Spherical law of cosines

Percentage Accurate: 73.9% → 94.3%

Time: 20.3s

Alternatives: 22

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: 73.9% 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: 94.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (fma
     (* (* (sin lambda1) (sin lambda2)) (cos phi2))
     (cos phi1)
     (* (* (cos phi2) (cos phi1)) (* (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(phi2)), cos(phi1), ((cos(phi2) * cos(phi1)) * (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(phi2)), cos(phi1), Float64(Float64(cos(phi2) * cos(phi1)) * 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[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 76.0%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-cos.f64 92.1

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

4. Applied egg-rr 92.1%

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

   1. lift-cos.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-rgt-in N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

6. Applied egg-rr 92.1%

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

7. Final simplification 92.1%

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

Alternative 2: 94.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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(((sin(phi1) * sin(phi2)) + ((cos(phi2) * cos(phi1)) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 76.0%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-cos.f64 92.1

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

4. Applied egg-rr 92.1%

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

5. Final simplification 92.1%

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

Alternative 3: 94.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (*
     (* (cos phi2) (cos phi1))
     (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(phi2) * cos(phi1)) * 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(phi2) * cos(phi1)) * 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[phi2], $MachinePrecision] * N[Cos[phi1], $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 76.0%

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

   1. cos-diff N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-sin.f64 92.1

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

4. Applied egg-rr 92.1%

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

5. Final simplification 92.1%

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

Alternative 4: 94.3% accurate, 0.7× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 76.0%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-cos.f64 92.1

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

4. Applied egg-rr 92.1%

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

5. Taylor expanded in phi1 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. cos-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. cos-neg N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-sin.f64 92.1

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

7. Simplified 92.1%

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

8. Final simplification 92.1%

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

Alternative 5: 94.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (acos
   (fma
    (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))
    (* (cos phi2) (cos phi1))
    (* (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(phi2) * cos(phi1)), (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(phi2) * cos(phi1)), 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[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-cos.f64 92.1

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

4. Applied egg-rr 92.1%

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

5. Taylor expanded in phi1 around 0

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

   1. lower-acos.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

7. Simplified 92.1%

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

8. Final simplification 92.1%

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

Alternative 6: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\lambda_1 + \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_2 + t\_0 \cdot \frac{1}{\frac{t\_1}{t\_1 \cdot t\_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(t\_3, t\_0, t\_2\right)\right) \cdot \left(-R\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (cos (+ lambda1 lambda2)))
        (t_2 (* (sin phi1) (sin phi2)))
        (t_3 (cos (- lambda1 lambda2))))
   (if (<= phi2 -3.6e-31)
     (* R (acos (+ t_2 (* t_0 (/ 1.0 (/ t_1 (* t_1 t_3)))))))
     (if (<= phi2 2.2e-13)
       (*
        R
        (acos
         (+
          t_2
          (*
           (cos phi1)
           (fma
            (cos lambda1)
            (cos lambda2)
            (* (sin lambda1) (sin lambda2)))))))
       (fma (* PI 0.5) R (* (asin (fma t_3 t_0 t_2)) (- R)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = cos((lambda1 + lambda2));
	double t_2 = sin(phi1) * sin(phi2);
	double t_3 = cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -3.6e-31) {
		tmp = R * acos((t_2 + (t_0 * (1.0 / (t_1 / (t_1 * t_3))))));
	} else if (phi2 <= 2.2e-13) {
		tmp = R * acos((t_2 + (cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = fma((((double) M_PI) * 0.5), R, (asin(fma(t_3, t_0, t_2)) * -R));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = cos(Float64(lambda1 + lambda2))
	t_2 = Float64(sin(phi1) * sin(phi2))
	t_3 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= -3.6e-31)
		tmp = Float64(R * acos(Float64(t_2 + Float64(t_0 * Float64(1.0 / Float64(t_1 / Float64(t_1 * t_3)))))));
	elseif (phi2 <= 2.2e-13)
		tmp = Float64(R * acos(Float64(t_2 + Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))))));
	else
		tmp = fma(Float64(pi * 0.5), R, Float64(asin(fma(t_3, t_0, t_2)) * Float64(-R)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 + lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -3.6e-31], N[(R * N[ArcCos[N[(t$95$2 + N[(t$95$0 * N[(1.0 / N[(t$95$1 / N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.2e-13], N[(R * N[ArcCos[N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] * R + N[(N[ArcSin[N[(t$95$3 * t$95$0 + t$95$2), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.60000000000000004e-31

   1. Initial program 84.2%

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

      1. cos-diff N/A

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

      2. flip-+ N/A

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

      3. cos-sum N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. difference-of-squares N/A

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

   4. Applied egg-rr 84.2%

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

   if -3.60000000000000004e-31 < phi2 < 2.19999999999999997e-13

   1. Initial program 68.7%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 84.6

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

   4. Applied egg-rr 84.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. cos-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 84.6

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

   7. Simplified 84.6%

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

   if 2.19999999999999997e-13 < phi2

   1. Initial program 81.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied egg-rr 81.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi \cdot 0.5, R, \left(-\sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right) \cdot R\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \frac{1}{\frac{\cos \left(\lambda_1 + \lambda_2\right)}{\cos \left(\lambda_1 + \lambda_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.18 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_0, t\_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(t\_0, \cos \phi_2 \cdot \cos \phi_1, t\_1\right)\right) \cdot \left(-R\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -1.18e-7)
     (* R (acos (fma (cos phi1) (* (cos phi2) t_0) t_1)))
     (if (<= phi2 2.2e-13)
       (*
        R
        (acos
         (+
          t_1
          (*
           (cos phi1)
           (fma
            (cos lambda1)
            (cos lambda2)
            (* (sin lambda1) (sin lambda2)))))))
       (fma
        (* PI 0.5)
        R
        (* (asin (fma t_0 (* (cos phi2) (cos phi1)) t_1)) (- R)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -1.18e-7) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * t_0), t_1));
	} else if (phi2 <= 2.2e-13) {
		tmp = R * acos((t_1 + (cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = fma((((double) M_PI) * 0.5), R, (asin(fma(t_0, (cos(phi2) * cos(phi1)), t_1)) * -R));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -1.18e-7)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * t_0), t_1)));
	elseif (phi2 <= 2.2e-13)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))))));
	else
		tmp = fma(Float64(pi * 0.5), R, Float64(asin(fma(t_0, Float64(cos(phi2) * cos(phi1)), t_1)) * Float64(-R)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.18e-7], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.2e-13], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] * R + N[(N[ArcSin[N[(t$95$0 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.18e-7

   1. Initial program 83.0%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 99.0

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

   4. Applied egg-rr 99.0%

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

   6. Applied egg-rr 83.0%

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

   if -1.18e-7 < phi2 < 2.19999999999999997e-13

   1. Initial program 69.9%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 85.2

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

   4. Applied egg-rr 85.2%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. cos-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 85.2

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

   7. Simplified 85.2%

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

   if 2.19999999999999997e-13 < phi2

   1. Initial program 81.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied egg-rr 81.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi \cdot 0.5, R, \left(-\sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right) \cdot R\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.18 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 1.1 \cdot 10^{+58}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 1.1e+58)
   (*
    R
    (acos
     (fma
      (cos phi1)
      (* (cos phi2) (cos (- lambda1 lambda2)))
      (* (sin phi1) (sin phi2)))))
   (*
    R
    (acos
     (+
      (* phi1 (sin phi2))
      (*
       (cos phi1)
       (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 (lambda1 <= 1.1e+58) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos((lambda1 - lambda2))), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= 1.1e+58)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, 1.1e+58], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 1.1e58

   1. Initial program 80.2%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 90.2

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

   4. Applied egg-rr 90.2%

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

   6. Applied egg-rr 80.2%

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

   if 1.1e58 < lambda1

   1. Initial program 60.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 42.9

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

   5. Simplified 42.9%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 29.9

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

   8. Simplified 29.9%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 40.3

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

   10. Applied egg-rr 40.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 1.1 \cdot 10^{+58}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2.35 \cdot 10^{+113}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 2.35e+113)
   (*
    R
    (acos
     (fma
      (cos phi1)
      (* (cos phi2) (cos (- lambda1 lambda2)))
      (* (sin phi1) (sin phi2)))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))
      (* phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= 2.35e+113) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos((lambda1 - lambda2))), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos(((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))) + (phi1 * phi2)));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= 2.35e+113)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2)))) + Float64(phi1 * phi2))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, 2.35e+113], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 2.3499999999999999e113

   1. Initial program 79.0%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 90.9

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

   4. Applied egg-rr 90.9%

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

   6. Applied egg-rr 79.0%

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

   if 2.3499999999999999e113 < lambda1

   1. Initial program 59.3%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 40.3

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

   5. Simplified 40.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 30.0

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

   8. Simplified 30.0%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 27.4

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

   11. Simplified 27.4%

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

       1. cos-diff N/A

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

       2. lift-cos.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift-sin.f64 N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 39.7

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

   13. Applied egg-rr 39.7%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2.35 \cdot 10^{+113}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.4e-8)
   (*
    R
    (acos
     (fma (sin phi1) (sin phi2) (* (cos phi1) (* (cos phi2) (cos lambda1))))))
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (cos phi2) (cos phi1)) (cos lambda2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.4e-8) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi2) * cos(phi1)) * cos(lambda2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.4e-8)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda1))))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi2) * cos(phi1)) * cos(lambda2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.4e-8], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $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[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.4e-8

   1. Initial program 82.2%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 70.8

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

   5. Simplified 70.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-acos.f64 70.8

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 70.9

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   7. Applied egg-rr 70.9%

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

   if 1.4e-8 < lambda2

   1. Initial program 60.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 60.5

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

   5. Simplified 60.5%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(0.5 \cdot 0 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 8.2e-7)
   (*
    R
    (acos
     (fma (sin phi1) (sin phi2) (* (cos phi1) (* (cos phi2) (cos lambda1))))))
   (*
    R
    (acos
     (+
      (* 0.5 0.0)
      (* (* (cos phi2) (cos phi1)) (cos (- lambda1 lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 8.2e-7) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	} else {
		tmp = R * acos(((0.5 * 0.0) + ((cos(phi2) * cos(phi1)) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 8.2e-7)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda1))))));
	else
		tmp = Float64(R * acos(Float64(Float64(0.5 * 0.0) + Float64(Float64(cos(phi2) * cos(phi1)) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 8.2e-7], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(0.5 * 0.0), $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 8.1999999999999998e-7

   1. Initial program 82.3%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 71.0

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

   5. Simplified 71.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-acos.f64 71.0

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 71.0

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   7. Applied egg-rr 71.0%

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

   if 8.1999999999999998e-7 < lambda2

   1. Initial program 59.5%

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

      1. sin-mult N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. metadata-eval 51.6

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

   4. Applied egg-rr 51.6%

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

   5. Taylor expanded in phi1 around 0

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

      1. cos-neg N/A

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

      2. +-inverses 51.6

         \[\leadsto \cos^{-1} \left(\color{blue}{0} \cdot 0.5 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Simplified 51.6%

      \[\leadsto \cos^{-1} \left(\color{blue}{0} \cdot 0.5 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(0.5 \cdot 0 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\lambda_2 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, t\_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(0.5 \cdot 0 + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1))))
   (if (<= lambda2 8.2e-7)
     (* R (acos (fma (cos lambda1) t_0 (* (sin phi1) (sin phi2)))))
     (* R (acos (+ (* 0.5 0.0) (* t_0 (cos (- lambda1 lambda2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double tmp;
	if (lambda2 <= 8.2e-7) {
		tmp = R * acos(fma(cos(lambda1), t_0, (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos(((0.5 * 0.0) + (t_0 * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	tmp = 0.0
	if (lambda2 <= 8.2e-7)
		tmp = Float64(R * acos(fma(cos(lambda1), t_0, Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(0.5 * 0.0) + Float64(t_0 * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, 8.2e-7], N[(R * N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * t$95$0 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(0.5 * 0.0), $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 8.1999999999999998e-7

   1. Initial program 82.3%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 71.0

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

   5. Simplified 71.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-acos.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 70.9

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

   8. Simplified 70.9%

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

   if 8.1999999999999998e-7 < lambda2

   1. Initial program 59.5%

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

      1. sin-mult N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. metadata-eval 51.6

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

   4. Applied egg-rr 51.6%

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

   5. Taylor expanded in phi1 around 0

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

      1. cos-neg N/A

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

      2. +-inverses 51.6

         \[\leadsto \cos^{-1} \left(\color{blue}{0} \cdot 0.5 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Simplified 51.6%

      \[\leadsto \cos^{-1} \left(\color{blue}{0} \cdot 0.5 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(0.5 \cdot 0 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;t\_1 \leq 0.9999999:\\ \;\;\;\;R \cdot \cos^{-1} \left(0.5 \cdot 0 + t\_0 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= t_1 0.9999999)
     (* R (acos (+ (* 0.5 0.0) (* t_0 t_1))))
     (* R (acos (+ (* (sin phi1) (sin phi2)) t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (t_1 <= 0.9999999) {
		tmp = R * acos(((0.5 * 0.0) + (t_0 * t_1)));
	} else {
		tmp = R * acos(((sin(phi1) * sin(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) :: t_1
    real(8) :: tmp
    t_0 = cos(phi2) * cos(phi1)
    t_1 = cos((lambda1 - lambda2))
    if (t_1 <= 0.9999999d0) then
        tmp = r * acos(((0.5d0 * 0.0d0) + (t_0 * t_1)))
    else
        tmp = r * acos(((sin(phi1) * sin(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(phi2) * Math.cos(phi1);
	double t_1 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (t_1 <= 0.9999999) {
		tmp = R * Math.acos(((0.5 * 0.0) + (t_0 * t_1)));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + t_0));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.cos(phi1)
	t_1 = math.cos((lambda1 - lambda2))
	tmp = 0
	if t_1 <= 0.9999999:
		tmp = R * math.acos(((0.5 * 0.0) + (t_0 * t_1)))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + t_0))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (t_1 <= 0.9999999)
		tmp = Float64(R * acos(Float64(Float64(0.5 * 0.0) + Float64(t_0 * t_1))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * cos(phi1);
	t_1 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (t_1 <= 0.9999999)
		tmp = R * acos(((0.5 * 0.0) + (t_0 * t_1)));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.9999999], N[(R * N[ArcCos[N[(N[(0.5 * 0.0), $MachinePrecision] + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.999999900000000053

   1. Initial program 76.2%

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

      1. sin-mult N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. metadata-eval 62.8

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

   4. Applied egg-rr 62.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. cos-neg N/A

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

      2. +-inverses 62.8

         \[\leadsto \cos^{-1} \left(\color{blue}{0} \cdot 0.5 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Simplified 62.8%

      \[\leadsto \cos^{-1} \left(\color{blue}{0} \cdot 0.5 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 0.999999900000000053 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 75.5%

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

   3. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 75.5

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

   8. Simplified 75.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.9999999:\\ \;\;\;\;R \cdot \cos^{-1} \left(0.5 \cdot 0 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. cos-diff N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-cos.f64 92.1

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

4. Applied egg-rr 92.1%

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

6. Applied egg-rr 76.0%

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

7. Final simplification 76.0%

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

Alternative 15: 40.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 -4.8e-7)
     (* R (acos (+ (* phi1 phi2) (* (cos phi1) t_0))))
     (* R (acos (+ (* (cos phi2) t_0) (* phi1 (sin phi2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -4.8e-7) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * t_0)));
	} else {
		tmp = R * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))));
	}
	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((lambda1 - lambda2))
    if (phi1 <= (-4.8d-7)) then
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * t_0)))
    else
        tmp = r * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))))
    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((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -4.8e-7) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * t_0)));
	} else {
		tmp = R * Math.acos(((Math.cos(phi2) * t_0) + (phi1 * Math.sin(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -4.8e-7:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * t_0)))
	else:
		tmp = R * math.acos(((math.cos(phi2) * t_0) + (phi1 * math.sin(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -4.8e-7)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * t_0) + Float64(phi1 * sin(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -4.8e-7)
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * t_0)));
	else
		tmp = R * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -4.8e-7], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.79999999999999957e-7

   1. Initial program 85.2%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 14.1

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

   5. Simplified 14.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 14.1

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

   8. Simplified 14.1%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 14.1

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

   11. Simplified 14.1%

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

   if -4.79999999999999957e-7 < phi1

   1. Initial program 72.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 53.7

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

   5. Simplified 53.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 47.2

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

   8. Simplified 47.2%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \cos^{-1} \left(0.5 \cdot 0 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (acos
   (+ (* 0.5 0.0) (* (* (cos phi2) (cos phi1)) (cos (- lambda1 lambda2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos(((0.5 * 0.0) + ((cos(phi2) * cos(phi1)) * cos((lambda1 - lambda2)))));
}

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(((0.5d0 * 0.0d0) + ((cos(phi2) * cos(phi1)) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.acos(((0.5 * 0.0) + ((Math.cos(phi2) * Math.cos(phi1)) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.acos(((0.5 * 0.0) + ((math.cos(phi2) * math.cos(phi1)) * math.cos((lambda1 - lambda2)))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(Float64(Float64(0.5 * 0.0) + Float64(Float64(cos(phi2) * cos(phi1)) * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * acos(((0.5 * 0.0) + ((cos(phi2) * cos(phi1)) * cos((lambda1 - lambda2)))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[ArcCos[N[(N[(0.5 * 0.0), $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

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

   1. sin-mult N/A

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

   2. div-inv N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. metadata-eval 59.7

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

4. Applied egg-rr 59.7%

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

5. Taylor expanded in phi1 around 0

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

   1. cos-neg N/A

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

   2. +-inverses 59.6

      \[\leadsto \cos^{-1} \left(\color{blue}{0} \cdot 0.5 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

7. Simplified 59.6%

   \[\leadsto \cos^{-1} \left(\color{blue}{0} \cdot 0.5 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

8. Final simplification 59.6%

   \[\leadsto R \cdot \cos^{-1} \left(0.5 \cdot 0 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]
9. Add Preprocessing

Alternative 17: 23.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* phi1 (sin phi2))))
   (if (<= lambda1 -5e-7)
     (* R (acos (+ t_0 (* (cos phi1) (cos lambda1)))))
     (* R (acos (+ t_0 (* (cos phi1) (cos lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * sin(phi2);
	double tmp;
	if (lambda1 <= -5e-7) {
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos((t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = phi1 * sin(phi2)
    if (lambda1 <= (-5d-7)) then
        tmp = r * acos((t_0 + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos((t_0 + (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 t_0 = phi1 * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -5e-7) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = phi1 * math.sin(phi2)
	tmp = 0
	if lambda1 <= -5e-7:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(lambda2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(phi1 * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -5e-7)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = phi1 * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -5e-7)
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -5e-7], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.99999999999999977e-7

   1. Initial program 66.3%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 33.4

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

   5. Simplified 33.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 25.2

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

   8. Simplified 25.2%

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

   9. Taylor expanded in lambda2 around 0

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

       1. lower-cos.f64 25.2

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

   11. Simplified 25.2%

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

   if -4.99999999999999977e-7 < lambda1

   1. Initial program 78.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 45.8

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

   5. Simplified 45.8%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 28.1

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

   8. Simplified 28.1%

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

   9. Taylor expanded in lambda1 around 0

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

       1. cos-neg N/A

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

       2. lower-cos.f64 22.1

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

   11. Simplified 22.1%

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

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 28.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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(((phi1 * sin(phi2)) + (cos(phi1) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 42.9

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

5. Simplified 42.9%

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

6. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 27.4

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

8. Simplified 27.4%

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

9. Final simplification 27.4%

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

Alternative 19: 21.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -5e-7)
   (* R (acos (+ (* phi1 phi2) (* (cos phi1) (cos lambda1)))))
   (* R (acos (+ (* phi1 phi2) (* (cos phi1) (cos lambda2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5e-7) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos(((phi1 * phi2) + (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 <= (-5d-7)) then
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos(((phi1 * phi2) + (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 <= -5e-7) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -5e-7:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * math.cos(lambda2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -5e-7)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -5e-7)
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -5e-7], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.99999999999999977e-7

   1. Initial program 66.3%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 33.4

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

   5. Simplified 33.4%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 25.2

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

   8. Simplified 25.2%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 24.6

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

   11. Simplified 24.6%

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

   12. Taylor expanded in lambda2 around 0

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

       1. lower-cos.f64 24.6

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

   14. Simplified 24.6%

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

   if -4.99999999999999977e-7 < lambda1

   1. Initial program 78.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 45.8

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

   5. Simplified 45.8%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 28.1

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

   8. Simplified 28.1%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 26.0

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

   11. Simplified 26.0%

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

   12. Taylor expanded in lambda1 around 0

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

       1. cos-neg N/A

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

       2. lower-cos.f64 20.2

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

   14. Simplified 20.2%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 20.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0065:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.0065)
   (* R (acos (+ (* phi1 phi2) (* (cos phi1) (cos lambda1)))))
   (*
    R
    (acos
     (+
      (* phi1 phi2)
      (* (fma -0.5 (* phi1 phi1) 1.0) (cos (- lambda2 lambda1))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0065) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos(((phi1 * phi2) + (fma(-0.5, (phi1 * phi1), 1.0) * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.0065)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(fma(-0.5, Float64(phi1 * phi1), 1.0) * cos(Float64(lambda2 - lambda1))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.0065], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0064999999999999997

   1. Initial program 85.2%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 14.1

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

   5. Simplified 14.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 14.1

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

   8. Simplified 14.1%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 14.1

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

   11. Simplified 14.1%

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

   12. Taylor expanded in lambda2 around 0

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

       1. lower-cos.f64 9.2

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

   14. Simplified 9.2%

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

   if -0.0064999999999999997 < phi1

   1. Initial program 72.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 53.7

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

   5. Simplified 53.7%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 32.4

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

   8. Simplified 32.4%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 30.0

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

   11. Simplified 30.0%

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

   12. Taylor expanded in phi1 around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

       3. lower-*.f64 N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. sub-neg N/A

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

       8. remove-double-neg N/A

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

       9. mul-1-neg N/A

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

       10. distribute-neg-in N/A

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

       11. +-commutative N/A

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

       12. cos-neg N/A

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

       13. lower-cos.f64 N/A

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

       14. mul-1-neg N/A

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

       15. unsub-neg N/A

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

       16. lower--.f64 23.0

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

   14. Simplified 23.0%

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

4. Final simplification 19.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0065:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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(((phi1 * phi2) + (cos(phi1) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 42.9

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

5. Simplified 42.9%

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

6. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 27.4

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

8. Simplified 27.4%

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

9. Taylor expanded in phi2 around 0

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

    1. lower-*.f64 25.7

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

11. Simplified 25.7%

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

12. Final simplification 25.7%

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

Alternative 22: 18.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 42.9

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

5. Simplified 42.9%

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

6. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 27.4

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

8. Simplified 27.4%

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

9. Taylor expanded in phi2 around 0

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

    1. lower-*.f64 25.7

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

11. Simplified 25.7%

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

12. Taylor expanded in phi1 around 0

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

    1. sub-neg N/A

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

    2. remove-double-neg N/A

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

    3. mul-1-neg N/A

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

    4. distribute-neg-in N/A

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

    5. +-commutative N/A

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

    6. cos-neg N/A

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

    7. lower-cos.f64 N/A

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

    8. mul-1-neg N/A

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

    9. unsub-neg N/A

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

    10. lower--.f64 17.7

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

14. Simplified 17.7%

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

15. Final simplification 17.7%

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

Reproduce

herbie shell --seed 2024219 
(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))