Spherical law of cosines

Percentage Accurate: 74.1% → 96.4%

Time: 25.2s

Alternatives: 27

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: 74.1% 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: 96.4% accurate, 0.4× speedup

Math

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

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 (<=
        (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        0.0)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (fma
        (* (cos phi2) (* (cos phi1) (sin lambda1)))
        (sin lambda2)
        (fma
         (cos phi2)
         (* (cos phi1) (* (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 = sin(phi1) * sin(phi2);
	double tmp;
	if (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos(fma((cos(phi2) * (cos(phi1) * sin(lambda1))), sin(lambda2), fma(cos(phi2), (cos(phi1) * (cos(lambda1) * cos(lambda2))), 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 (acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(fma(Float64(cos(phi2) * Float64(cos(phi1) * sin(lambda1))), sin(lambda2), fma(cos(phi2), Float64(cos(phi1) * Float64(cos(lambda1) * cos(lambda2))), 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[N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $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))))) < 0.0

   1. Initial program 3.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 3.4

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

   8. Applied rewrites 3.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff N/A

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

      13. lift--.f64 N/A

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

      14. acos-cos-s 46.8

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

   10. Applied rewrites 46.8%

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

   if 0.0 < (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 77.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 98.6%

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

4. Final simplification 96.4%

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

Alternative 2: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(\cos \phi_1 \cdot \sin \lambda_1\right) \cdot \sin \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (<=
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
      0.0)
   (* (- lambda2 lambda1) R)
   (*
    R
    (acos
     (fma
      (* (cos phi1) (cos lambda1))
      (* (cos phi2) (cos lambda2))
      (fma
       (sin phi1)
       (sin phi2)
       (* (cos phi2) (* (* (cos phi1) (sin lambda1)) (sin 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 (acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos(fma((cos(phi1) * cos(lambda1)), (cos(phi2) * cos(lambda2)), fma(sin(phi1), sin(phi2), (cos(phi2) * ((cos(phi1) * sin(lambda1)) * sin(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 (acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(fma(Float64(cos(phi1) * cos(lambda1)), Float64(cos(phi2) * cos(lambda2)), fma(sin(phi1), sin(phi2), Float64(cos(phi2) * Float64(Float64(cos(phi1) * sin(lambda1)) * sin(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[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], 0.0], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] * N[Sin[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))))) < 0.0

   1. Initial program 3.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 3.4

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

   8. Applied rewrites 3.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff N/A

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

      13. lift--.f64 N/A

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

      14. acos-cos-s 46.8

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

   10. Applied rewrites 46.8%

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

   if 0.0 < (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 77.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 98.6%

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

   8. Applied rewrites 98.6%

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

4. Final simplification 96.4%

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

Alternative 3: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right), \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (<=
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
      0.0)
   (* (- lambda2 lambda1) R)
   (*
    R
    (acos
     (fma
      (sin phi2)
      (sin phi1)
      (fma
       (cos phi1)
       (* (cos phi2) (* (sin lambda1) (sin lambda2)))
       (* (cos lambda1) (* (cos phi1) (* (cos phi2) (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 (acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos(fma(sin(phi2), sin(phi1), fma(cos(phi1), (cos(phi2) * (sin(lambda1) * sin(lambda2))), (cos(lambda1) * (cos(phi1) * (cos(phi2) * 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 (acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), fma(cos(phi1), Float64(cos(phi2) * Float64(sin(lambda1) * sin(lambda2))), Float64(cos(lambda1) * Float64(cos(phi1) * Float64(cos(phi2) * cos(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[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], 0.0], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $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))))) < 0.0

   1. Initial program 3.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 3.4

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

   8. Applied rewrites 3.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff N/A

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

      13. lift--.f64 N/A

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

      14. acos-cos-s 46.8

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

   10. Applied rewrites 46.8%

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

   if 0.0 < (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 77.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 98.6%

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

   8. Applied rewrites 98.6%

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

   10. Applied rewrites 98.6%

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

4. Final simplification 96.4%

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

Alternative 4: 96.4% accurate, 0.4× speedup

Math

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

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 (<=
        (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        0.0)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (fma
        (cos phi2)
        (fma
         (cos phi1)
         (* (sin lambda1) (sin lambda2))
         (* (cos phi1) (* (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 = sin(phi1) * sin(phi2);
	double tmp;
	if (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos(fma(cos(phi2), fma(cos(phi1), (sin(lambda1) * sin(lambda2)), (cos(phi1) * (cos(lambda1) * cos(lambda2)))), 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 (acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(fma(cos(phi2), fma(cos(phi1), Float64(sin(lambda1) * sin(lambda2)), Float64(cos(phi1) * Float64(cos(lambda1) * cos(lambda2)))), 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[N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $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))))) < 0.0

   1. Initial program 3.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 3.4

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

   8. Applied rewrites 3.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff N/A

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

      13. lift--.f64 N/A

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

      14. acos-cos-s 46.8

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

   10. Applied rewrites 46.8%

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

   if 0.0 < (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 77.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 98.6%

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

   8. Applied rewrites 98.6%

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

4. Final simplification 96.4%

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

Alternative 5: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + t\_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

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))))) 0.0)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (+
        t_0
        (*
         t_1
         (fma
          (sin lambda2)
          (sin lambda1)
          (* (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))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos((t_0 + (t_1 * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * 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))))) <= 0.0)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(t_1 * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(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]}, 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], 0.0], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $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))))) < 0.0

   1. Initial program 3.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 3.4

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

   8. Applied rewrites 3.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff N/A

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

      13. lift--.f64 N/A

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

      14. acos-cos-s 46.8

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

   10. Applied rewrites 46.8%

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

   if 0.0 < (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 77.0%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 98.6

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

   4. Applied rewrites 98.6%

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

4. Final simplification 96.4%

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

Alternative 6: 96.4% accurate, 0.4× speedup

Math

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

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 (<=
        (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        0.0)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (fma
        (cos phi2)
        (*
         (cos phi1)
         (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin 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 = sin(phi1) * sin(phi2);
	double tmp;
	if (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))), 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 (acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))), 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[N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $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))))) < 0.0

   1. Initial program 3.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 3.4

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

   8. Applied rewrites 3.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff N/A

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

      13. lift--.f64 N/A

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

      14. acos-cos-s 46.8

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

   10. Applied rewrites 46.8%

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

   if 0.0 < (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 77.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 98.6%

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in phi1 around inf

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out N/A

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

       7. lower-fma.f64 N/A

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

   11. Applied rewrites 98.6%

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

4. Final simplification 96.4%

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

Alternative 7: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(0.5, \pi, \mathsf{fma}\left(-0.5, \pi, \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.7 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 -3.5e-14)
   (*
    R
    (fma
     0.5
     PI
     (fma
      -0.5
      PI
      (acos
       (fma
        (* (cos phi1) (cos phi2))
        (cos (- lambda2 lambda1))
        (* (sin phi1) (sin phi2)))))))
   (if (<= phi2 4.7e-8)
     (*
      R
      (acos
       (fma
        (cos phi1)
        (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2)))
        (* (sin phi1) phi2))))
     (*
      R
      (acos
       (fma
        (sin phi2)
        (sin phi1)
        (* (cos phi1) (* (cos phi2) (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 <= -3.5e-14) {
		tmp = R * fma(0.5, ((double) M_PI), fma(-0.5, ((double) M_PI), acos(fma((cos(phi1) * cos(phi2)), cos((lambda2 - lambda1)), (sin(phi1) * sin(phi2))))));
	} else if (phi2 <= 4.7e-8) {
		tmp = R * acos(fma(cos(phi1), fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2))), (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(fma(sin(phi2), sin(phi1), (cos(phi1) * (cos(phi2) * 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 <= -3.5e-14)
		tmp = Float64(R * fma(0.5, pi, fma(-0.5, pi, acos(fma(Float64(cos(phi1) * cos(phi2)), cos(Float64(lambda2 - lambda1)), Float64(sin(phi1) * sin(phi2)))))));
	elseif (phi2 <= 4.7e-8)
		tmp = Float64(R * acos(fma(cos(phi1), fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))), Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * Float64(cos(phi2) * 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, -3.5e-14], N[(R * N[(0.5 * Pi + N[(-0.5 * Pi + N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 4.7e-8], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.5000000000000002e-14

   1. Initial program 79.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied rewrites 79.9%

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

   6. Applied rewrites 79.4%

      \[\leadsto \mathsf{fma}\left(\pi \cdot 0.5, R, \left(-\color{blue}{\mathsf{fma}\left({\pi}^{0.6666666666666666}, \sqrt[3]{\pi} \cdot 0.5, -\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}\right) \cdot R\right) \]

   7. Taylor expanded in R around -inf

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

   9. Applied rewrites 80.0%

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

   if -3.5000000000000002e-14 < phi2 < 4.6999999999999997e-8

   1. Initial program 67.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 89.3%

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

   6. Applied rewrites 89.4%

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

   7. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 89.4

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

   9. Applied rewrites 89.4%

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

   if 4.6999999999999997e-8 < phi2

   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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.0

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

   4. Applied rewrites 77.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(0.5, \pi, \mathsf{fma}\left(-0.5, \pi, \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.7 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(0.5, \pi, \mathsf{fma}\left(-0.5, \pi, \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.7 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \sin \phi_1, \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 -3.5e-14)
   (*
    R
    (fma
     0.5
     PI
     (fma
      -0.5
      PI
      (acos
       (fma
        (* (cos phi1) (cos phi2))
        (cos (- lambda2 lambda1))
        (* (sin phi1) (sin phi2)))))))
   (if (<= phi2 4.7e-8)
     (*
      R
      (acos
       (fma
        phi2
        (sin phi1)
        (*
         (cos phi1)
         (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))))
     (*
      R
      (acos
       (fma
        (sin phi2)
        (sin phi1)
        (* (cos phi1) (* (cos phi2) (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 <= -3.5e-14) {
		tmp = R * fma(0.5, ((double) M_PI), fma(-0.5, ((double) M_PI), acos(fma((cos(phi1) * cos(phi2)), cos((lambda2 - lambda1)), (sin(phi1) * sin(phi2))))));
	} else if (phi2 <= 4.7e-8) {
		tmp = R * acos(fma(phi2, sin(phi1), (cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = R * acos(fma(sin(phi2), sin(phi1), (cos(phi1) * (cos(phi2) * 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 <= -3.5e-14)
		tmp = Float64(R * fma(0.5, pi, fma(-0.5, pi, acos(fma(Float64(cos(phi1) * cos(phi2)), cos(Float64(lambda2 - lambda1)), Float64(sin(phi1) * sin(phi2)))))));
	elseif (phi2 <= 4.7e-8)
		tmp = Float64(R * acos(fma(phi2, sin(phi1), Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))))));
	else
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * Float64(cos(phi2) * 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, -3.5e-14], N[(R * N[(0.5 * Pi + N[(-0.5 * Pi + N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 4.7e-8], N[(R * N[ArcCos[N[(phi2 * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.5000000000000002e-14

   1. Initial program 79.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied rewrites 79.9%

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

   6. Applied rewrites 79.4%

      \[\leadsto \mathsf{fma}\left(\pi \cdot 0.5, R, \left(-\color{blue}{\mathsf{fma}\left({\pi}^{0.6666666666666666}, \sqrt[3]{\pi} \cdot 0.5, -\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}\right) \cdot R\right) \]

   7. Taylor expanded in R around -inf

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

   9. Applied rewrites 80.0%

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

   if -3.5000000000000002e-14 < phi2 < 4.6999999999999997e-8

   1. Initial program 67.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if 4.6999999999999997e-8 < phi2

   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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.0

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

   4. Applied rewrites 77.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(0.5, \pi, \mathsf{fma}\left(-0.5, \pi, \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.7 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \sin \phi_1, \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(0.5, \pi, \mathsf{fma}\left(-0.5, \pi, \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 -3.5e-14)
   (*
    R
    (fma
     0.5
     PI
     (fma
      -0.5
      PI
      (acos
       (fma
        (* (cos phi1) (cos phi2))
        (cos (- lambda2 lambda1))
        (* (sin phi1) (sin phi2)))))))
   (if (<= phi2 1.05e-8)
     (*
      R
      (acos
       (*
        (cos phi1)
        (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))))
     (*
      R
      (acos
       (fma
        (sin phi2)
        (sin phi1)
        (* (cos phi1) (* (cos phi2) (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 <= -3.5e-14) {
		tmp = R * fma(0.5, ((double) M_PI), fma(-0.5, ((double) M_PI), acos(fma((cos(phi1) * cos(phi2)), cos((lambda2 - lambda1)), (sin(phi1) * sin(phi2))))));
	} else if (phi2 <= 1.05e-8) {
		tmp = R * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos(fma(sin(phi2), sin(phi1), (cos(phi1) * (cos(phi2) * 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 <= -3.5e-14)
		tmp = Float64(R * fma(0.5, pi, fma(-0.5, pi, acos(fma(Float64(cos(phi1) * cos(phi2)), cos(Float64(lambda2 - lambda1)), Float64(sin(phi1) * sin(phi2)))))));
	elseif (phi2 <= 1.05e-8)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * Float64(cos(phi2) * 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, -3.5e-14], N[(R * N[(0.5 * Pi + N[(-0.5 * Pi + N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.05e-8], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.5000000000000002e-14

   1. Initial program 79.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied rewrites 79.9%

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

   6. Applied rewrites 79.4%

      \[\leadsto \mathsf{fma}\left(\pi \cdot 0.5, R, \left(-\color{blue}{\mathsf{fma}\left({\pi}^{0.6666666666666666}, \sqrt[3]{\pi} \cdot 0.5, -\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}\right) \cdot R\right) \]

   7. Taylor expanded in R around -inf

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

   9. Applied rewrites 80.0%

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

   if -3.5000000000000002e-14 < phi2 < 1.04999999999999997e-8

   1. Initial program 67.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 89.3%

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

   6. Applied rewrites 89.4%

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

   7. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 89.0

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

   9. Applied rewrites 89.0%

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

   if 1.04999999999999997e-8 < phi2

   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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.0

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

   4. Applied rewrites 77.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(0.5, \pi, \mathsf{fma}\left(-0.5, \pi, \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, t\_0\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

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 (<= lambda2 -4e+17)
     (*
      R
      (acos
       (*
        (cos phi1)
        (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
     (if (<= lambda2 5.6e-8)
       (* R (acos (fma (cos phi2) (* (cos phi1) (cos lambda1)) t_0)))
       (if (<= lambda2 1.65e+190)
         (* R (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos lambda2)))))
         (*
          R
          (acos
           (*
            (cos phi2)
            (fma
             (sin lambda2)
             (sin lambda1)
             (* (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 (lambda2 <= -4e+17) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	} else if (lambda2 <= 5.6e-8) {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda1)), t_0));
	} else if (lambda2 <= 1.65e+190) {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	} else {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * 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 (lambda2 <= -4e+17)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	elseif (lambda2 <= 5.6e-8)
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda1)), t_0)));
	elseif (lambda2 <= 1.65e+190)
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(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[lambda2, -4e+17], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 5.6e-8], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 1.65e+190], 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[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda2 < -4e17

   1. Initial program 55.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 50.1

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

   7. Applied rewrites 50.1%

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

   if -4e17 < lambda2 < 5.5999999999999999e-8

   1. Initial program 90.0%

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

   3. Taylor expanded in lambda2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 5.5999999999999999e-8 < lambda2 < 1.65e190

   1. Initial program 60.7%

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

   3. Taylor expanded in lambda1 around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if 1.65e190 < lambda2

   1. Initial program 54.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 43.1

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

   5. Applied rewrites 43.1%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 74.2

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

   7. Applied rewrites 74.2%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, t\_0\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

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 (<= lambda2 -4e+17)
     (*
      R
      (acos
       (*
        (cos phi1)
        (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
     (if (<= lambda2 5.6e-8)
       (* R (acos (fma (cos phi2) (* (cos phi1) (cos lambda1)) t_0)))
       (if (<= lambda2 1.65e+190)
         (* R (acos (fma (cos phi2) (* (cos phi1) (cos lambda2)) t_0)))
         (*
          R
          (acos
           (*
            (cos phi2)
            (fma
             (sin lambda2)
             (sin lambda1)
             (* (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 (lambda2 <= -4e+17) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	} else if (lambda2 <= 5.6e-8) {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda1)), t_0));
	} else if (lambda2 <= 1.65e+190) {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda2)), t_0));
	} else {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * 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 (lambda2 <= -4e+17)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	elseif (lambda2 <= 5.6e-8)
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda1)), t_0)));
	elseif (lambda2 <= 1.65e+190)
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda2)), t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(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[lambda2, -4e+17], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 5.6e-8], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 1.65e+190], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda2 < -4e17

   1. Initial program 55.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 50.1

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

   7. Applied rewrites 50.1%

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

   if -4e17 < lambda2 < 5.5999999999999999e-8

   1. Initial program 90.0%

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

   3. Taylor expanded in lambda2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 5.5999999999999999e-8 < lambda2 < 1.65e190

   1. Initial program 60.7%

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

   3. Taylor expanded in lambda1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if 1.65e190 < lambda2

   1. Initial program 54.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 43.1

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

   5. Applied rewrites 43.1%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 74.2

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

   7. Applied rewrites 74.2%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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
           (fma
            (sin phi2)
            (sin phi1)
            (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))))
   (if (<= phi2 -3.5e-14)
     t_0
     (if (<= phi2 1.05e-8)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (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(fma(sin(phi2), sin(phi1), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	double tmp;
	if (phi2 <= -3.5e-14) {
		tmp = t_0;
	} else if (phi2 <= 1.05e-8) {
		tmp = R * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * 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(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))))
	tmp = 0.0
	if (phi2 <= -3.5e-14)
		tmp = t_0;
	elseif (phi2 <= 1.05e-8)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.5e-14], t$95$0, If[LessEqual[phi2, 1.05e-8], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $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 < -3.5000000000000002e-14 or 1.04999999999999997e-8 < phi2

   1. Initial program 78.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 78.6

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 78.6

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

   4. Applied rewrites 78.6%

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

   if -3.5000000000000002e-14 < phi2 < 1.04999999999999997e-8

   1. Initial program 67.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 89.3%

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

   6. Applied rewrites 89.4%

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

   7. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 89.0

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

   9. Applied rewrites 89.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.0039:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= lambda2 -4e+17)
   (*
    R
    (acos
     (*
      (cos phi1)
      (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
   (if (<= lambda2 0.0039)
     (*
      R
      (acos
       (fma
        (cos phi2)
        (* (cos phi1) (cos lambda1))
        (* (sin phi1) (sin phi2)))))
     (*
      R
      (acos
       (*
        (cos phi2)
        (fma (sin lambda2) (sin lambda1) (* (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 tmp;
	if (lambda2 <= -4e+17) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	} else if (lambda2 <= 0.0039) {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda1)), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * 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 <= -4e+17)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	elseif (lambda2 <= 0.0039)
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda1)), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(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[lambda2, -4e+17], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 0.0039], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -4e17

   1. Initial program 55.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-diff N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 50.1

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

   7. Applied rewrites 50.1%

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

   if -4e17 < lambda2 < 0.0038999999999999998

   1. Initial program 89.6%

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

   3. Taylor expanded in lambda2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if 0.0038999999999999998 < lambda2

   1. Initial program 57.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 42.8

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

   5. Applied rewrites 42.8%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R \]

      3. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R \]

      5. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

      6. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \cdot R \]

      8. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \cdot R \]

      9. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

      11. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\sin \lambda_1 \cdot \sin \lambda_2} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      12. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      13. lower-fma.f64 67.2

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

   7. Applied rewrites 67.2%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.0039:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_1 \leq -1.56 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t\_0\right)\right)\\ \end{array} \end{array} \]

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 (* (cos lambda1) (cos lambda2))))
   (if (<= phi1 -1.56e-6)
     (* R (acos (* (cos phi1) (fma (sin lambda1) (sin lambda2) t_0))))
     (* R (acos (* (cos phi2) (fma (sin lambda2) (sin lambda1) 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) * cos(lambda2);
	double tmp;
	if (phi1 <= -1.56e-6) {
		tmp = R * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), t_0)));
	} else {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), 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(cos(lambda1) * cos(lambda2))
	tmp = 0.0
	if (phi1 <= -1.56e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), t_0))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), 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[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.56e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.5600000000000001e-6

   1. Initial program 74.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      8. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]

      9. associate-+r+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

      10. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) \cdot R \]

      11. lower-+.f64 N/A

          \[\leadsto \cos^{-1} \color{blue}{\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)} \cdot R \]

   4. Applied rewrites 98.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)} \cdot R \]

   6. Applied rewrites 98.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \sin \lambda_1\right), \sin \lambda_2, \mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \cdot R \]

   7. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right) + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right)\right)} \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1} + \cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right)\right) \cdot R \]

      3. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1}\right) \cdot R \]

      4. distribute-rgt-out N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \cdot R \]

      5. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_1} \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      7. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      9. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \color{blue}{\sin \lambda_2}, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \color{blue}{\cos \lambda_1 \cdot \cos \lambda_2}\right)\right) \cdot R \]

      11. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)\right) \cdot R \]

      12. lower-cos.f64 59.4

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)\right) \cdot R \]

   9. Applied rewrites 59.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \cdot R \]

   if -1.5600000000000001e-6 < phi1

   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 phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 50.8

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 50.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   6. Step-by-step derivation

      1. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R \]

      3. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R \]

      5. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

      6. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \cdot R \]

      8. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \cdot R \]

      9. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

      11. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\sin \lambda_1 \cdot \sin \lambda_2} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      12. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      13. lower-fma.f64 64.9

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

   7. Applied rewrites 64.9%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.56 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.102:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 0.102)
   (*
    R
    (acos
     (*
      (cos phi1)
      (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))))
   (* R (acos (* (cos phi2) (cos (- lambda2 lambda1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 0.102) {
		tmp = R * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - 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)
	tmp = 0.0
	if (phi2 <= 0.102)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

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, 0.102], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.101999999999999993

   1. Initial program 72.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      8. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]

      9. associate-+r+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

      10. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) \cdot R \]

      11. lower-+.f64 N/A

          \[\leadsto \cos^{-1} \color{blue}{\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)} \cdot R \]

   4. Applied rewrites 93.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)} \cdot R \]

   6. Applied rewrites 93.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \sin \lambda_1\right), \sin \lambda_2, \mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \cdot R \]

   7. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right) + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right)\right)} \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1} + \cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right)\right) \cdot R \]

      3. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1}\right) \cdot R \]

      4. distribute-rgt-out N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \cdot R \]

      5. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_1} \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      7. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\color{blue}{\sin \lambda_1}, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      9. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \color{blue}{\sin \lambda_2}, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \color{blue}{\cos \lambda_1 \cdot \cos \lambda_2}\right)\right) \cdot R \]

      11. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)\right) \cdot R \]

      12. lower-cos.f64 59.6

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)\right) \cdot R \]

   9. Applied rewrites 59.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \cdot R \]

   if 0.101999999999999993 < phi2

   1. Initial program 77.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 52.6

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 52.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.102:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.102:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 0.102)
   (*
    R
    (acos
     (*
      (cos phi1)
      (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
   (* R (acos (* (cos phi2) (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) {
	double tmp;
	if (phi2 <= 0.102) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - 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)
	tmp = 0.0
	if (phi2 <= 0.102)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	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, 0.102], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.101999999999999993

   1. Initial program 72.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      8. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]

      9. associate-+r+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

      10. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) \cdot R \]

      11. lower-+.f64 N/A

          \[\leadsto \cos^{-1} \color{blue}{\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)} \cdot R \]

   4. Applied rewrites 93.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)} \cdot R \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right) + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1} + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1 + \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1}\right) \cdot R \]

      3. distribute-rgt-out N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_1} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      6. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_1}, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      11. lower-sin.f64 59.6

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right)\right) \cdot R \]

   7. Applied rewrites 59.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

   if 0.101999999999999993 < phi2

   1. Initial program 77.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 52.6

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 52.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.102:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -0.028:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.102:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2, \phi_2 \cdot -0.5, 1\right), \cos \phi_1 \cdot t\_0, \sin \phi_1 \cdot \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

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 -0.028)
     (* R (acos (fma (cos phi2) (cos phi1) (* (sin phi1) (sin phi2)))))
     (if (<= phi2 0.102)
       (*
        R
        (acos
         (fma
          (fma phi2 (* phi2 -0.5) 1.0)
          (* (cos phi1) t_0)
          (* (sin phi1) phi2))))
       (* 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 <= -0.028) {
		tmp = R * acos(fma(cos(phi2), cos(phi1), (sin(phi1) * sin(phi2))));
	} else if (phi2 <= 0.102) {
		tmp = R * acos(fma(fma(phi2, (phi2 * -0.5), 1.0), (cos(phi1) * t_0), (sin(phi1) * phi2)));
	} else {
		tmp = R * acos((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 <= -0.028)
		tmp = Float64(R * acos(fma(cos(phi2), cos(phi1), Float64(sin(phi1) * sin(phi2)))));
	elseif (phi2 <= 0.102)
		tmp = Float64(R * acos(fma(fma(phi2, Float64(phi2 * -0.5), 1.0), Float64(cos(phi1) * t_0), Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * 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[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.028], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.102], N[(R * N[ArcCos[N[(N[(phi2 * N[(phi2 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0280000000000000006

   1. Initial program 79.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      10. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      12. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      13. lower-sin.f64 58.7

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 58.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      8. lower-sin.f64 41.4

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   8. Applied rewrites 41.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if -0.0280000000000000006 < phi2 < 0.101999999999999993

   1. Initial program 68.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_2 \cdot \left(\sin \phi_1 + \frac{-1}{2} \cdot \left(\phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\phi_2 \cdot \sin \phi_1 + \phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)} + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. associate-+l+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\phi_2 \cdot \sin \phi_1 + \left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      3. +-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \phi_2 \cdot \sin \phi_1\right)} \cdot R \]

      4. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\phi_2 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \phi_2\right) \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \phi_2 \cdot \sin \phi_1\right) \cdot R \]

      5. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\color{blue}{\left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \phi_2\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \phi_2 \cdot \sin \phi_1\right) \cdot R \]

      6. distribute-lft1-in N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \phi_2\right) + 1\right) \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} + \phi_2 \cdot \sin \phi_1\right) \cdot R \]

      7. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \phi_2\right) + 1, \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right), \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   5. Applied rewrites 68.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2, \phi_2 \cdot -0.5, 1\right), \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if 0.101999999999999993 < phi2

   1. Initial program 77.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 52.6

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 52.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.028:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.102:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2, \phi_2 \cdot -0.5, 1\right), \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -8.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -8.8e-7)
     (* 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 (phi1 <= -8.8e-7) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi1 <= (-8.8d-7)) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -8.8e-7) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -8.8e-7:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -8.8e-7)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -8.8e-7)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -8.8e-7], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.8000000000000004e-7

   1. Initial program 74.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 49.4

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 49.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if -8.8000000000000004e-7 < phi1

   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 phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 50.8

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 50.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 53.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00028:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.00028)
   (* R (acos (* (cos phi1) (cos lambda2))))
   (* R (acos (* (cos phi2) (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) {
	double tmp;
	if (phi1 <= -0.00028) {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-0.00028d0)) then
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    else
        tmp = r * acos((cos(phi2) * cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.00028) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.00028:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.00028)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.00028)
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	else
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.00028], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.7999999999999998e-4

   1. Initial program 74.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      10. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      12. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      13. lower-sin.f64 62.2

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 62.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 44.4

         \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   8. Applied rewrites 44.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]

   if -2.7999999999999998e-4 < phi1

   1. Initial program 73.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 50.8

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 50.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00028:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;t\_0 \leq 0.99:\\ \;\;\;\;R \cdot \cos^{-1} t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \end{array} \]

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 0.99)
     (* 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 <= 0.99) {
		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 <= 0.99) {
		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 <= 0.99:
		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, 0.99], 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)) < 0.98999999999999999

   1. Initial program 71.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 45.5

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 45.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 30.8

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 30.8%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   if 0.98999999999999999 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 79.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 35.3

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 35.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 7.0

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 7.0%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      2. acos-cos N/A

         \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      3. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      4. lower-remainder.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right| \cdot R \]

      5. lift-PI.f64 N/A

         \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right| \cdot R \]

      6. *-commutative N/A

         \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)\right| \cdot R \]

      7. lower-*.f64 17.5

         \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\pi \cdot 2\right)}\right)\right| \cdot R \]

   10. Applied rewrites 17.5%

       \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.99:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.15e-8)
   (* R (acos (* (cos phi1) (cos lambda2))))
   (* R (acos (* (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 tmp;
	if (phi2 <= 2.15e-8) {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= 2.15d-8) then
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    else
        tmp = r * acos((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 tmp;
	if (phi2 <= 2.15e-8) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.15e-8:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.15e-8)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 <= 2.15e-8)
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	else
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.15e-8], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.1500000000000001e-8

   1. Initial program 72.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      10. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      12. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      13. lower-sin.f64 53.9

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 53.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 35.7

         \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   8. Applied rewrites 35.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]

   if 2.1500000000000001e-8 < phi2

   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 phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 52.2

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 52.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]

      2. lower-cos.f64 39.9

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]

   8. Applied rewrites 39.9%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 37.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

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 (<= phi1 -4.8e-6)
   (* R (acos (* (cos phi1) (cos lambda2))))
   (* 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 (phi1 <= -4.8e-6) {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	} 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 (phi1 <= (-4.8d-6)) then
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    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 (phi1 <= -4.8e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	} 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 phi1 <= -4.8e-6:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	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 (phi1 <= -4.8e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	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 (phi1 <= -4.8e-6)
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	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[phi1, -4.8e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $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 phi1 < -4.7999999999999998e-6

   1. Initial program 74.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      10. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      12. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      13. lower-sin.f64 61.6

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 61.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 43.9

         \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   8. Applied rewrites 43.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]

   if -4.7999999999999998e-6 < phi1

   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 phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 50.8

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 50.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 26.1

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 26.1%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 24.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.65:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

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 1.65)
   (* R (fabs (remainder (- lambda1 lambda2) (* PI 2.0))))
   (* 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 <= 1.65) {
		tmp = R * fabs(remainder((lambda1 - lambda2), (((double) M_PI) * 2.0)));
	} else {
		tmp = R * acos(cos(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 tmp;
	if (lambda2 <= 1.65) {
		tmp = R * Math.abs(Math.IEEEremainder((lambda1 - lambda2), (Math.PI * 2.0)));
	} 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 <= 1.65:
		tmp = R * math.fabs(math.remainder((lambda1 - lambda2), (math.pi * 2.0)))
	else:
		tmp = R * math.acos(math.cos(lambda2))
	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_] := If[LessEqual[lambda2, 1.65], 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], N[(R * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.6499999999999999

   1. Initial program 79.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 41.9

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 41.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 21.1

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 21.1%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      2. acos-cos N/A

         \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      3. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      4. lower-remainder.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right| \cdot R \]

      5. lift-PI.f64 N/A

         \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right| \cdot R \]

      6. *-commutative N/A

         \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)\right| \cdot R \]

      7. lower-*.f64 19.1

         \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\pi \cdot 2\right)}\right)\right| \cdot R \]

   10. Applied rewrites 19.1%

       \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|} \cdot R \]

   if 1.6499999999999999 < lambda2

   1. Initial program 57.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 42.8

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 42.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 28.9

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 28.9%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot R \]
   10. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]

       2. lower-cos.f64 29.0

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]

   11. Applied rewrites 29.0%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.65:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 24.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -13000000000000:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \end{array} \]

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
 (if (<= lambda1 -13000000000000.0)
   (* R (acos (cos lambda1)))
   (* 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 tmp;
	if (lambda1 <= -13000000000000.0) {
		tmp = R * acos(cos(lambda1));
	} 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 tmp;
	if (lambda1 <= -13000000000000.0) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} 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):
	tmp = 0
	if lambda1 <= -13000000000000.0:
		tmp = R * math.acos(math.cos(lambda1))
	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_] := If[LessEqual[lambda1, -13000000000000.0], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $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 lambda1 < -1.3e13

   1. Initial program 59.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 41.5

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 41.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 28.0

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 28.0%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]
   10. Step-by-step derivation

       1. lower-cos.f64 28.1

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   11. Applied rewrites 28.1%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if -1.3e13 < lambda1

   1. Initial program 78.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 42.3

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 42.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 21.5

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 21.5%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      2. acos-cos N/A

         \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      3. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      4. lower-remainder.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right| \cdot R \]

      5. lift-PI.f64 N/A

         \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right| \cdot R \]

      6. *-commutative N/A

         \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)\right| \cdot R \]

      7. lower-*.f64 19.2

         \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\pi \cdot 2\right)}\right)\right| \cdot R \]

   10. Applied rewrites 19.2%

       \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -13000000000000:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 19.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right| \end{array} \]

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 (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) {
	return R * fabs(remainder((lambda1 - lambda2), (((double) M_PI) * 2.0)));
}

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.abs(Math.IEEEremainder((lambda1 - lambda2), (Math.PI * 2.0)));
}

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.fabs(math.remainder((lambda1 - lambda2), (math.pi * 2.0)))

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[Abs[N[With[{TMP1 = N[(lambda1 - lambda2), $MachinePrecision], TMP2 = N[(Pi * 2.0), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   12. lower--.f64 42.1

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

5. Applied rewrites 42.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 23.0

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Applied rewrites 23.0%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
9. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   2. acos-cos N/A

      \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

   3. lower-fabs.f64 N/A

      \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

   4. lower-remainder.f64 N/A

      \[\leadsto \left|\color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right| \cdot R \]

   5. lift-PI.f64 N/A

      \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right| \cdot R \]

   6. *-commutative N/A

      \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)\right| \cdot R \]

   7. lower-*.f64 19.3

      \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\pi \cdot 2\right)}\right)\right| \cdot R \]

10. Applied rewrites 19.3%

    \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|} \cdot R \]

11. Final simplification 19.3%

    \[\leadsto R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right| \]
12. Add Preprocessing

Alternative 26: 9.1% accurate, 69.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \left(\lambda_2 - \lambda_1\right) \cdot R \end{array} \]

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 73.9%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   12. lower--.f64 42.1

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

5. Applied rewrites 42.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 23.0

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Applied rewrites 23.0%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
9. Step-by-step derivation

   1. cos-diff N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot R \]

   2. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   4. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   5. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   6. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   7. lift-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right) \cdot R \]

   8. lift-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right) \cdot R \]

   9. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right) \cdot R \]

   10. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right) \cdot R \]

   11. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right) \cdot R \]

   12. cos-diff N/A

       \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]

   13. lift--.f64 N/A

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

   14. acos-cos-s 5.9

       \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

10. Applied rewrites 5.9%

    \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]
11. Add Preprocessing

Alternative 27: 3.6% accurate, 104.5× 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 73.9%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   12. lower--.f64 42.1

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

5. Applied rewrites 42.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 23.0

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Applied rewrites 23.0%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

9. Taylor expanded in lambda1 around inf

   \[\leadsto \color{blue}{R \cdot \lambda_1} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{\lambda_1 \cdot R} \]

    2. lower-*.f64 4.7

       \[\leadsto \color{blue}{\lambda_1 \cdot R} \]

11. Applied rewrites 4.7%

    \[\leadsto \color{blue}{\lambda_1 \cdot R} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))