64 41.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\phi_2\right)\right)\right), \mathsf{*.f64}\left(\phi_1, \mathsf{sin.f64}\left(\phi_2\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 41.6%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\frac{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(2 \cdot \cos \phi_2\right)}{\phi_1 \cdot \sin \phi_2}}}{2 \cdot \frac{1}{\sin \phi_1 \cdot \sin \phi_2}}\right) \cdot R \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\frac{\frac{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(2 \cdot \cos \phi_2\right)}{\phi_1 \cdot \sin \phi_2}}{2 \cdot \frac{1}{\sin \phi_1 \cdot \sin \phi_2}}\right), \color{blue}{R}\right) \]

   9. Applied egg-rr 41.7%

      \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\frac{\sin \phi_1 \cdot \sin \phi_2}{2}}{\frac{\phi_1}{\cos \left(\lambda_1 - \lambda_2\right) \cdot 2} \cdot \tan \phi_2}\right) \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.9 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{\frac{\sin \phi_1 \cdot \sin \phi_2}{2}}{\frac{\phi_1}{\cos \left(\lambda_1 - \lambda_2\right) \cdot 2} \cdot \tan \phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;t\_0 \leq 0.93:\\ \;\;\;\;R \cdot \cos^{-1} t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (- lambda1 lambda2))))
   (if (<= t_0 0.93)
     (* R (acos t_0))
     (* R (fabs (remainder (- lambda1 lambda2) (* PI 2.0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.93) {
		tmp = R * acos(t_0);
	} else {
		tmp = R * fabs(remainder((lambda1 - lambda2), (((double) M_PI) * 2.0)));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.93) {
		tmp = R * Math.acos(t_0);
	} else {
		tmp = R * Math.abs(Math.IEEEremainder((lambda1 - lambda2), (Math.PI * 2.0)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if t_0 <= 0.93:
		tmp = R * math.acos(t_0)
	else:
		tmp = R * math.fabs(math.remainder((lambda1 - lambda2), (math.pi * 2.0)))
	return tmp

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.93], N[(R * N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(R * N[Abs[N[With[{TMP1 = N[(lambda1 - lambda2), $MachinePrecision], TMP2 = N[(Pi * 2.0), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.930000000000000049

   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 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. 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 36.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 36.0%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   if 0.930000000000000049 < (cos.f64 (-.f64 lambda1 lambda2))

   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. 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 33.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 33.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   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 9.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 9.0%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. cos-acos N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right), R\right) \]

      2. acos-cos N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|\right), R\right) \]

      3. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

      4. remainder-lowering-remainder.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

      5. acos-cos-s N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\left(\lambda_1 - \lambda_2\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{*.f64}\left(2, \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

      8. PI-lowering-PI.f64 22.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right)\right), R\right) \]

   10. Applied egg-rr 22.0%

       \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.93:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 9.5e-6)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 9.5e-6) {
		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.
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 <= 9.5d-6) 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;
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 <= 9.5e-6) {
		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])
[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 <= 9.5e-6:
		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])
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 <= 9.5e-6)
		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])){:}
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 <= 9.5e-6)
		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.
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, 9.5e-6], 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 phi2 < 9.5000000000000005e-6

   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. 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 54.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 54.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   if 9.5000000000000005e-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 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 41.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 41.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;t\_0 \leq 0.99999:\\ \;\;\;\;R \cdot \cos^{-1} t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (- lambda1 lambda2))))
   (if (<= t_0 0.99999) (* R (acos t_0)) (* (- lambda2 lambda1) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.99999) {
		tmp = R * acos(t_0);
	} else {
		tmp = (lambda2 - lambda1) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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((lambda1 - lambda2))
    if (t_0 <= 0.99999d0) then
        tmp = r * acos(t_0)
    else
        tmp = (lambda2 - lambda1) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.99999) {
		tmp = R * Math.acos(t_0);
	} else {
		tmp = (lambda2 - lambda1) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if t_0 <= 0.99999:
		tmp = R * math.acos(t_0)
	else:
		tmp = (lambda2 - lambda1) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (t_0 <= 0.99999)
		tmp = Float64(R * acos(t_0));
	else
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (t_0 <= 0.99999)
		tmp = R * acos(t_0);
	else
		tmp = (lambda2 - lambda1) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.99999], N[(R * N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.999990000000000046

   1. Initial program 73.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 49.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 49.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   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 34.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 34.0%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   if 0.999990000000000046 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 74.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 33.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 33.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   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 5.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 5.0%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 5.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   10. Applied egg-rr 5.0%

       \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

       2. cos-diff N/A

          \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

       3. acos-cos-s N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

       4. --lowering--.f64 16.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

   12. Applied egg-rr 16.4%

       \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.99999:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 1.45e-11)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi1) (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 1.45e-11) {
		tmp = R * acos((cos(phi1) * 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.
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 <= 1.45d-11) then
        tmp = r * acos((cos(phi1) * 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;
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 <= 1.45e-11) {
		tmp = R * Math.acos((Math.cos(phi1) * 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])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.45e-11:
		tmp = R * math.acos((math.cos(phi1) * 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])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.45e-11)
		tmp = Float64(R * acos(Float64(cos(phi1) * 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])){:}
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 <= 1.45e-11)
		tmp = R * acos((cos(phi1) * 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.
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, 1.45e-11], N[(R * N[ArcCos[N[(N[Cos[phi1], $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 lambda2 < 1.45e-11

   1. Initial program 77.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 45.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 45.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \lambda_1\right)\right), R\right) \]

      2. cos-lowering-cos.f64 35.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right), R\right) \]

   8. Simplified 35.1%

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]

   if 1.45e-11 < lambda2

   1. Initial program 62.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 46.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 46.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\cos \lambda_2}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-lowering-cos.f64 46.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right), R\right) \]

   8. Simplified 46.1%

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \lambda_2}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 37.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.85 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 1.85e-20)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (cos (- lambda1 lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 1.85e-20) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} 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.
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 <= 1.85d-20) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos(cos((lambda1 - lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= 1.85e-20) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos(Math.cos((lambda1 - lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.85e-20:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos(math.cos((lambda1 - lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.85e-20)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	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])){:}
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 <= 1.85e-20)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	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.
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, 1.85e-20], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $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 lambda2 < 1.85e-20

   1. Initial program 76.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 46.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 46.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \lambda_1\right)\right), R\right) \]

      2. cos-lowering-cos.f64 35.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right), R\right) \]

   8. Simplified 35.2%

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]

   if 1.85e-20 < lambda2

   1. Initial program 63.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 45.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 45.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   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 29.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 29.3%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.85 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 42.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (acos (* (cos phi1) (cos (- lambda2 lambda1))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos((cos(phi1) * cos((lambda2 - lambda1))));
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 * acos((cos(phi1) * cos((lambda2 - lambda1))))
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))))
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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 45.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

5. Simplified 45.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Final simplification 45.9%

   \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \]
7. Add Preprocessing

Alternative 25: 26.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 0.0005:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 2e-145)
   (* R (acos (cos lambda1)))
   (if (<= lambda2 0.0005)
     (* (- lambda2 lambda1) R)
     (* R (acos (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 2e-145) {
		tmp = R * acos(cos(lambda1));
	} else if (lambda2 <= 0.0005) {
		tmp = (lambda2 - lambda1) * R;
	} 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.
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 <= 2d-145) then
        tmp = r * acos(cos(lambda1))
    else if (lambda2 <= 0.0005d0) then
        tmp = (lambda2 - lambda1) * r
    else
        tmp = r * acos(cos(lambda2))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= 2e-145) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else if (lambda2 <= 0.0005) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * Math.acos(Math.cos(lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 2e-145:
		tmp = R * math.acos(math.cos(lambda1))
	elif lambda2 <= 0.0005:
		tmp = (lambda2 - lambda1) * R
	else:
		tmp = R * math.acos(math.cos(lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 2e-145)
		tmp = Float64(R * acos(cos(lambda1)));
	elseif (lambda2 <= 0.0005)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(cos(lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 <= 2e-145)
		tmp = R * acos(cos(lambda1));
	elseif (lambda2 <= 0.0005)
		tmp = (lambda2 - lambda1) * R;
	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.
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, 2e-145], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 0.0005], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 1.99999999999999983e-145

   1. Initial program 76.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 48.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 48.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   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 29.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 29.2%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   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 17.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right)\right), R\right) \]

   11. Simplified 17.8%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if 1.99999999999999983e-145 < lambda2 < 5.0000000000000001e-4

   1. Initial program 77.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 30.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 30.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   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 15.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 15.8%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 15.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   10. Applied egg-rr 15.8%

       \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

       2. cos-diff N/A

          \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

       3. acos-cos-s N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

       4. --lowering--.f64 21.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

   12. Applied egg-rr 21.5%

       \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

   if 5.0000000000000001e-4 < lambda2

   1. Initial program 62.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 46.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 46.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   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 29.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 29.9%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   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 30.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right)\right), R\right) \]

   11. Simplified 30.0%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 0.0005:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 20.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -20:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -20.0)
   (* R (acos (cos lambda1)))
   (* (- lambda2 lambda1) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -20.0) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = (lambda2 - lambda1) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 - lambda2) <= (-20.0d0)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = (lambda2 - lambda1) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -20.0) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = (lambda2 - lambda1) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= -20.0:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = (lambda2 - lambda1) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -20.0)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -20.0)
		tmp = R * acos(cos(lambda1));
	else
		tmp = (lambda2 - lambda1) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -20.0], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -20

   1. Initial program 73.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 50.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 50.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in 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 32.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 32.0%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   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 18.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right)\right), R\right) \]

   11. Simplified 18.2%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if -20 < (-.f64 lambda1 lambda2)

   1. Initial program 73.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 43.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 43.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   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 25.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 25.6%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 29.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   10. Applied egg-rr 29.8%

       \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

       2. cos-diff N/A

          \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

       3. acos-cos-s N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

       4. --lowering--.f64 6.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

   12. Applied egg-rr 6.4%

       \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -20:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 9.2% accurate, 122.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \left(\lambda_2 - \lambda_1\right) \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (- lambda2 lambda1) R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (lambda2 - lambda1) * R;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = (lambda2 - lambda1) * r
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (lambda2 - lambda1) * R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return (lambda2 - lambda1) * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(Float64(lambda2 - lambda1) * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = (lambda2 - lambda1) * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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 45.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

5. Simplified 45.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

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. Step-by-step derivation

   1. cos-diff N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

   3. fma-define N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

   4. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right), R\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right), R\right) \]

   9. sin-lowering-sin.f64 33.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

10. Applied egg-rr 33.4%

    \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

    2. cos-diff N/A

       \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

    3. acos-cos-s N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

    4. --lowering--.f64 6.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

12. Applied egg-rr 6.1%

    \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]
13. Add Preprocessing

Alternative 28: 3.7% accurate, 204.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_1 \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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);
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.
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;
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])
[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])
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])){:}
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.
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 73.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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 45.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

5. Simplified 45.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

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 lambda1 around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\lambda_1}, R\right) \]
10. Step-by-step derivation

11. Simplified 4.8%

    \[\leadsto \color{blue}{\lambda_1} \cdot R \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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))