Spherical law of cosines

Percentage Accurate: 73.6% → 93.6%

Time: 19.7s

Alternatives: 25

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: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.6% accurate, 0.7× speedup

Math

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

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (acos
   (fma
    (*
     (cos phi2)
     (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))
    (cos phi1)
    (* (sin phi1) (sin phi2))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 93.8

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

4. Applied rewrites 93.8%

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

5. Taylor expanded in lambda1 around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

7. Applied rewrites 93.9%

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

8. Final simplification 93.9%

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

Alternative 2: 93.6% accurate, 0.7× speedup

Math

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

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (fma
    (sin phi2)
    (sin phi1)
    (*
     (cos phi1)
     (*
      (cos phi2)
      (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))))))
  R))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 93.8

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

4. Applied rewrites 93.8%

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

5. Taylor expanded in lambda1 around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

7. Applied rewrites 93.9%

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

8. Taylor expanded in lambda1 around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

10. Applied rewrites 93.8%

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

11. Final simplification 93.8%

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

Alternative 3: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (* (cos (- lambda2 lambda1)) (cos phi2))
            (cos phi1)
            (* (sin phi1) (sin phi2))))
          R)))
   (if (<= phi2 -3.2e-8)
     t_0
     (if (<= phi2 9.2e-7)
       (*
        (acos
         (fma
          (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))
          (cos phi1)
          (* (sin phi1) phi2)))
        R)
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(fma((cos((lambda2 - lambda1)) * cos(phi2)), cos(phi1), (sin(phi1) * sin(phi2)))) * R;
	double tmp;
	if (phi2 <= -3.2e-8) {
		tmp = t_0;
	} else if (phi2 <= 9.2e-7) {
		tmp = acos(fma(fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))), cos(phi1), (sin(phi1) * phi2))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2)), cos(phi1), Float64(sin(phi1) * sin(phi2)))) * R)
	tmp = 0.0
	if (phi2 <= -3.2e-8)
		tmp = t_0;
	elseif (phi2 <= 9.2e-7)
		tmp = Float64(acos(fma(fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))), cos(phi1), Float64(sin(phi1) * phi2))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -3.2e-8], t$95$0, If[LessEqual[phi2, 9.2e-7], N[(N[ArcCos[N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.2000000000000002e-8 or 9.1999999999999998e-7 < phi2

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.5

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. cos-diff N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. cos-diff N/A

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

      16. lower-cos.f64 N/A

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

      17. lower--.f64 81.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 81.5

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

   4. Applied rewrites 81.5%

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

   if -3.2000000000000002e-8 < phi2 < 9.1999999999999998e-7

   1. Initial program 71.4%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 87.4

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. cos-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 87.5

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

   7. Applied rewrites 87.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -6.7 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (* (cos (- lambda2 lambda1)) (cos phi2))
            (cos phi1)
            (* (sin phi1) (sin phi2))))
          R)))
   (if (<= phi2 -6.7e-10)
     t_0
     (if (<= phi2 5.9e-8)
       (*
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))))
        R)
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(fma((cos((lambda2 - lambda1)) * cos(phi2)), cos(phi1), (sin(phi1) * sin(phi2)))) * R;
	double tmp;
	if (phi2 <= -6.7e-10) {
		tmp = t_0;
	} else if (phi2 <= 5.9e-8) {
		tmp = acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2)), cos(phi1), Float64(sin(phi1) * sin(phi2)))) * R)
	tmp = 0.0
	if (phi2 <= -6.7e-10)
		tmp = t_0;
	elseif (phi2 <= 5.9e-8)
		tmp = Float64(acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -6.7e-10], t$95$0, If[LessEqual[phi2, 5.9e-8], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -6.6999999999999996e-10 or 5.8999999999999999e-8 < phi2

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.5

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. cos-diff N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. cos-diff N/A

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

      16. lower-cos.f64 N/A

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

      17. lower--.f64 81.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 81.5

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

   4. Applied rewrites 81.5%

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

   if -6.6999999999999996e-10 < phi2 < 5.8999999999999999e-8

   1. Initial program 71.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 87.4

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 87.4

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

   7. Applied rewrites 87.4%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.7 \cdot 10^{-10}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -6.7 \cdot 10^{-10}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(t\_0 \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(t\_0 \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 -6.7e-10)
     (* (acos (fma (sin phi2) (sin phi1) (* (* t_0 (cos phi1)) (cos phi2)))) R)
     (if (<= phi2 5.9e-8)
       (*
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))))
        R)
       (*
        (acos (fma (sin phi2) (sin phi1) (* (* t_0 (cos phi2)) (cos phi1))))
        R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= -6.7e-10) {
		tmp = acos(fma(sin(phi2), sin(phi1), ((t_0 * cos(phi1)) * cos(phi2)))) * R;
	} else if (phi2 <= 5.9e-8) {
		tmp = acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))))) * R;
	} else {
		tmp = acos(fma(sin(phi2), sin(phi1), ((t_0 * cos(phi2)) * cos(phi1)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= -6.7e-10)
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(t_0 * cos(phi1)) * cos(phi2)))) * R);
	elseif (phi2 <= 5.9e-8)
		tmp = Float64(acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))) * R);
	else
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(t_0 * cos(phi2)) * cos(phi1)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -6.7e-10], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 5.9e-8], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.6999999999999996e-10

   1. Initial program 78.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 78.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 78.1

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

      11. lift-cos.f64 N/A

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

      12. lift--.f64 N/A

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

      13. cos-diff N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. cos-diff N/A

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

      17. lower-cos.f64 N/A

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

      18. lower--.f64 78.1

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

   4. Applied rewrites 78.1%

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

   if -6.6999999999999996e-10 < phi2 < 5.8999999999999999e-8

   1. Initial program 71.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 87.4

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 87.4

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

   7. Applied rewrites 87.4%

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

   if 5.8999999999999999e-8 < phi2

   1. Initial program 84.9%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. cos-diff N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 84.9

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

   6. Applied rewrites 84.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.7 \cdot 10^{-10}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -6.7 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (sin phi2)
            (sin phi1)
            (* (* (cos (- lambda2 lambda1)) (cos phi1)) (cos phi2))))
          R)))
   (if (<= phi2 -6.7e-10)
     t_0
     (if (<= phi2 5.9e-8)
       (*
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))))
        R)
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(fma(sin(phi2), sin(phi1), ((cos((lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R;
	double tmp;
	if (phi2 <= -6.7e-10) {
		tmp = t_0;
	} else if (phi2 <= 5.9e-8) {
		tmp = acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R)
	tmp = 0.0
	if (phi2 <= -6.7e-10)
		tmp = t_0;
	elseif (phi2 <= 5.9e-8)
		tmp = Float64(acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -6.7e-10], t$95$0, If[LessEqual[phi2, 5.9e-8], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -6.6999999999999996e-10 or 5.8999999999999999e-8 < phi2

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 81.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 81.5

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

      11. lift-cos.f64 N/A

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

      12. lift--.f64 N/A

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

      13. cos-diff N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. cos-diff N/A

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

      17. lower-cos.f64 N/A

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

      18. lower--.f64 81.5

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

   4. Applied rewrites 81.5%

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

   if -6.6999999999999996e-10 < phi2 < 5.8999999999999999e-8

   1. Initial program 71.2%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 87.4

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 87.4

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

   7. Applied rewrites 87.4%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.7 \cdot 10^{-10}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_1, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -8.5e-8)
   (*
    (acos
     (*
      (cos phi2)
      (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))))
    R)
   (if (<= lambda2 2.9e-6)
     (*
      (acos
       (fma (* (cos phi2) (cos lambda1)) (cos phi1) (* (sin phi1) (sin phi2))))
      R)
     (*
      (acos
       (fma (sin phi2) (sin phi1) (* (* (cos phi2) (cos lambda2)) (cos phi1))))
      R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -8.5e-8) {
		tmp = acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))))) * R;
	} else if (lambda2 <= 2.9e-6) {
		tmp = acos(fma((cos(phi2) * cos(lambda1)), cos(phi1), (sin(phi1) * sin(phi2)))) * R;
	} else {
		tmp = acos(fma(sin(phi2), sin(phi1), ((cos(phi2) * cos(lambda2)) * cos(phi1)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -8.5e-8)
		tmp = Float64(acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))) * R);
	elseif (lambda2 <= 2.9e-6)
		tmp = Float64(acos(fma(Float64(cos(phi2) * cos(lambda1)), cos(phi1), Float64(sin(phi1) * sin(phi2)))) * R);
	else
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi2) * cos(lambda2)) * cos(phi1)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -8.5e-8], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 2.9e-6], N[(N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -8.49999999999999935e-8

   1. Initial program 65.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 67.8

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

   7. Applied rewrites 67.8%

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

   if -8.49999999999999935e-8 < lambda2 < 2.9000000000000002e-6

   1. Initial program 88.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. cos-mult N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. cos-diff N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. cos-diff N/A

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

      16. lower-cos.f64 N/A

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

      17. lower--.f64 N/A

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

      18. clear-num N/A

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

      19. cos-mult N/A

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

      20. lift-cos.f64 N/A

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

      21. lift-cos.f64 N/A

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

      22. lift-*.f64 N/A

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in lambda2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cos-neg N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 88.9

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

   7. Applied rewrites 88.9%

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

   if 2.9000000000000002e-6 < lambda2

   1. Initial program 57.8%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_1, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -8.5e-8)
   (*
    (acos
     (*
      (cos phi2)
      (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))))
    R)
   (if (<= lambda2 2.9e-6)
     (*
      (acos
       (fma (* (cos phi1) (cos phi2)) (cos lambda1) (* (sin phi1) (sin phi2))))
      R)
     (*
      (acos
       (fma (sin phi2) (sin phi1) (* (* (cos phi2) (cos lambda2)) (cos phi1))))
      R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -8.5e-8) {
		tmp = acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))))) * R;
	} else if (lambda2 <= 2.9e-6) {
		tmp = acos(fma((cos(phi1) * cos(phi2)), cos(lambda1), (sin(phi1) * sin(phi2)))) * R;
	} else {
		tmp = acos(fma(sin(phi2), sin(phi1), ((cos(phi2) * cos(lambda2)) * cos(phi1)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -8.5e-8)
		tmp = Float64(acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))) * R);
	elseif (lambda2 <= 2.9e-6)
		tmp = Float64(acos(fma(Float64(cos(phi1) * cos(phi2)), cos(lambda1), Float64(sin(phi1) * sin(phi2)))) * R);
	else
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi2) * cos(lambda2)) * cos(phi1)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -8.5e-8], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 2.9e-6], N[(N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -8.49999999999999935e-8

   1. Initial program 65.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cos-neg N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 67.8

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

   7. Applied rewrites 67.8%

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

   if -8.49999999999999935e-8 < lambda2 < 2.9000000000000002e-6

   1. Initial program 88.9%

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

   3. Taylor expanded in lambda2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if 2.9000000000000002e-6 < lambda2

   1. Initial program 57.8%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.0015:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -0.0015)
   (*
    (acos
     (*
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))
      (cos phi1)))
    R)
   (if (<= lambda2 2.9e-6)
     (*
      (acos
       (fma (* (cos phi1) (cos phi2)) (cos lambda1) (* (sin phi1) (sin phi2))))
      R)
     (*
      (acos
       (fma (sin phi2) (sin phi1) (* (* (cos phi2) (cos lambda2)) (cos phi1))))
      R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -0.0015) {
		tmp = acos((fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))) * cos(phi1))) * R;
	} else if (lambda2 <= 2.9e-6) {
		tmp = acos(fma((cos(phi1) * cos(phi2)), cos(lambda1), (sin(phi1) * sin(phi2)))) * R;
	} else {
		tmp = acos(fma(sin(phi2), sin(phi1), ((cos(phi2) * cos(lambda2)) * cos(phi1)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -0.0015)
		tmp = Float64(acos(Float64(fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))) * cos(phi1))) * R);
	elseif (lambda2 <= 2.9e-6)
		tmp = Float64(acos(fma(Float64(cos(phi1) * cos(phi2)), cos(lambda1), Float64(sin(phi1) * sin(phi2)))) * R);
	else
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi2) * cos(lambda2)) * cos(phi1)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -0.0015], N[(N[ArcCos[N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 2.9e-6], N[(N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -0.0015

   1. Initial program 65.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 33.4

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

   5. Applied rewrites 33.4%

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

   7. Applied rewrites 49.4%

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

   if -0.0015 < lambda2 < 2.9000000000000002e-6

   1. Initial program 88.6%

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

   3. Taylor expanded in lambda2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if 2.9000000000000002e-6 < lambda2

   1. Initial program 57.8%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.0015:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (sin phi2)
            (sin phi1)
            (* (* (cos phi2) (cos lambda2)) (cos phi1))))
          R)))
   (if (<= phi2 -9.5e-9)
     t_0
     (if (<= phi2 1.8e-6)
       (*
        (acos
         (*
          (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))
          (cos phi1)))
        R)
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(fma(sin(phi2), sin(phi1), ((cos(phi2) * cos(lambda2)) * cos(phi1)))) * R;
	double tmp;
	if (phi2 <= -9.5e-9) {
		tmp = t_0;
	} else if (phi2 <= 1.8e-6) {
		tmp = acos((fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))) * cos(phi1))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi2) * cos(lambda2)) * cos(phi1)))) * R)
	tmp = 0.0
	if (phi2 <= -9.5e-9)
		tmp = t_0;
	elseif (phi2 <= 1.8e-6)
		tmp = Float64(acos(Float64(fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))) * cos(phi1))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -9.5e-9], t$95$0, If[LessEqual[phi2, 1.8e-6], N[(N[ArcCos[N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -9.5000000000000007e-9 or 1.79999999999999992e-6 < phi2

   1. Initial program 81.5%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if -9.5000000000000007e-9 < phi2 < 1.79999999999999992e-6

   1. Initial program 71.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 71.2

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

   5. Applied rewrites 71.2%

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

   7. Applied rewrites 87.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -9.5e-9)
   (* (acos (fma (sin phi2) (sin phi1) (* (cos phi1) (cos phi2)))) R)
   (if (<= phi2 9.2e-6)
     (*
      (acos
       (*
        (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))
        (cos phi1)))
      R)
     (*
      (acos
       (+ (* (cos (- lambda1 lambda2)) (cos phi2)) (* (sin phi1) (sin phi2))))
      R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -9.5e-9) {
		tmp = acos(fma(sin(phi2), sin(phi1), (cos(phi1) * cos(phi2)))) * R;
	} else if (phi2 <= 9.2e-6) {
		tmp = acos((fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))) * cos(phi1))) * R;
	} else {
		tmp = acos(((cos((lambda1 - lambda2)) * cos(phi2)) + (sin(phi1) * sin(phi2)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -9.5e-9)
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * cos(phi2)))) * R);
	elseif (phi2 <= 9.2e-6)
		tmp = Float64(acos(Float64(fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(Float64(cos(Float64(lambda1 - lambda2)) * cos(phi2)) + Float64(sin(phi1) * sin(phi2)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -9.5e-9], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 9.2e-6], N[(N[ArcCos[N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.5000000000000007e-9

   1. Initial program 78.2%

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

   3. Taylor expanded in lambda2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 44.5%

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

   if -9.5000000000000007e-9 < phi2 < 9.2e-6

   1. Initial program 70.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 87.0%

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

   if 9.2e-6 < phi2

   1. Initial program 85.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 49.1

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

   5. Applied rewrites 49.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -9.5e-9)
   (* (acos (fma (sin phi2) (sin phi1) (* (cos phi1) (cos phi2)))) R)
   (if (<= phi2 4.8e-8)
     (* (fma PI 0.5 (- (asin (* (cos (- lambda2 lambda1)) (cos phi1))))) R)
     (*
      (acos
       (+ (* (cos (- lambda1 lambda2)) (cos phi2)) (* (sin phi1) (sin phi2))))
      R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -9.5e-9) {
		tmp = acos(fma(sin(phi2), sin(phi1), (cos(phi1) * cos(phi2)))) * R;
	} else if (phi2 <= 4.8e-8) {
		tmp = fma(((double) M_PI), 0.5, -asin((cos((lambda2 - lambda1)) * cos(phi1)))) * R;
	} else {
		tmp = acos(((cos((lambda1 - lambda2)) * cos(phi2)) + (sin(phi1) * sin(phi2)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -9.5e-9)
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * cos(phi2)))) * R);
	elseif (phi2 <= 4.8e-8)
		tmp = Float64(fma(pi, 0.5, Float64(-asin(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1))))) * R);
	else
		tmp = Float64(acos(Float64(Float64(cos(Float64(lambda1 - lambda2)) * cos(phi2)) + Float64(sin(phi1) * sin(phi2)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -9.5e-9], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 4.8e-8], N[(N[(Pi * 0.5 + (-N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.5000000000000007e-9

   1. Initial program 78.2%

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

   3. Taylor expanded in lambda2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 44.5%

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

   if -9.5000000000000007e-9 < phi2 < 4.79999999999999997e-8

   1. Initial program 71.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 71.2

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

   5. Applied rewrites 71.2%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-asin.f64 71.2

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

   7. Applied rewrites 71.2%

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

   if 4.79999999999999997e-8 < phi2

   1. Initial program 84.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 48.6

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

   5. Applied rewrites 48.6%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -9.5e-9)
   (* (acos (fma (sin phi2) (sin phi1) (* (cos phi1) (cos phi2)))) R)
   (if (<= phi2 4.8e-8)
     (* (fma PI 0.5 (- (asin (* (cos (- lambda2 lambda1)) (cos phi1))))) R)
     (* (acos (* (cos (- lambda1 lambda2)) (cos phi2))) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -9.5e-9) {
		tmp = acos(fma(sin(phi2), sin(phi1), (cos(phi1) * cos(phi2)))) * R;
	} else if (phi2 <= 4.8e-8) {
		tmp = fma(((double) M_PI), 0.5, -asin((cos((lambda2 - lambda1)) * cos(phi1)))) * R;
	} else {
		tmp = acos((cos((lambda1 - lambda2)) * cos(phi2))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -9.5e-9)
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * cos(phi2)))) * R);
	elseif (phi2 <= 4.8e-8)
		tmp = Float64(fma(pi, 0.5, Float64(-asin(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1))))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(lambda1 - lambda2)) * cos(phi2))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -9.5e-9], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 4.8e-8], N[(N[(Pi * 0.5 + (-N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.5000000000000007e-9

   1. Initial program 78.2%

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

   3. Taylor expanded in lambda2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 44.5%

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

   if -9.5000000000000007e-9 < phi2 < 4.79999999999999997e-8

   1. Initial program 71.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 71.2

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

   5. Applied rewrites 71.2%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-asin.f64 71.2

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

   7. Applied rewrites 71.2%

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

   if 4.79999999999999997e-8 < phi2

   1. Initial program 84.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 49.1

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

   5. Applied rewrites 49.1%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.000105:\\ \;\;\;\;\cos^{-1} \left(\frac{1}{\frac{1}{\cos \left(\lambda_2 - \lambda_1\right)}} \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.000105)
   (* (acos (* (/ 1.0 (/ 1.0 (cos (- lambda2 lambda1)))) (cos phi1))) R)
   (* (acos (* (cos (- lambda1 lambda2)) (cos phi2))) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.000105) {
		tmp = acos(((1.0 / (1.0 / cos((lambda2 - lambda1)))) * cos(phi1))) * R;
	} else {
		tmp = acos((cos((lambda1 - lambda2)) * cos(phi2))) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-0.000105d0)) then
        tmp = acos(((1.0d0 / (1.0d0 / cos((lambda2 - lambda1)))) * cos(phi1))) * r
    else
        tmp = acos((cos((lambda1 - lambda2)) * cos(phi2))) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.000105) {
		tmp = Math.acos(((1.0 / (1.0 / Math.cos((lambda2 - lambda1)))) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((Math.cos((lambda1 - lambda2)) * Math.cos(phi2))) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.000105:
		tmp = math.acos(((1.0 / (1.0 / math.cos((lambda2 - lambda1)))) * math.cos(phi1))) * R
	else:
		tmp = math.acos((math.cos((lambda1 - lambda2)) * math.cos(phi2))) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.000105)
		tmp = Float64(acos(Float64(Float64(1.0 / Float64(1.0 / cos(Float64(lambda2 - lambda1)))) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(lambda1 - lambda2)) * cos(phi2))) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.000105)
		tmp = acos(((1.0 / (1.0 / cos((lambda2 - lambda1)))) * cos(phi1))) * R;
	else
		tmp = acos((cos((lambda1 - lambda2)) * cos(phi2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.000105], N[(N[ArcCos[N[(N[(1.0 / N[(1.0 / N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.05e-4

   1. Initial program 87.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 45.7

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 45.8%

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

   if -1.05e-4 < phi1

   1. Initial program 73.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 53.9

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

   5. Applied rewrites 53.9%

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

Alternative 15: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.5 \cdot 10^{-56}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 0.00325:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.5e-56)
   (* (acos (* (cos lambda1) (cos phi1))) R)
   (if (<= phi2 0.00325)
     (* (acos (* (cos lambda2) (cos phi1))) R)
     (* (acos (* (cos phi2) (cos lambda1))) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.5e-56) {
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	} else if (phi2 <= 0.00325) {
		tmp = acos((cos(lambda2) * cos(phi1))) * R;
	} else {
		tmp = acos((cos(phi2) * cos(lambda1))) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.5d-56) then
        tmp = acos((cos(lambda1) * cos(phi1))) * r
    else if (phi2 <= 0.00325d0) then
        tmp = acos((cos(lambda2) * cos(phi1))) * r
    else
        tmp = acos((cos(phi2) * cos(lambda1))) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.5e-56) {
		tmp = Math.acos((Math.cos(lambda1) * Math.cos(phi1))) * R;
	} else if (phi2 <= 0.00325) {
		tmp = Math.acos((Math.cos(lambda2) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((Math.cos(phi2) * Math.cos(lambda1))) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.5e-56:
		tmp = math.acos((math.cos(lambda1) * math.cos(phi1))) * R
	elif phi2 <= 0.00325:
		tmp = math.acos((math.cos(lambda2) * math.cos(phi1))) * R
	else:
		tmp = math.acos((math.cos(phi2) * math.cos(lambda1))) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.5e-56)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R);
	elseif (phi2 <= 0.00325)
		tmp = Float64(acos(Float64(cos(lambda2) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi2) * cos(lambda1))) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.5e-56)
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	elseif (phi2 <= 0.00325)
		tmp = acos((cos(lambda2) * cos(phi1))) * R;
	else
		tmp = acos((cos(phi2) * cos(lambda1))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.5e-56], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 0.00325], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.49999999999999995e-56

   1. Initial program 73.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 36.0%

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

   if 1.49999999999999995e-56 < phi2 < 0.00324999999999999985

   1. Initial program 79.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 25.5%

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

   if 0.00324999999999999985 < phi2

   1. Initial program 85.8%

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

   3. Taylor expanded in lambda2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 37.9%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.5 \cdot 10^{-56}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 0.00325:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.7 \cdot 10^{-85}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 0.00052:\\ \;\;\;\;\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.7e-85)
   (* (acos (* (cos lambda1) (cos phi1))) R)
   (if (<= phi2 0.00052)
     (* (acos (cos (- lambda1 lambda2))) R)
     (* (acos (* (cos phi2) (cos lambda1))) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.7e-85) {
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	} else if (phi2 <= 0.00052) {
		tmp = acos(cos((lambda1 - lambda2))) * R;
	} else {
		tmp = acos((cos(phi2) * cos(lambda1))) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 7.7d-85) then
        tmp = acos((cos(lambda1) * cos(phi1))) * r
    else if (phi2 <= 0.00052d0) then
        tmp = acos(cos((lambda1 - lambda2))) * r
    else
        tmp = acos((cos(phi2) * cos(lambda1))) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.7e-85) {
		tmp = Math.acos((Math.cos(lambda1) * Math.cos(phi1))) * R;
	} else if (phi2 <= 0.00052) {
		tmp = Math.acos(Math.cos((lambda1 - lambda2))) * R;
	} else {
		tmp = Math.acos((Math.cos(phi2) * Math.cos(lambda1))) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 7.7e-85:
		tmp = math.acos((math.cos(lambda1) * math.cos(phi1))) * R
	elif phi2 <= 0.00052:
		tmp = math.acos(math.cos((lambda1 - lambda2))) * R
	else:
		tmp = math.acos((math.cos(phi2) * math.cos(lambda1))) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 7.7e-85)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R);
	elseif (phi2 <= 0.00052)
		tmp = Float64(acos(cos(Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi2) * cos(lambda1))) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 7.7e-85)
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	elseif (phi2 <= 0.00052)
		tmp = acos(cos((lambda1 - lambda2))) * R;
	else
		tmp = acos((cos(phi2) * cos(lambda1))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 7.7e-85], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 0.00052], N[(N[ArcCos[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 7.70000000000000031e-85

   1. Initial program 73.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. lower-cos.f64 49.4

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 49.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 35.9%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   if 7.70000000000000031e-85 < phi2 < 5.19999999999999954e-4

   1. Initial program 74.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. lower-cos.f64 74.2

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 74.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 52.9%

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]

   if 5.19999999999999954e-4 < phi2

   1. Initial program 85.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 62.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 62.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 37.9%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.7 \cdot 10^{-85}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 0.00052:\\ \;\;\;\;\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.000105:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 -0.000105)
     (* (acos (* t_0 (cos phi1))) R)
     (* (acos (* t_0 (cos phi2))) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.000105) {
		tmp = acos((t_0 * cos(phi1))) * R;
	} else {
		tmp = acos((t_0 * cos(phi2))) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    if (phi1 <= (-0.000105d0)) then
        tmp = acos((t_0 * cos(phi1))) * r
    else
        tmp = acos((t_0 * cos(phi2))) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.000105) {
		tmp = Math.acos((t_0 * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((t_0 * Math.cos(phi2))) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -0.000105:
		tmp = math.acos((t_0 * math.cos(phi1))) * R
	else:
		tmp = math.acos((t_0 * math.cos(phi2))) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -0.000105)
		tmp = Float64(acos(Float64(t_0 * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(t_0 * cos(phi2))) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -0.000105)
		tmp = acos((t_0 * cos(phi1))) * R;
	else
		tmp = acos((t_0 * cos(phi2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.000105], N[(N[ArcCos[N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.05e-4

   1. Initial program 87.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. lower-cos.f64 45.7

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 45.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   if -1.05e-4 < phi1

   1. Initial program 73.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      9. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      10. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_2\right) \cdot R \]

      15. lower-cos.f64 53.9

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   5. Applied rewrites 53.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 53.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0034:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.0034)
   (* (acos (* (cos (- lambda1 lambda2)) (cos phi1))) R)
   (* (acos (* (cos phi2) (cos lambda1))) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0034) {
		tmp = acos((cos((lambda1 - lambda2)) * cos(phi1))) * R;
	} else {
		tmp = acos((cos(phi2) * cos(lambda1))) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 0.0034d0) then
        tmp = acos((cos((lambda1 - lambda2)) * cos(phi1))) * r
    else
        tmp = acos((cos(phi2) * cos(lambda1))) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0034) {
		tmp = Math.acos((Math.cos((lambda1 - lambda2)) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((Math.cos(phi2) * Math.cos(lambda1))) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.0034:
		tmp = math.acos((math.cos((lambda1 - lambda2)) * math.cos(phi1))) * R
	else:
		tmp = math.acos((math.cos(phi2) * math.cos(lambda1))) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.0034)
		tmp = Float64(acos(Float64(cos(Float64(lambda1 - lambda2)) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi2) * cos(lambda1))) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.0034)
		tmp = acos((cos((lambda1 - lambda2)) * cos(phi1))) * R;
	else
		tmp = acos((cos(phi2) * cos(lambda1))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.0034], N[(N[ArcCos[N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.00339999999999999981

   1. Initial program 73.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. lower-cos.f64 50.8

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 50.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   if 0.00339999999999999981 < phi2

   1. Initial program 85.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 62.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 62.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 37.9%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0034:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 34.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.0022:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 0.0022)
   (* (acos (* (cos phi2) (cos lambda1))) R)
   (* (acos (cos (- lambda1 lambda2))) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.0022) {
		tmp = acos((cos(phi2) * cos(lambda1))) * R;
	} else {
		tmp = acos(cos((lambda1 - lambda2))) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 0.0022d0) then
        tmp = acos((cos(phi2) * cos(lambda1))) * r
    else
        tmp = acos(cos((lambda1 - lambda2))) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.0022) {
		tmp = Math.acos((Math.cos(phi2) * Math.cos(lambda1))) * R;
	} else {
		tmp = Math.acos(Math.cos((lambda1 - lambda2))) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 0.0022:
		tmp = math.acos((math.cos(phi2) * math.cos(lambda1))) * R
	else:
		tmp = math.acos(math.cos((lambda1 - lambda2))) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 0.0022)
		tmp = Float64(acos(Float64(cos(phi2) * cos(lambda1))) * R);
	else
		tmp = Float64(acos(cos(Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 0.0022)
		tmp = acos((cos(phi2) * cos(lambda1))) * R;
	else
		tmp = acos(cos((lambda1 - lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 0.0022], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.00220000000000000013

   1. Initial program 81.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 66.8

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 66.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 37.9%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   if 0.00220000000000000013 < lambda2

   1. Initial program 57.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. lower-cos.f64 46.1

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 46.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 34.2%

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.0022:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 19.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -100:\\ \;\;\;\;\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\left(\left(\cos \lambda_2 \cdot \lambda_1\right) \cdot \lambda_1\right) \cdot -0.5\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -100.0)
   (* (fma PI 0.5 (- (asin (cos (- lambda2 lambda1))))) R)
   (* (acos (* (* (* (cos lambda2) lambda1) lambda1) -0.5)) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -100.0) {
		tmp = fma(((double) M_PI), 0.5, -asin(cos((lambda2 - lambda1)))) * R;
	} else {
		tmp = acos((((cos(lambda2) * lambda1) * lambda1) * -0.5)) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -100.0)
		tmp = Float64(fma(pi, 0.5, Float64(-asin(cos(Float64(lambda2 - lambda1))))) * R);
	else
		tmp = Float64(acos(Float64(Float64(Float64(cos(lambda2) * lambda1) * lambda1) * -0.5)) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -100.0], N[(N[(Pi * 0.5 + (-N[ArcSin[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * lambda1), $MachinePrecision] * lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -100

   1. Initial program 74.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. lower-cos.f64 49.6

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 49.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 34.3%

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
   9. Step-by-step derivation

      1. lift-acos.f64 N/A

         \[\leadsto \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

      2. acos-asin N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \cdot R \]

      4. div-inv N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      6. lower-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)}, \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \color{blue}{\frac{1}{2}}, \mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \color{blue}{\mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot R \]

   10. Applied rewrites 34.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   if -100 < (-.f64 lambda1 lambda2)

   1. Initial program 78.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. lower-cos.f64 38.6

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 38.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 22.8%

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 10.6%

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2 \cdot \lambda_1, -0.5, \sin \lambda_2\right), \lambda_1, \cos \lambda_2\right)\right) \cdot R \]

   12. Taylor expanded in lambda1 around inf

       \[\leadsto \cos^{-1} \left(\frac{-1}{2} \cdot \left({\lambda_1}^{2} \cdot \cos \lambda_2\right)\right) \cdot R \]
   13. Step-by-step derivation

   14. Applied rewrites 11.2%

       \[\leadsto \cos^{-1} \left(\left(\left(\cos \lambda_2 \cdot \lambda_1\right) \cdot \lambda_1\right) \cdot -0.5\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 26.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (fma PI 0.5 (- (asin (cos (- lambda2 lambda1))))) R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return fma(((double) M_PI), 0.5, -asin(cos((lambda2 - lambda1)))) * R;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(fma(pi, 0.5, Float64(-asin(cos(Float64(lambda2 - lambda1))))) * R)
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(Pi * 0.5 + (-N[ArcSin[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   8. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   10. distribute-neg-in N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   11. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   12. remove-double-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   13. sub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   14. lower--.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   15. lower-cos.f64 42.1

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

5. Applied rewrites 42.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
7. Step-by-step derivation

8. Applied rewrites 26.4%

   \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
9. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   2. acos-asin N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \cdot R \]

   4. div-inv N/A

      \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

   6. lower-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)}, \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \color{blue}{\frac{1}{2}}, \mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   8. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \color{blue}{\mathsf{neg}\left(\sin^{-1} \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot R \]

10. Applied rewrites 26.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
11. Add Preprocessing

Alternative 22: 21.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \cos \lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \lambda_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.15e-8)
   (* (acos (cos lambda1)) R)
   (* (acos (cos lambda2)) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.15e-8) {
		tmp = acos(cos(lambda1)) * R;
	} else {
		tmp = acos(cos(lambda2)) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 1.15d-8) then
        tmp = acos(cos(lambda1)) * r
    else
        tmp = acos(cos(lambda2)) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.15e-8) {
		tmp = Math.acos(Math.cos(lambda1)) * R;
	} else {
		tmp = Math.acos(Math.cos(lambda2)) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.15e-8:
		tmp = math.acos(math.cos(lambda1)) * R
	else:
		tmp = math.acos(math.cos(lambda2)) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.15e-8)
		tmp = Float64(acos(cos(lambda1)) * R);
	else
		tmp = Float64(acos(cos(lambda2)) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.15e-8)
		tmp = acos(cos(lambda1)) * R;
	else
		tmp = acos(cos(lambda2)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.15e-8], N[(N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.15e-8

   1. Initial program 81.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. lower-cos.f64 41.1

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 41.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 24.5%

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 18.7%

       \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

   if 1.15e-8 < lambda2

   1. Initial program 57.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      13. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      14. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. lower-cos.f64 46.1

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 46.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 34.2%

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 34.0%

       \[\leadsto \cos^{-1} \cos \lambda_2 \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 26.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (cos (- lambda1 lambda2))) R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(cos((lambda1 - lambda2))) * R;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(cos((lambda1 - lambda2))) * r
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(Math.cos((lambda1 - lambda2))) * R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(math.cos((lambda1 - lambda2))) * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(cos(Float64(lambda1 - lambda2))) * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(cos((lambda1 - lambda2))) * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   8. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   10. distribute-neg-in N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   11. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   12. remove-double-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   13. sub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   14. lower--.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   15. lower-cos.f64 42.1

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

5. Applied rewrites 42.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
7. Step-by-step derivation

8. Applied rewrites 26.4%

   \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
9. Add Preprocessing

Alternative 24: 17.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \cos^{-1} \cos \lambda_1 \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (cos lambda1)) R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(cos(lambda1)) * R;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(cos(lambda1)) * r
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(Math.cos(lambda1)) * R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(math.cos(lambda1)) * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(cos(lambda1)) * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(cos(lambda1)) * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   8. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   10. distribute-neg-in N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   11. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   12. remove-double-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   13. sub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   14. lower--.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   15. lower-cos.f64 42.1

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

5. Applied rewrites 42.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
7. Step-by-step derivation

8. Applied rewrites 26.4%

   \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]

9. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]
10. Step-by-step derivation

11. Applied rewrites 17.5%

    \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]
12. Add Preprocessing

Alternative 25: 2.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \cos^{-1} \left(\mathsf{fma}\left(\lambda_1 \cdot \lambda_1, -0.5, 1\right)\right) \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (fma (* lambda1 lambda1) -0.5 1.0)) R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(fma((lambda1 * lambda1), -0.5, 1.0)) * R;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(fma(Float64(lambda1 * lambda1), -0.5, 1.0)) * R)
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(lambda1 * lambda1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   8. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \lambda_1 + \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   10. distribute-neg-in N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   11. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   12. remove-double-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   13. sub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   14. lower--.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   15. lower-cos.f64 42.1

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

5. Applied rewrites 42.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
7. Step-by-step derivation

8. Applied rewrites 26.4%

   \[\leadsto \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]

9. Taylor expanded in lambda1 around 0

   \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(\lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]
10. Step-by-step derivation

11. Applied rewrites 12.0%

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2 \cdot \lambda_1, -0.5, \sin \lambda_2\right), \lambda_1, \cos \lambda_2\right)\right) \cdot R \]

12. Taylor expanded in lambda2 around 0

    \[\leadsto \cos^{-1} \left(1 + \frac{-1}{2} \cdot {\lambda_1}^{\color{blue}{2}}\right) \cdot R \]
13. Step-by-step derivation

14. Applied rewrites 2.9%

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_1 \cdot \lambda_1, -0.5, 1\right)\right) \cdot R \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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))