Spherical law of cosines

Percentage Accurate: 73.7% → 94.0%

Time: 20.6s

Alternatives: 30

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% 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: 94.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + fma(Float64(Float64(sin(lambda1) * sin(lambda2)) * cos(phi1)), cos(phi2), Float64(cos(phi1) * Float64(cos(phi2) * Float64(cos(lambda1) * cos(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[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 74.7%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

4. Applied rewrites 94.7%

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

Alternative 2: 94.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

4. Applied rewrites 94.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

6. Applied rewrites 94.7%

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

7. Final simplification 94.7%

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

Alternative 3: 94.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 94.7

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

4. Applied rewrites 94.7%

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

5. Final simplification 94.7%

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

Alternative 4: 94.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

4. Applied rewrites 94.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

6. Applied rewrites 94.7%

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

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. cos-neg N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. cos-neg N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 53.8

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

9. Applied rewrites 53.8%

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

10. Taylor expanded in lambda1 around inf

    \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]
11. 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

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

    3. *-commutative N/A

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. distribute-rgt-in N/A

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

12. Applied rewrites 94.6%

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

13. Final simplification 94.6%

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

Alternative 5: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -0.052:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 50000000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + t\_0 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1
         (*
          R
          (acos
           (+ (* (sin phi1) (sin phi2)) (* t_0 (cos (- lambda1 lambda2))))))))
   (if (<= phi2 -0.052)
     t_1
     (if (<= phi2 50000000000.0)
       (*
        R
        (acos
         (+
          (* (sin phi1) (fma phi2 (* -0.16666666666666666 (* phi2 phi2)) phi2))
          (*
           t_0
           (fma
            (sin lambda2)
            (sin lambda1)
            (* (cos lambda1) (cos lambda2)))))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = R * acos(((sin(phi1) * sin(phi2)) + (t_0 * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi2 <= -0.052) {
		tmp = t_1;
	} else if (phi2 <= 50000000000.0) {
		tmp = R * acos(((sin(phi1) * fma(phi2, (-0.16666666666666666 * (phi2 * phi2)), phi2)) + (t_0 * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(t_0 * cos(Float64(lambda1 - lambda2))))))
	tmp = 0.0
	if (phi2 <= -0.052)
		tmp = t_1;
	elseif (phi2 <= 50000000000.0)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * fma(phi2, Float64(-0.16666666666666666 * Float64(phi2 * phi2)), phi2)) + Float64(t_0 * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.052], t$95$1, If[LessEqual[phi2, 50000000000.0], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[(phi2 * N[(-0.16666666666666666 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision] + phi2), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.0519999999999999976 or 5e10 < phi2

   1. Initial program 82.5%

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

   if -0.0519999999999999976 < phi2 < 5e10

   1. Initial program 67.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({\phi_2}^{2} \cdot \sin \phi_1\right) + \sin \phi_1\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\phi_2 \cdot \left(\frac{-1}{6} \cdot \left({\phi_2}^{2} \cdot \sin \phi_1\right)\right) + \phi_2 \cdot \sin \phi_1\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\phi_2 \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \sin \phi_1\right)} + \phi_2 \cdot \sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\color{blue}{\left(\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right)\right) \cdot \sin \phi_1} + \phi_2 \cdot \sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. distribute-rgt-out N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \left(\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) + \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \left(\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) + \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1} \cdot \left(\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) + \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      8. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot {\phi_2}^{2}, \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \color{blue}{\frac{-1}{6} \cdot {\phi_2}^{2}}, \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      10. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \color{blue}{\left(\phi_2 \cdot \phi_2\right)}, \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-*.f64 67.0

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \color{blue}{\left(\phi_2 \cdot \phi_2\right)}, \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   5. Applied rewrites 67.0%

      \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_1} \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \color{blue}{\cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      6. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right)\right) \cdot R \]

      9. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

      11. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \lambda_1 \cdot \sin \lambda_2} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      12. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      13. lower-fma.f64 89.1

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

   7. Applied rewrites 89.1%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.052:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 50000000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_1\right)\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.11:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \left(\sin \phi_2 \cdot \mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t\_0 \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= phi1 -5.2e-13)
     (fma
      (* PI 0.5)
      R
      (*
       (asin (fma (sin phi1) (sin phi2) (* (cos phi1) (* (cos phi2) t_1))))
       (- R)))
     (if (<= phi1 0.11)
       (*
        R
        (acos
         (+
          (*
           t_0
           (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))
          (*
           phi1
           (* (sin phi2) (fma -0.16666666666666666 (* phi1 phi1) 1.0))))))
       (* R (acos (+ (* (sin phi1) (sin phi2)) (* t_0 t_1))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -5.2e-13) {
		tmp = fma((((double) M_PI) * 0.5), R, (asin(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * t_1)))) * -R));
	} else if (phi1 <= 0.11) {
		tmp = R * acos(((t_0 * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))) + (phi1 * (sin(phi2) * fma(-0.16666666666666666, (phi1 * phi1), 1.0)))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (t_0 * t_1)));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -5.2e-13)
		tmp = fma(Float64(pi * 0.5), R, Float64(asin(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * t_1)))) * Float64(-R)));
	elseif (phi1 <= 0.11)
		tmp = Float64(R * acos(Float64(Float64(t_0 * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * Float64(sin(phi2) * fma(-0.16666666666666666, Float64(phi1 * phi1), 1.0))))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(t_0 * t_1))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -5.2e-13], N[(N[(Pi * 0.5), $MachinePrecision] * R + N[(N[ArcSin[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.11], N[(R * N[ArcCos[N[(N[(t$95$0 * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[(N[Sin[phi2], $MachinePrecision] * N[(-0.16666666666666666 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.2000000000000001e-13

   1. Initial program 80.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-acos.f64 N/A

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

      4. acos-asin N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

   if -5.2000000000000001e-13 < phi1 < 0.110000000000000001

   1. Initial program 67.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-cos.f64 90.9

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({\phi_1}^{2} \cdot \sin \phi_2\right) + \sin \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\color{blue}{\left({\phi_1}^{2} \cdot \sin \phi_2\right) \cdot \frac{-1}{6}} + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      3. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\color{blue}{{\phi_1}^{2} \cdot \left(\sin \phi_2 \cdot \frac{-1}{6}\right)} + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left({\phi_1}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin \phi_2\right)} + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      5. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\color{blue}{\left(\phi_1 \cdot \phi_1\right)} \cdot \left(\frac{-1}{6} \cdot \sin \phi_2\right) + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      6. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \left(\frac{-1}{6} \cdot \sin \phi_2\right)\right)} + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\phi_1 \cdot \left(\phi_1 \cdot \color{blue}{\left(\sin \phi_2 \cdot \frac{-1}{6}\right)}\right) + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      8. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\phi_1 \cdot \color{blue}{\left(\left(\phi_1 \cdot \sin \phi_2\right) \cdot \frac{-1}{6}\right)} + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      9. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\phi_1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\phi_1 \cdot \sin \phi_2\right)\right)} + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \left(\frac{-1}{6} \cdot \left(\phi_1 \cdot \sin \phi_2\right)\right) + \sin \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      11. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\phi_1 \cdot \color{blue}{\left(\left(\phi_1 \cdot \sin \phi_2\right) \cdot \frac{-1}{6}\right)} + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      12. associate-*l* N/A

          \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\phi_1 \cdot \color{blue}{\left(\phi_1 \cdot \left(\sin \phi_2 \cdot \frac{-1}{6}\right)\right)} + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      13. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\phi_1 \cdot \left(\phi_1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin \phi_2\right)}\right) + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      14. associate-*l* N/A

          \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\color{blue}{\left(\phi_1 \cdot \phi_1\right) \cdot \left(\frac{-1}{6} \cdot \sin \phi_2\right)} + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

      15. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\color{blue}{{\phi_1}^{2}} \cdot \left(\frac{-1}{6} \cdot \sin \phi_2\right) + \sin \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 90.6%

      \[\leadsto \cos^{-1} \left(\color{blue}{\phi_1 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right) \cdot \sin \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

   if 0.110000000000000001 < phi1

   1. Initial program 85.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.11:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \left(\sin \phi_2 \cdot \mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_1\right)\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= phi1 -5.2e-13)
     (fma
      (* PI 0.5)
      R
      (*
       (asin (fma (sin phi1) (sin phi2) (* (cos phi1) (* (cos phi2) t_1))))
       (- R)))
     (if (<= phi1 0.105)
       (*
        R
        (acos
         (+
          t_0
          (*
           (fma -0.5 (* phi1 phi1) 1.0)
           (*
            (cos phi2)
            (fma
             (cos lambda1)
             (cos lambda2)
             (* (sin lambda1) (sin lambda2))))))))
       (* R (acos (+ t_0 (* (* (cos phi1) (cos phi2)) t_1))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -5.2e-13) {
		tmp = fma((((double) M_PI) * 0.5), R, (asin(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * t_1)))) * -R));
	} else if (phi1 <= 0.105) {
		tmp = R * acos((t_0 + (fma(-0.5, (phi1 * phi1), 1.0) * (cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))))));
	} else {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * t_1)));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -5.2e-13)
		tmp = fma(Float64(pi * 0.5), R, Float64(asin(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * t_1)))) * Float64(-R)));
	elseif (phi1 <= 0.105)
		tmp = Float64(R * acos(Float64(t_0 + Float64(fma(-0.5, Float64(phi1 * phi1), 1.0) * Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * t_1))));
	end
	return tmp
end

Wolfram

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[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -5.2e-13], N[(N[(Pi * 0.5), $MachinePrecision] * R + N[(N[ArcSin[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.105], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Cos[phi2], $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]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.2000000000000001e-13

   1. Initial program 80.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-acos.f64 N/A

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

      4. acos-asin N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

   if -5.2000000000000001e-13 < phi1 < 0.104999999999999996

   1. Initial program 67.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-cos.f64 90.9

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 90.6

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

   7. Applied rewrites 90.6%

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

   if 0.104999999999999996 < phi1

   1. Initial program 85.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 -5.2e-13)
     (fma
      (* PI 0.5)
      R
      (*
       (asin (fma (sin phi1) (sin phi2) (* (cos phi1) (* (cos phi2) t_0))))
       (- R)))
     (if (<= phi1 0.105)
       (*
        R
        (acos
         (fma
          phi1
          (sin phi2)
          (*
           (fma -0.5 (* phi1 phi1) 1.0)
           (*
            (cos phi2)
            (fma
             (cos lambda1)
             (cos lambda2)
             (* (sin lambda1) (sin lambda2))))))))
       (*
        R
        (acos
         (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) t_0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -5.2e-13) {
		tmp = fma((((double) M_PI) * 0.5), R, (asin(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * t_0)))) * -R));
	} else if (phi1 <= 0.105) {
		tmp = R * acos(fma(phi1, sin(phi2), (fma(-0.5, (phi1 * phi1), 1.0) * (cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * t_0)));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -5.2e-13)
		tmp = fma(Float64(pi * 0.5), R, Float64(asin(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * t_0)))) * Float64(-R)));
	elseif (phi1 <= 0.105)
		tmp = Float64(R * acos(fma(phi1, sin(phi2), Float64(fma(-0.5, Float64(phi1 * phi1), 1.0) * Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -5.2e-13], N[(N[(Pi * 0.5), $MachinePrecision] * R + N[(N[ArcSin[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.105], N[(R * N[ArcCos[N[(phi1 * N[Sin[phi2], $MachinePrecision] + N[(N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Cos[phi2], $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]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.2000000000000001e-13

   1. Initial program 80.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-acos.f64 N/A

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

      4. acos-asin N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

   if -5.2000000000000001e-13 < phi1 < 0.104999999999999996

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 90.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

   6. Applied rewrites 90.9%

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

   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. cos-neg N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 48.3

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

   9. Applied rewrites 48.3%

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

   10. Taylor expanded in phi1 around 0

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

   12. Applied rewrites 90.5%

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

   if 0.104999999999999996 < phi1

   1. Initial program 85.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 -5.2e-13)
     (fma
      (* PI 0.5)
      R
      (*
       (asin (fma (sin phi1) (sin phi2) (* (cos phi1) (* (cos phi2) t_0))))
       (- R)))
     (if (<= phi1 0.105)
       (*
        R
        (acos
         (fma
          (cos phi2)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))
          (* phi1 (sin phi2)))))
       (*
        R
        (acos
         (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) t_0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -5.2e-13) {
		tmp = fma((((double) M_PI) * 0.5), R, (asin(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * t_0)))) * -R));
	} else if (phi1 <= 0.105) {
		tmp = R * acos(fma(cos(phi2), fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))), (phi1 * sin(phi2))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * t_0)));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -5.2e-13)
		tmp = fma(Float64(pi * 0.5), R, Float64(asin(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * t_0)))) * Float64(-R)));
	elseif (phi1 <= 0.105)
		tmp = Float64(R * acos(fma(cos(phi2), fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))), Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -5.2e-13], N[(N[(Pi * 0.5), $MachinePrecision] * R + N[(N[ArcSin[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.105], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.2000000000000001e-13

   1. Initial program 80.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-acos.f64 N/A

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

      4. acos-asin N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 80.3%

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

   if -5.2000000000000001e-13 < phi1 < 0.104999999999999996

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 90.4

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

   7. Applied rewrites 90.4%

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

   if 0.104999999999999996 < phi1

   1. Initial program 85.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := \sin \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_1, \cos \phi_1, \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          R
          (acos
           (+
            (* (sin phi1) (sin phi2))
            (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))
        (t_1 (* (sin lambda1) (sin lambda2))))
   (if (<= phi2 -3.7e-14)
     t_0
     (if (<= phi2 8e-15)
       (*
        R
        (acos
         (fma t_1 (cos phi1) (* (cos phi1) (* (cos lambda1) (cos lambda2))))))
       (if (<= phi2 6.5e+109)
         (* R (acos (* (cos phi2) (fma (cos lambda1) (cos lambda2) t_1))))
         t_0)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double t_1 = sin(lambda1) * sin(lambda2);
	double tmp;
	if (phi2 <= -3.7e-14) {
		tmp = t_0;
	} else if (phi2 <= 8e-15) {
		tmp = R * acos(fma(t_1, cos(phi1), (cos(phi1) * (cos(lambda1) * cos(lambda2)))));
	} else if (phi2 <= 6.5e+109) {
		tmp = R * acos((cos(phi2) * fma(cos(lambda1), cos(lambda2), t_1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))))
	t_1 = Float64(sin(lambda1) * sin(lambda2))
	tmp = 0.0
	if (phi2 <= -3.7e-14)
		tmp = t_0;
	elseif (phi2 <= 8e-15)
		tmp = Float64(R * acos(fma(t_1, cos(phi1), Float64(cos(phi1) * Float64(cos(lambda1) * cos(lambda2))))));
	elseif (phi2 <= 6.5e+109)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), t_1))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.7e-14], t$95$0, If[LessEqual[phi2, 8e-15], N[(R * N[ArcCos[N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 6.5e+109], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.70000000000000001e-14 or 6.5e109 < phi2

   1. Initial program 83.8%

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

   if -3.70000000000000001e-14 < phi2 < 8.0000000000000006e-15

   1. Initial program 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 89.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

   6. Applied rewrites 89.9%

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

   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. cos-neg N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 89.9

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

   9. Applied rewrites 89.9%

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

   11. Applied rewrites 89.9%

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

   if 8.0000000000000006e-15 < phi2 < 6.5e109

   1. Initial program 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 72.4

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

   7. Applied rewrites 72.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_1 \cdot \sin \lambda_2, \cos \phi_1, \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_1\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          R
          (acos
           (+
            (* (sin phi1) (sin phi2))
            (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))
        (t_1
         (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
   (if (<= phi2 -3.7e-14)
     t_0
     (if (<= phi2 8e-15)
       (* R (acos (* (cos phi1) t_1)))
       (if (<= phi2 6.5e+109) (* R (acos (* (cos phi2) t_1))) t_0)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double t_1 = fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)));
	double tmp;
	if (phi2 <= -3.7e-14) {
		tmp = t_0;
	} else if (phi2 <= 8e-15) {
		tmp = R * acos((cos(phi1) * t_1));
	} else if (phi2 <= 6.5e+109) {
		tmp = R * acos((cos(phi2) * t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))))
	t_1 = fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))
	tmp = 0.0
	if (phi2 <= -3.7e-14)
		tmp = t_0;
	elseif (phi2 <= 8e-15)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_1)));
	elseif (phi2 <= 6.5e+109)
		tmp = Float64(R * acos(Float64(cos(phi2) * t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.7e-14], t$95$0, If[LessEqual[phi2, 8e-15], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 6.5e+109], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.70000000000000001e-14 or 6.5e109 < phi2

   1. Initial program 83.8%

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

   if -3.70000000000000001e-14 < phi2 < 8.0000000000000006e-15

   1. Initial program 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 89.9%

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

   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. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 89.9

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

   7. Applied rewrites 89.9%

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

   if 8.0000000000000006e-15 < phi2 < 6.5e109

   1. Initial program 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 72.4

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

   7. Applied rewrites 72.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ t_1 := \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_1\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(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))))))))
        (t_1
         (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
   (if (<= phi2 -3.7e-14)
     t_0
     (if (<= phi2 8e-15)
       (* R (acos (* (cos phi1) t_1)))
       (if (<= phi2 6.5e+109) (* R (acos (* (cos phi2) t_1))) t_0)))))

C

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 t_1 = fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)));
	double tmp;
	if (phi2 <= -3.7e-14) {
		tmp = t_0;
	} else if (phi2 <= 8e-15) {
		tmp = R * acos((cos(phi1) * t_1));
	} else if (phi2 <= 6.5e+109) {
		tmp = R * acos((cos(phi2) * t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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)))))))
	t_1 = fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))
	tmp = 0.0
	if (phi2 <= -3.7e-14)
		tmp = t_0;
	elseif (phi2 <= 8e-15)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_1)));
	elseif (phi2 <= 6.5e+109)
		tmp = Float64(R * acos(Float64(cos(phi2) * t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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]}, Block[{t$95$1 = N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.7e-14], t$95$0, If[LessEqual[phi2, 8e-15], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 6.5e+109], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.70000000000000001e-14 or 6.5e109 < phi2

   1. Initial program 83.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lower-fma.f64 83.8

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 83.8

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

   4. Applied rewrites 83.8%

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

   if -3.70000000000000001e-14 < phi2 < 8.0000000000000006e-15

   1. Initial program 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 89.9%

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

   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. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 89.9

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

   7. Applied rewrites 89.9%

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

   if 8.0000000000000006e-15 < phi2 < 6.5e109

   1. Initial program 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 72.4

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

   7. Applied rewrites 72.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1300000.0)
   (*
    R
    (acos
     (*
      (cos phi1)
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))))
   (if (<= phi1 0.105)
     (*
      R
      (acos
       (*
        (cos phi2)
        (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
     (*
      R
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (* (* (cos phi1) (cos phi2)) (cos lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1300000.0) {
		tmp = R * acos((cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))));
	} else if (phi1 <= 0.105) {
		tmp = R * acos((cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1300000.0)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))));
	elseif (phi1 <= 0.105)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1300000.0], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.105], N[(R * N[ArcCos[N[(N[Cos[phi2], $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[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.3e6

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

   6. Applied rewrites 99.1%

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

   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. cos-neg N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 62.4

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

   9. Applied rewrites 62.4%

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

   if -1.3e6 < phi1 < 0.104999999999999996

   1. Initial program 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 87.8

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

   7. Applied rewrites 87.8%

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

   if 0.104999999999999996 < phi1

   1. Initial program 85.1%

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

   3. Taylor expanded in 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 60.2

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

   5. Applied rewrites 60.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1300000.0)
   (*
    R
    (acos
     (*
      (cos phi1)
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))))
   (if (<= phi1 0.105)
     (*
      R
      (acos
       (*
        (cos phi2)
        (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
     (*
      R
      (acos
       (fma
        (cos lambda2)
        (* (cos phi1) (cos phi2))
        (* (sin phi1) (sin phi2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1300000.0) {
		tmp = R * acos((cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))));
	} else if (phi1 <= 0.105) {
		tmp = R * acos((cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(fma(cos(lambda2), (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1300000.0)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))));
	elseif (phi1 <= 0.105)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(fma(cos(lambda2), Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1300000.0], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.105], N[(R * N[ArcCos[N[(N[Cos[phi2], $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[lambda2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.3e6

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

   6. Applied rewrites 99.1%

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

   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. cos-neg N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. cos-neg N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 62.4

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

   9. Applied rewrites 62.4%

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

   if -1.3e6 < phi1 < 0.104999999999999996

   1. Initial program 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 87.8

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

   7. Applied rewrites 87.8%

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

   if 0.104999999999999996 < phi1

   1. Initial program 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 60.2

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

   7. Applied rewrites 60.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
   (if (<= phi1 -1300000.0)
     (* R (acos (* (cos phi1) t_0)))
     (if (<= phi1 0.105)
       (* R (acos (* (cos phi2) t_0)))
       (*
        R
        (acos
         (fma
          (cos lambda2)
          (* (cos phi1) (cos phi2))
          (* (sin phi1) (sin phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)));
	double tmp;
	if (phi1 <= -1300000.0) {
		tmp = R * acos((cos(phi1) * t_0));
	} else if (phi1 <= 0.105) {
		tmp = R * acos((cos(phi2) * t_0));
	} else {
		tmp = R * acos(fma(cos(lambda2), (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))
	tmp = 0.0
	if (phi1 <= -1300000.0)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	elseif (phi1 <= 0.105)
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	else
		tmp = Float64(R * acos(fma(cos(lambda2), Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1300000.0], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.105], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.3e6

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   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. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 62.4

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

   7. Applied rewrites 62.4%

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

   if -1.3e6 < phi1 < 0.104999999999999996

   1. Initial program 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 87.8

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

   7. Applied rewrites 87.8%

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

   if 0.104999999999999996 < phi1

   1. Initial program 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 60.2

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

   7. Applied rewrites 60.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
   (if (<= phi1 -1300000.0)
     (* R (acos (* (cos phi1) t_0)))
     (if (<= phi1 0.105)
       (* R (acos (* (cos phi2) t_0)))
       (*
        R
        (acos
         (fma
          (cos phi2)
          (* (cos phi1) (cos lambda2))
          (* (sin phi1) (sin phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)));
	double tmp;
	if (phi1 <= -1300000.0) {
		tmp = R * acos((cos(phi1) * t_0));
	} else if (phi1 <= 0.105) {
		tmp = R * acos((cos(phi2) * t_0));
	} else {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda2)), (sin(phi1) * sin(phi2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))
	tmp = 0.0
	if (phi1 <= -1300000.0)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	elseif (phi1 <= 0.105)
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	else
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda2)), Float64(sin(phi1) * sin(phi2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1300000.0], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.105], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.3e6

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   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. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 62.4

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

   7. Applied rewrites 62.4%

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

   if -1.3e6 < phi1 < 0.104999999999999996

   1. Initial program 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lift-*.f64 N/A

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      6. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_1}, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      9. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

      12. lower-sin.f64 87.8

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

   7. Applied rewrites 87.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)} \cdot R \]

   if 0.104999999999999996 < phi1

   1. Initial program 85.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 60.2

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 60.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.000195:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, t\_0\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -0.000195)
     (* R (acos (fma (cos phi2) (* (cos phi1) (cos lambda1)) t_0)))
     (if (<= lambda1 7.5e-14)
       (* R (acos (fma (cos phi2) (* (cos phi1) (cos lambda2)) t_0)))
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma
           (cos lambda1)
           (cos lambda2)
           (* (sin lambda1) (sin lambda2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -0.000195) {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda1)), t_0));
	} else if (lambda1 <= 7.5e-14) {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda2)), t_0));
	} else {
		tmp = R * acos((cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -0.000195)
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda1)), t_0)));
	elseif (lambda1 <= 7.5e-14)
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda2)), t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.000195], 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[lambda1, 7.5e-14], 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[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]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.94999999999999996e-4

   1. Initial program 69.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   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 69.6

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 69.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if -1.94999999999999996e-4 < lambda1 < 7.4999999999999996e-14

   1. Initial program 88.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 88.5

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 88.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if 7.4999999999999996e-14 < lambda1

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      4. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      5. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

      6. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      8. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      9. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      10. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1}, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      13. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      14. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      15. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

   4. Applied rewrites 99.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right) \cdot R \]

   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. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

      12. lower-sin.f64 59.5

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

   7. Applied rewrites 59.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.000195:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.000195:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -0.000195)
     (* R (acos (fma (cos phi2) (* (cos phi1) (cos lambda1)) t_0)))
     (* R (acos (fma (cos phi2) (* (cos phi1) (cos lambda2)) t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -0.000195) {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda1)), t_0));
	} else {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda2)), t_0));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -0.000195)
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda1)), t_0)));
	else
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda2)), t_0)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.000195], 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], 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]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.94999999999999996e-4

   1. Initial program 69.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   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 69.6

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 69.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if -1.94999999999999996e-4 < lambda1

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 59.9

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 59.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.000195:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.06 \cdot 10^{+30}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \phi_2 \cdot -0.16666666666666666, \phi_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 4.6e-18)
   (*
    R
    (acos
     (fma (cos phi2) (* (cos phi1) (cos lambda1)) (* (sin phi1) (sin phi2)))))
   (if (<= lambda2 1.06e+30)
     (*
      R
      (fma
       PI
       0.5
       (-
        (asin
         (fma
          (cos phi1)
          (* (cos phi2) (cos (- lambda2 lambda1)))
          (*
           (sin phi1)
           (fma (* phi2 phi2) (* phi2 -0.16666666666666666) phi2)))))))
     (*
      R
      (acos
       (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 4.6e-18) {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda1)), (sin(phi1) * sin(phi2))));
	} else if (lambda2 <= 1.06e+30) {
		tmp = R * fma(((double) M_PI), 0.5, -asin(fma(cos(phi1), (cos(phi2) * cos((lambda2 - lambda1))), (sin(phi1) * fma((phi2 * phi2), (phi2 * -0.16666666666666666), phi2)))));
	} else {
		tmp = R * acos(fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 4.6e-18)
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda1)), Float64(sin(phi1) * sin(phi2)))));
	elseif (lambda2 <= 1.06e+30)
		tmp = Float64(R * fma(pi, 0.5, Float64(-asin(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))), Float64(sin(phi1) * fma(Float64(phi2 * phi2), Float64(phi2 * -0.16666666666666666), phi2)))))));
	else
		tmp = Float64(R * acos(fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 4.6e-18], 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], If[LessEqual[lambda2, 1.06e+30], N[(R * N[(Pi * 0.5 + (-N[ArcSin[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[(N[(phi2 * phi2), $MachinePrecision] * N[(phi2 * -0.16666666666666666), $MachinePrecision] + phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 4.6000000000000002e-18

   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. 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 66.5

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 66.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if 4.6000000000000002e-18 < lambda2 < 1.06e30

   1. Initial program 88.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\color{blue}{\phi_2 \cdot \left(\sin \phi_1 + \frac{-1}{6} \cdot \left({\phi_2}^{2} \cdot \sin \phi_1\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({\phi_2}^{2} \cdot \sin \phi_1\right) + \sin \phi_1\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\phi_2 \cdot \left(\frac{-1}{6} \cdot \left({\phi_2}^{2} \cdot \sin \phi_1\right)\right) + \phi_2 \cdot \sin \phi_1\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\phi_2 \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \sin \phi_1\right)} + \phi_2 \cdot \sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\color{blue}{\left(\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right)\right) \cdot \sin \phi_1} + \phi_2 \cdot \sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. distribute-rgt-out N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \left(\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) + \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \left(\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) + \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1} \cdot \left(\phi_2 \cdot \left(\frac{-1}{6} \cdot {\phi_2}^{2}\right) + \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      8. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot {\phi_2}^{2}, \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \color{blue}{\frac{-1}{6} \cdot {\phi_2}^{2}}, \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      10. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \frac{-1}{6} \cdot \color{blue}{\left(\phi_2 \cdot \phi_2\right)}, \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-*.f64 44.6

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \color{blue}{\left(\phi_2 \cdot \phi_2\right)}, \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   5. Applied rewrites 44.6%

      \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \phi_2 \cdot -0.16666666666666666, \phi_2\right)\right)\right)\right)} \cdot R \]

   if 1.06e30 < lambda2

   1. Initial program 57.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 33.1

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 33.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 26.4%

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 37.6%

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.06 \cdot 10^{+30}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \phi_2 \cdot -0.16666666666666666, \phi_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 50.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 8e-15)
     (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) t_0))))
     (* R (acos (* (cos phi2) t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 8e-15) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0)));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 8d-15) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0)))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 8e-15) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * t_0)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 8e-15:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * t_0)))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 8e-15)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 8e-15)
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0)));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 8e-15], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.0000000000000006e-15

   1. Initial program 72.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1}\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 51.0

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 51.0%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1}\right) \cdot R \]

   if 8.0000000000000006e-15 < phi2

   1. Initial program 80.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 47.7

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 47.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 46.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_0, \cos \phi_1 \cdot \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 8e-15)
     (*
      R
      (acos
       (fma
        t_0
        (* (cos phi1) (fma -0.5 (* phi2 phi2) 1.0))
        (*
         (sin phi1)
         (fma phi2 (* -0.16666666666666666 (* phi2 phi2)) phi2)))))
     (* R (acos (* (cos phi2) t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 8e-15) {
		tmp = R * acos(fma(t_0, (cos(phi1) * fma(-0.5, (phi2 * phi2), 1.0)), (sin(phi1) * fma(phi2, (-0.16666666666666666 * (phi2 * phi2)), phi2))));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 8e-15)
		tmp = Float64(R * acos(fma(t_0, Float64(cos(phi1) * fma(-0.5, Float64(phi2 * phi2), 1.0)), Float64(sin(phi1) * fma(phi2, Float64(-0.16666666666666666 * Float64(phi2 * phi2)), phi2)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 8e-15], N[(R * N[ArcCos[N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[(phi2 * N[(-0.16666666666666666 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision] + phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.0000000000000006e-15

   1. Initial program 72.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_2 \cdot \left(\sin \phi_1 + \phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{6} \cdot \left(\phi_2 \cdot \sin \phi_1\right)\right)\right) + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Applied rewrites 45.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \cos \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right)\right)\right)} \cdot R \]

   if 8.0000000000000006e-15 < phi2

   1. Initial program 80.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 47.7

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 47.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1 \cdot \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 36.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.1 \cdot 10^{+69}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -0.00025:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.1e+69)
   (* R (acos (* (cos phi2) (cos lambda1))))
   (if (<= lambda1 -0.00025)
     (* R (acos (* (cos phi1) (cos lambda1))))
     (* R (acos (* (cos phi2) (cos lambda2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.1e+69) {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	} else if (lambda1 <= -0.00025) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (lambda1 <= (-1.1d+69)) then
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    else if (lambda1 <= (-0.00025d0)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi2) * cos(lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.1e+69) {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	} else if (lambda1 <= -0.00025) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.1e+69:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	elif lambda1 <= -0.00025:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.1e+69)
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	elseif (lambda1 <= -0.00025)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.1e+69)
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	elseif (lambda1 <= -0.00025)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.1e+69], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -0.00025], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.1000000000000001e69

   1. Initial program 71.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 46.0

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 46.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 46.2%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R \]

   if -1.1000000000000001e69 < lambda1 < -2.5000000000000001e-4

   1. Initial program 64.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      4. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      5. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

      6. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      8. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      9. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      10. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1}, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      13. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      14. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      15. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

   4. Applied rewrites 99.4%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right) \cdot R \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)} \cdot R \]

      2. +-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

      3. lift-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. associate-+l+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      5. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right)} \cdot \cos \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1} + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   6. Applied rewrites 99.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \cdot R \]

   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. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right)\right) \cdot R \]

      4. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right) \cdot \cos \phi_1}\right) \cdot R \]

      5. distribute-rgt-out N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right)} \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_1} \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      9. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)}\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\color{blue}{\sin \lambda_2}, \sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2}\right)\right) \cdot R \]

      13. cos-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)\right) \cdot R \]

      14. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)\right) \cdot R \]

      15. lower-cos.f64 58.5

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)\right) \cdot R \]

   9. Applied rewrites 58.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \cdot R \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R \]
   11. Step-by-step derivation

   12. Applied rewrites 44.5%

       \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R \]

   if -2.5000000000000001e-4 < lambda1

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 44.8

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 34.8%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.1 \cdot 10^{+69}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -0.00025:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 50.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 8e-15)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 8e-15) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 8d-15) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 8e-15) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 8e-15:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 8e-15)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 8e-15)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 8e-15], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.0000000000000006e-15

   1. Initial program 72.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 51.5

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 51.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if 8.0000000000000006e-15 < phi2

   1. Initial program 80.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 47.7

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 47.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 48.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1300000.0)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi2) (cos (- lambda2 lambda1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1300000.0) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (phi1 <= (-1300000.0d0)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi2) * cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1300000.0) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1300000.0:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1300000.0)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1300000.0)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1300000.0], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.3e6

   1. Initial program 79.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      4. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      5. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

      6. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      8. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      9. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      10. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1}, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      13. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      14. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      15. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

   4. Applied rewrites 99.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right) \cdot R \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)} \cdot R \]

      2. +-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

      3. lift-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. associate-+l+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      5. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right)} \cdot \cos \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1} + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   6. Applied rewrites 99.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \cdot R \]

   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. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right)\right) \cdot R \]

      4. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right) \cdot \cos \phi_1}\right) \cdot R \]

      5. distribute-rgt-out N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right)} \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_1} \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      9. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)}\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\color{blue}{\sin \lambda_2}, \sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2}\right)\right) \cdot R \]

      13. cos-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)\right) \cdot R \]

      14. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)\right) \cdot R \]

      15. lower-cos.f64 62.4

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)\right) \cdot R \]

   9. Applied rewrites 62.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \cdot R \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R \]
   11. Step-by-step derivation

   12. Applied rewrites 38.1%

       \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R \]

   if -1.3e6 < phi1

   1. Initial program 73.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 51.3

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 51.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1300000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 36.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -7.6 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -7.6e-44)
   (* R (acos (* (cos phi2) (cos lambda1))))
   (* R (acos (* (cos phi1) (cos lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -7.6e-44) {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (lambda1 <= (-7.6d-44)) then
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -7.6e-44) {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -7.6e-44:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -7.6e-44)
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -7.6e-44)
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -7.6e-44], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -7.6000000000000002e-44

   1. Initial program 72.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 44.6

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 43.3%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R \]

   if -7.6000000000000002e-44 < lambda1

   1. Initial program 75.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      4. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      5. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

      6. distribute-lft-in N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      8. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      9. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      10. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1}, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      13. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      14. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right) \cdot \cos \phi_1, \cos \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

      15. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R \]

   4. Applied rewrites 92.6%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right) \cdot R \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)} \cdot R \]

      2. +-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

      3. lift-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. associate-+l+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      5. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right)} \cdot \cos \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)} + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1} + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   6. Applied rewrites 92.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \cdot R \]

   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. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right)\right) \cdot R \]

      4. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1 + \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right) \cdot \cos \phi_1}\right) \cdot R \]

      5. distribute-rgt-out N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right)} \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_1} \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      9. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)}\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\color{blue}{\sin \lambda_2}, \sin \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \color{blue}{\sin \lambda_1}, \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2\right)\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \lambda_2}\right)\right) \cdot R \]

      13. cos-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)\right) \cdot R \]

      14. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_1} \cdot \cos \lambda_2\right)\right) \cdot R \]

      15. lower-cos.f64 49.8

          \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \color{blue}{\cos \lambda_2}\right)\right) \cdot R \]

   9. Applied rewrites 49.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} \cdot R \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right) \cdot R \]
   11. Step-by-step derivation

   12. Applied rewrites 31.1%

       \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -7.6 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 36.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.000235:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -0.000235)
   (* R (acos (* (cos phi2) (cos lambda1))))
   (* R (acos (* (cos phi2) (cos lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.000235) {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (lambda1 <= (-0.000235d0)) then
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi2) * cos(lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.000235) {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -0.000235:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -0.000235)
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -0.000235)
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -0.000235], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.34999999999999993e-4

   1. Initial program 69.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 44.7

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 44.6%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R \]

   if -2.34999999999999993e-4 < lambda1

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 44.8

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 34.8%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.000235:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 34.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1860000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1860000000.0)
   (* R (acos (* (cos phi2) (cos lambda1))))
   (* R (acos (cos (- lambda2 lambda1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1860000000.0) {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	} else {
		tmp = R * acos(cos((lambda2 - lambda1)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (lambda2 <= 1860000000.0d0) then
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    else
        tmp = r * acos(cos((lambda2 - lambda1)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1860000000.0) {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos(Math.cos((lambda2 - lambda1)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1860000000.0:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	else:
		tmp = R * math.acos(math.cos((lambda2 - lambda1)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1860000000.0)
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	else
		tmp = Float64(R * acos(cos(Float64(lambda2 - lambda1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1860000000.0)
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	else
		tmp = R * acos(cos((lambda2 - lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1860000000.0], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.86e9

   1. Initial program 78.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 47.1

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 47.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 40.0%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R \]

   if 1.86e9 < lambda2

   1. Initial program 60.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 36.2

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 36.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 27.3%

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1860000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 21.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.000195:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -0.000195)
   (* R (acos (cos lambda1)))
   (* R (acos (cos lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.000195) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * acos(cos(lambda2));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (lambda1 <= (-0.000195d0)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * acos(cos(lambda2))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.000195) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * Math.acos(Math.cos(lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -0.000195:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * math.acos(math.cos(lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -0.000195)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * acos(cos(lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -0.000195)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * acos(cos(lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -0.000195], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.94999999999999996e-4

   1. Initial program 69.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 44.7

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 33.2%

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 33.3%

       \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

   if -1.94999999999999996e-4 < lambda1

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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 44.8

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 25.2%

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \cos \lambda_2 \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 18.1%

       \[\leadsto \cos^{-1} \cos \lambda_2 \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.000195:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 26.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (acos (cos (- lambda2 lambda1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos(cos((lambda2 - lambda1)));
}

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 = r * acos(cos((lambda2 - lambda1)))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.acos(Math.cos((lambda2 - lambda1)));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.acos(math.cos((lambda2 - lambda1)))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(cos(Float64(lambda2 - lambda1))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * acos(cos((lambda2 - lambda1)));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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 44.7

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

5. Applied rewrites 44.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
7. Step-by-step derivation

8. Applied rewrites 27.4%

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

9. Final simplification 27.4%

   \[\leadsto R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \]
10. Add Preprocessing

Alternative 30: 17.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ R \cdot \cos^{-1} \cos \lambda_1 \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (acos (cos lambda1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos(cos(lambda1));
}

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 = r * acos(cos(lambda1))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.acos(Math.cos(lambda1));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.acos(math.cos(lambda1))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(cos(lambda1)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * acos(cos(lambda1));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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 44.7

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

5. Applied rewrites 44.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
7. Step-by-step derivation

8. Applied rewrites 27.4%

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

9. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot R \]
10. Step-by-step derivation

11. Applied rewrites 20.0%

    \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

12. Final simplification 20.0%

    \[\leadsto R \cdot \cos^{-1} \cos \lambda_1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(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))