Spherical law of cosines

Percentage Accurate: 73.5% → 94.2%

Time: 14.3s

Alternatives: 29

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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: 73.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 0.6× speedup

Math

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

FPCore

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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(lambda2) * cos(lambda1)) - (-sin(lambda2) * sin(lambda1)))))) * 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(lambda2) * Math.cos(lambda1)) - (-Math.sin(lambda2) * Math.sin(lambda1)))))) * 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(lambda2) * math.cos(lambda1)) - (-math.sin(lambda2) * math.sin(lambda1)))))) * R

Julia

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

MATLAB

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

Derivation

1. Initial program 73.5%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. sub-flip N/A

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

   4. +-commutative N/A

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

   5. cos-sum N/A

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

   6. cos-neg N/A

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

   7. cos-neg-rev N/A

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

   8. mul-1-neg N/A

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

   9. lower--.f64 N/A

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

   10. cos-neg N/A

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

   11. mul-1-neg N/A

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

   12. cos-neg-rev N/A

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

   13. lower-*.f64 N/A

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

   14. cos-neg N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. sin-neg N/A

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

   19. lower-neg.f64 N/A

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

   20. lower-sin.f64 N/A

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

   21. lower-sin.f64 94.2

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

3. Applied rewrites 94.2%

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

Alternative 2: 94.2% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 73.5%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. sub-flip N/A

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

   4. +-commutative N/A

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

   5. cos-sum N/A

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

   6. cos-neg N/A

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

   7. cos-neg-rev N/A

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

   8. mul-1-neg N/A

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

   9. lower--.f64 N/A

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

   10. cos-neg N/A

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

   11. mul-1-neg N/A

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

   12. cos-neg-rev N/A

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

   13. lower-*.f64 N/A

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

   14. cos-neg N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. sin-neg N/A

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

   19. lower-neg.f64 N/A

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

   20. lower-sin.f64 N/A

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

   21. lower-sin.f64 94.2

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

3. Applied rewrites 94.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. sub-flip N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-neg.f64 N/A

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

   16. lift-sin.f64 N/A

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

   17. lift-*.f64 94.2

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

5. Applied rewrites 94.2%

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

6. Taylor expanded in lambda1 around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 94.2

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

8. Applied rewrites 94.2%

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

Alternative 3: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.36:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 55000000:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333 - 0.0001984126984126984 \cdot \left(\phi_2 \cdot \phi_2\right), -0.16666666666666666\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (/ PI 2.0)
           (-
            (* PI 0.5)
            (acos
             (fma
              (cos (- lambda2 lambda1))
              (* (cos phi2) (cos phi1))
              (* (sin phi2) (sin phi1))))))
          R)))
   (if (<= phi2 -0.36)
     t_0
     (if (<= phi2 55000000.0)
       (*
        (acos
         (+
          (*
           (sin phi1)
           (*
            phi2
            (+
             1.0
             (*
              (* phi2 phi2)
              (fma
               (* phi2 phi2)
               (- 0.008333333333333333 (* 0.0001984126984126984 (* phi2 phi2)))
               -0.16666666666666666)))))
          (*
           (* (cos phi1) (cos phi2))
           (-
            (* (cos lambda2) (cos lambda1))
            (* (- (sin lambda2)) (sin lambda1))))))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(cos((lambda2 - lambda1)), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))))) * R;
	double tmp;
	if (phi2 <= -0.36) {
		tmp = t_0;
	} else if (phi2 <= 55000000.0) {
		tmp = acos(((sin(phi1) * (phi2 * (1.0 + ((phi2 * phi2) * fma((phi2 * phi2), (0.008333333333333333 - (0.0001984126984126984 * (phi2 * phi2))), -0.16666666666666666))))) + ((cos(phi1) * cos(phi2)) * ((cos(lambda2) * cos(lambda1)) - (-sin(lambda2) * sin(lambda1)))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(cos(Float64(lambda2 - lambda1)), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))))) * R)
	tmp = 0.0
	if (phi2 <= -0.36)
		tmp = t_0;
	elseif (phi2 <= 55000000.0)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * Float64(phi2 * Float64(1.0 + Float64(Float64(phi2 * phi2) * fma(Float64(phi2 * phi2), Float64(0.008333333333333333 - Float64(0.0001984126984126984 * Float64(phi2 * phi2))), -0.16666666666666666))))) + Float64(Float64(cos(phi1) * cos(phi2)) * Float64(Float64(cos(lambda2) * cos(lambda1)) - Float64(Float64(-sin(lambda2)) * sin(lambda1)))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -0.36], t$95$0, If[LessEqual[phi2, 55000000.0], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[(phi2 * N[(1.0 + N[(N[(phi2 * phi2), $MachinePrecision] * N[(N[(phi2 * phi2), $MachinePrecision] * N[(0.008333333333333333 - N[(0.0001984126984126984 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda2], $MachinePrecision]) * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.35999999999999999 or 5.5e7 < phi2

   1. Initial program 78.4%

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

      1. lift-acos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. acos-asin N/A

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

      13. lower--.f64 N/A

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

   3. Applied rewrites 78.3%

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

      1. lift-asin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. asin-acos N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. acos-asin-rev N/A

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

   5. Applied rewrites 78.4%

      \[\leadsto \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right) \cdot R \]

   if -0.35999999999999999 < phi2 < 5.5e7

   1. Initial program 68.7%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 89.3

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

   3. Applied rewrites 89.3%

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

   4. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      2. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{{\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)}\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \color{blue}{\left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)}\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      4. pow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\color{blue}{{\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right)} - \frac{1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      5. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\color{blue}{{\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right)} - \frac{1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      6. sub-flip N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      7. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) + \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      8. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left({\phi_2}^{2}, \color{blue}{\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}}, \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      9. pow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {\phi_2}^{2}, \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      10. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {\phi_2}^{2}, \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\phi_2}^{2}}, \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\phi_2}^{2}}, \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      13. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120} - \frac{1}{5040} \cdot {\color{blue}{\phi_2}}^{2}, \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      14. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120} - \frac{1}{5040} \cdot \color{blue}{{\phi_2}^{2}}, \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      15. pow2 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120} - \frac{1}{5040} \cdot \left(\phi_2 \cdot \color{blue}{\phi_2}\right), \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      16. lift-*.f64 88.6

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333 - 0.0001984126984126984 \cdot \left(\phi_2 \cdot \color{blue}{\phi_2}\right), -0.16666666666666666\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

   6. Applied rewrites 88.6%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333 - 0.0001984126984126984 \cdot \left(\phi_2 \cdot \phi_2\right), -0.16666666666666666\right)\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.34:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 55000000:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(0.008333333333333333, \phi_2 \cdot \phi_2, -0.16666666666666666\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (/ PI 2.0)
           (-
            (* PI 0.5)
            (acos
             (fma
              (cos (- lambda2 lambda1))
              (* (cos phi2) (cos phi1))
              (* (sin phi2) (sin phi1))))))
          R)))
   (if (<= phi2 -0.34)
     t_0
     (if (<= phi2 55000000.0)
       (*
        (acos
         (+
          (*
           (sin phi1)
           (*
            phi2
            (+
             1.0
             (*
              (* phi2 phi2)
              (fma 0.008333333333333333 (* phi2 phi2) -0.16666666666666666)))))
          (*
           (* (cos phi1) (cos phi2))
           (-
            (* (cos lambda2) (cos lambda1))
            (* (- (sin lambda2)) (sin lambda1))))))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(cos((lambda2 - lambda1)), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))))) * R;
	double tmp;
	if (phi2 <= -0.34) {
		tmp = t_0;
	} else if (phi2 <= 55000000.0) {
		tmp = acos(((sin(phi1) * (phi2 * (1.0 + ((phi2 * phi2) * fma(0.008333333333333333, (phi2 * phi2), -0.16666666666666666))))) + ((cos(phi1) * cos(phi2)) * ((cos(lambda2) * cos(lambda1)) - (-sin(lambda2) * sin(lambda1)))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(cos(Float64(lambda2 - lambda1)), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))))) * R)
	tmp = 0.0
	if (phi2 <= -0.34)
		tmp = t_0;
	elseif (phi2 <= 55000000.0)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * Float64(phi2 * Float64(1.0 + Float64(Float64(phi2 * phi2) * fma(0.008333333333333333, Float64(phi2 * phi2), -0.16666666666666666))))) + Float64(Float64(cos(phi1) * cos(phi2)) * Float64(Float64(cos(lambda2) * cos(lambda1)) - Float64(Float64(-sin(lambda2)) * sin(lambda1)))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -0.34], t$95$0, If[LessEqual[phi2, 55000000.0], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[(phi2 * N[(1.0 + N[(N[(phi2 * phi2), $MachinePrecision] * N[(0.008333333333333333 * N[(phi2 * phi2), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda2], $MachinePrecision]) * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.340000000000000024 or 5.5e7 < phi2

   1. Initial program 78.4%

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

      1. lift-acos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. acos-asin N/A

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

      13. lower--.f64 N/A

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

   3. Applied rewrites 78.3%

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

      1. lift-asin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. asin-acos N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. acos-asin-rev N/A

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

   5. Applied rewrites 78.4%

      \[\leadsto \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right) \cdot R \]

   if -0.340000000000000024 < phi2 < 5.5e7

   1. Initial program 68.7%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 89.3

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

   3. Applied rewrites 89.3%

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

   4. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      2. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \color{blue}{{\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)}\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)}\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      4. pow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {\phi_2}^{2}} - \frac{1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      5. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {\phi_2}^{2}} - \frac{1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      6. sub-flip N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      7. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} + \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      8. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{{\phi_2}^{2}}, \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      9. pow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \phi_2 \cdot \color{blue}{\phi_2}, \frac{-1}{6}\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      10. lift-*.f64 88.5

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(0.008333333333333333, \phi_2 \cdot \color{blue}{\phi_2}, -0.16666666666666666\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

   6. Applied rewrites 88.5%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(0.008333333333333333, \phi_2 \cdot \phi_2, -0.16666666666666666\right)\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 55000000:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \left(\phi_2 \cdot \left(1 - 0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (/ PI 2.0)
           (-
            (* PI 0.5)
            (acos
             (fma
              (cos (- lambda2 lambda1))
              (* (cos phi2) (cos phi1))
              (* (sin phi2) (sin phi1))))))
          R)))
   (if (<= phi2 -0.3)
     t_0
     (if (<= phi2 55000000.0)
       (*
        (acos
         (+
          (* (sin phi1) (* phi2 (- 1.0 (* 0.16666666666666666 (* phi2 phi2)))))
          (*
           (* (cos phi1) (cos phi2))
           (-
            (* (cos lambda2) (cos lambda1))
            (* (- (sin lambda2)) (sin lambda1))))))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(cos((lambda2 - lambda1)), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))))) * R;
	double tmp;
	if (phi2 <= -0.3) {
		tmp = t_0;
	} else if (phi2 <= 55000000.0) {
		tmp = acos(((sin(phi1) * (phi2 * (1.0 - (0.16666666666666666 * (phi2 * phi2))))) + ((cos(phi1) * cos(phi2)) * ((cos(lambda2) * cos(lambda1)) - (-sin(lambda2) * sin(lambda1)))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(cos(Float64(lambda2 - lambda1)), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))))) * R)
	tmp = 0.0
	if (phi2 <= -0.3)
		tmp = t_0;
	elseif (phi2 <= 55000000.0)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * Float64(phi2 * Float64(1.0 - Float64(0.16666666666666666 * Float64(phi2 * phi2))))) + Float64(Float64(cos(phi1) * cos(phi2)) * Float64(Float64(cos(lambda2) * cos(lambda1)) - Float64(Float64(-sin(lambda2)) * sin(lambda1)))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -0.3], t$95$0, If[LessEqual[phi2, 55000000.0], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[(phi2 * N[(1.0 - N[(0.16666666666666666 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda2], $MachinePrecision]) * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.299999999999999989 or 5.5e7 < phi2

   1. Initial program 78.4%

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

      1. lift-acos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. acos-asin N/A

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

      13. lower--.f64 N/A

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

   3. Applied rewrites 78.3%

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

      1. lift-asin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. asin-acos N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. acos-asin-rev N/A

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

   5. Applied rewrites 78.4%

      \[\leadsto \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right) \cdot R \]

   if -0.299999999999999989 < phi2 < 5.5e7

   1. Initial program 68.7%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 89.3

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

   3. Applied rewrites 89.3%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 88.5

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

   6. Applied rewrites 88.5%

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

Alternative 6: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.00175:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 55000000:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (/ PI 2.0)
           (-
            (* PI 0.5)
            (acos
             (fma
              (cos (- lambda2 lambda1))
              (* (cos phi2) (cos phi1))
              (* (sin phi2) (sin phi1))))))
          R)))
   (if (<= phi2 -0.00175)
     t_0
     (if (<= phi2 55000000.0)
       (*
        (acos
         (+
          (* (sin phi1) phi2)
          (*
           (* (cos phi1) (cos phi2))
           (-
            (* (cos lambda2) (cos lambda1))
            (* (- (sin lambda2)) (sin lambda1))))))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(cos((lambda2 - lambda1)), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))))) * R;
	double tmp;
	if (phi2 <= -0.00175) {
		tmp = t_0;
	} else if (phi2 <= 55000000.0) {
		tmp = acos(((sin(phi1) * phi2) + ((cos(phi1) * cos(phi2)) * ((cos(lambda2) * cos(lambda1)) - (-sin(lambda2) * sin(lambda1)))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(cos(Float64(lambda2 - lambda1)), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))))) * R)
	tmp = 0.0
	if (phi2 <= -0.00175)
		tmp = t_0;
	elseif (phi2 <= 55000000.0)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * phi2) + Float64(Float64(cos(phi1) * cos(phi2)) * Float64(Float64(cos(lambda2) * cos(lambda1)) - Float64(Float64(-sin(lambda2)) * sin(lambda1)))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -0.00175], t$95$0, If[LessEqual[phi2, 55000000.0], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda2], $MachinePrecision]) * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.00175000000000000004 or 5.5e7 < phi2

   1. Initial program 78.4%

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

      1. lift-acos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. acos-asin N/A

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

      13. lower--.f64 N/A

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

   3. Applied rewrites 78.3%

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

      1. lift-asin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. asin-acos N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. acos-asin-rev N/A

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

   5. Applied rewrites 78.4%

      \[\leadsto \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right) \cdot R \]

   if -0.00175000000000000004 < phi2 < 5.5e7

   1. Initial program 68.6%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 89.3

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

   3. Applied rewrites 89.3%

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

   4. Taylor expanded in phi2 around 0

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

   6. Applied rewrites 88.4%

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

Alternative 7: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -1.15 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 2.7 \cdot 10^{-18}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \left(-\sin \lambda_1\right) \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (/ PI 2.0)
           (-
            (* PI 0.5)
            (acos
             (fma
              (cos (- lambda2 lambda1))
              (* (cos phi2) (cos phi1))
              (* (sin phi2) (sin phi1))))))
          R)))
   (if (<= phi2 -1.15e-14)
     t_0
     (if (<= phi2 2.7e-18)
       (*
        (acos
         (*
          (cos phi1)
          (-
           (* (cos lambda1) (cos lambda2))
           (* (- (sin lambda1)) (sin lambda2)))))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(cos((lambda2 - lambda1)), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))))) * R;
	double tmp;
	if (phi2 <= -1.15e-14) {
		tmp = t_0;
	} else if (phi2 <= 2.7e-18) {
		tmp = acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) - (-sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(cos(Float64(lambda2 - lambda1)), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))))) * R)
	tmp = 0.0
	if (phi2 <= -1.15e-14)
		tmp = t_0;
	elseif (phi2 <= 2.7e-18)
		tmp = Float64(acos(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) - Float64(Float64(-sin(lambda1)) * sin(lambda2))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -1.15e-14], t$95$0, If[LessEqual[phi2, 2.7e-18], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda1], $MachinePrecision]) * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.14999999999999999e-14 or 2.69999999999999989e-18 < phi2

   1. Initial program 77.7%

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

      1. lift-acos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. acos-asin N/A

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

      13. lower--.f64 N/A

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

   3. Applied rewrites 77.6%

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

      1. lift-asin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. asin-acos N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. acos-asin-rev N/A

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

   5. Applied rewrites 77.7%

      \[\leadsto \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right) \cdot R \]

   if -1.14999999999999999e-14 < phi2 < 2.69999999999999989e-18

   1. Initial program 68.8%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 89.2

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

   3. Applied rewrites 89.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sub-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-*.f64 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-sin.f64 89.2

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

   8. Applied rewrites 89.2%

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

Alternative 8: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -6.6 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 2.7 \cdot 10^{-18}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (/ PI 2.0)
           (-
            (* PI 0.5)
            (acos
             (fma
              (cos (- lambda2 lambda1))
              (* (cos phi2) (cos phi1))
              (* (sin phi2) (sin phi1))))))
          R)))
   (if (<= phi2 -6.6e-7)
     t_0
     (if (<= phi2 2.7e-18)
       (*
        (acos
         (+
          (* (sin phi1) (sin phi2))
          (*
           (cos phi1)
           (-
            (* (cos lambda2) (cos lambda1))
            (* (- (sin lambda2)) (sin lambda1))))))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(cos((lambda2 - lambda1)), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))))) * R;
	double tmp;
	if (phi2 <= -6.6e-7) {
		tmp = t_0;
	} else if (phi2 <= 2.7e-18) {
		tmp = acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * ((cos(lambda2) * cos(lambda1)) - (-sin(lambda2) * sin(lambda1)))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(cos(Float64(lambda2 - lambda1)), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))))) * R)
	tmp = 0.0
	if (phi2 <= -6.6e-7)
		tmp = t_0;
	elseif (phi2 <= 2.7e-18)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) - Float64(Float64(-sin(lambda2)) * sin(lambda1)))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -6.6e-7], t$95$0, If[LessEqual[phi2, 2.7e-18], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda2], $MachinePrecision]) * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -6.6000000000000003e-7 or 2.69999999999999989e-18 < phi2

   1. Initial program 77.8%

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

      1. lift-acos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. acos-asin N/A

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

      13. lower--.f64 N/A

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

   3. Applied rewrites 77.7%

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

      1. lift-asin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. asin-acos N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. acos-asin-rev N/A

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

   5. Applied rewrites 77.8%

      \[\leadsto \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right) \cdot R \]

   if -6.6000000000000003e-7 < phi2 < 2.69999999999999989e-18

   1. Initial program 68.9%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 89.2

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

   3. Applied rewrites 89.2%

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

   4. Taylor expanded in phi2 around 0

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

      1. lift-cos.f64 89.1

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

   6. Applied rewrites 89.1%

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

Alternative 9: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -1.15 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \left(-\sin \lambda_1\right) \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (+
            (* (sin phi1) (sin phi2))
            (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
          R)))
   (if (<= phi2 -1.15e-14)
     t_0
     (if (<= phi2 1.05e-47)
       (*
        (acos
         (*
          (cos phi1)
          (-
           (* (cos lambda1) (cos lambda2))
           (* (- (sin lambda1)) (sin lambda2)))))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
	double tmp;
	if (phi2 <= -1.15e-14) {
		tmp = t_0;
	} else if (phi2 <= 1.05e-47) {
		tmp = acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) - (-sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
    if (phi2 <= (-1.15d-14)) then
        tmp = t_0
    else if (phi2 <= 1.05d-47) then
        tmp = acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) - (-sin(lambda1) * sin(lambda2))))) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R;
	double tmp;
	if (phi2 <= -1.15e-14) {
		tmp = t_0;
	} else if (phi2 <= 1.05e-47) {
		tmp = Math.acos((Math.cos(phi1) * ((Math.cos(lambda1) * Math.cos(lambda2)) - (-Math.sin(lambda1) * Math.sin(lambda2))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R
	tmp = 0
	if phi2 <= -1.15e-14:
		tmp = t_0
	elif phi2 <= 1.05e-47:
		tmp = math.acos((math.cos(phi1) * ((math.cos(lambda1) * math.cos(lambda2)) - (-math.sin(lambda1) * math.sin(lambda2))))) * R
	else:
		tmp = t_0
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
	tmp = 0.0
	if (phi2 <= -1.15e-14)
		tmp = t_0;
	elseif (phi2 <= 1.05e-47)
		tmp = Float64(acos(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) - Float64(Float64(-sin(lambda1)) * sin(lambda2))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
	tmp = 0.0;
	if (phi2 <= -1.15e-14)
		tmp = t_0;
	elseif (phi2 <= 1.05e-47)
		tmp = acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) - (-sin(lambda1) * sin(lambda2))))) * R;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = 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]}, If[LessEqual[phi2, -1.15e-14], t$95$0, If[LessEqual[phi2, 1.05e-47], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda1], $MachinePrecision]) * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.14999999999999999e-14 or 1.05e-47 < phi2

   1. Initial program 77.2%

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

   if -1.14999999999999999e-14 < phi2 < 1.05e-47

   1. Initial program 69.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 89.3

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

   3. Applied rewrites 89.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sub-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-sin.f64 89.2

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

   8. Applied rewrites 89.2%

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

Alternative 10: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -1.15 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \left(-\sin \lambda_1\right) \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (sin phi2)
            (sin phi1)
            (* (* (cos (- lambda2 lambda1)) (cos phi2)) (cos phi1))))
          R)))
   (if (<= phi2 -1.15e-14)
     t_0
     (if (<= phi2 1.05e-47)
       (*
        (acos
         (*
          (cos phi1)
          (-
           (* (cos lambda1) (cos lambda2))
           (* (- (sin lambda1)) (sin lambda2)))))
        R)
       t_0))))

C

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(phi2)) * cos(phi1)))) * R;
	double tmp;
	if (phi2 <= -1.15e-14) {
		tmp = t_0;
	} else if (phi2 <= 1.05e-47) {
		tmp = acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) - (-sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi2)) * cos(phi1)))) * R)
	tmp = 0.0
	if (phi2 <= -1.15e-14)
		tmp = t_0;
	elseif (phi2 <= 1.05e-47)
		tmp = Float64(acos(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) - Float64(Float64(-sin(lambda1)) * sin(lambda2))))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -1.15e-14], t$95$0, If[LessEqual[phi2, 1.05e-47], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda1], $MachinePrecision]) * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.14999999999999999e-14 or 1.05e-47 < phi2

   1. Initial program 77.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   3. Applied rewrites 77.2%

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

   if -1.14999999999999999e-14 < phi2 < 1.05e-47

   1. Initial program 69.0%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 89.3

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

   3. Applied rewrites 89.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sub-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-sin.f64 89.2

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

   8. Applied rewrites 89.2%

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

Alternative 11: 73.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2 - \left(-\sin \lambda_1\right) \cdot \sin \lambda_2\\ \mathbf{if}\;\lambda_1 \leq -6.8 \cdot 10^{+288}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq -1.16 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq 0.054:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot t\_0\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (-
          (* (cos lambda1) (cos lambda2))
          (* (- (sin lambda1)) (sin lambda2)))))
   (if (<= lambda1 -6.8e+288)
     (* (acos (* (cos phi2) t_0)) R)
     (if (<= lambda1 -1.16e-6)
       (*
        (acos
         (fma
          (cos lambda1)
          (* (cos phi2) (cos phi1))
          (* (sin phi2) (sin phi1))))
        R)
       (if (<= lambda1 0.054)
         (*
          (acos
           (fma
            (cos lambda2)
            (* (cos phi1) (cos phi2))
            (* (sin phi1) (sin phi2))))
          R)
         (* (acos (* (cos phi1) t_0)) R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos(lambda1) * cos(lambda2)) - (-sin(lambda1) * sin(lambda2));
	double tmp;
	if (lambda1 <= -6.8e+288) {
		tmp = acos((cos(phi2) * t_0)) * R;
	} else if (lambda1 <= -1.16e-6) {
		tmp = acos(fma(cos(lambda1), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))) * R;
	} else if (lambda1 <= 0.054) {
		tmp = acos(fma(cos(lambda2), (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))) * R;
	} else {
		tmp = acos((cos(phi1) * t_0)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(lambda1) * cos(lambda2)) - Float64(Float64(-sin(lambda1)) * sin(lambda2)))
	tmp = 0.0
	if (lambda1 <= -6.8e+288)
		tmp = Float64(acos(Float64(cos(phi2) * t_0)) * R);
	elseif (lambda1 <= -1.16e-6)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))) * R);
	elseif (lambda1 <= 0.054)
		tmp = Float64(acos(fma(cos(lambda2), Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi1) * t_0)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda1], $MachinePrecision]) * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -6.8e+288], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda1, -1.16e-6], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda1, 0.054], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -6.80000000000000038e288

   1. Initial program 49.4%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 98.8

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

   3. Applied rewrites 98.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sub-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-sin.f64 61.2

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

   8. Applied rewrites 61.2%

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

   if -6.80000000000000038e288 < lambda1 < -1.1599999999999999e-6

   1. Initial program 60.1%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 59.7

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

   4. Applied rewrites 59.7%

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

   if -1.1599999999999999e-6 < lambda1 < 0.0539999999999999994

   1. Initial program 88.5%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 88.8

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

   3. Applied rewrites 88.8%

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

   4. Taylor expanded in lambda1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-*.f64 88.2

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

   6. Applied rewrites 88.2%

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

   if 0.0539999999999999994 < lambda1

   1. Initial program 59.8%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 99.2

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

   3. Applied rewrites 99.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sub-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-sin.f64 59.6

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

   8. Applied rewrites 59.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \left(-\sin \lambda_1\right) \cdot \sin \lambda_2\right)\right)} \cdot R \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \left(-\sin \lambda_1\right) \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+250}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq -1.16 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq 0.054:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (*
            (cos phi1)
            (-
             (* (cos lambda1) (cos lambda2))
             (* (- (sin lambda1)) (sin lambda2)))))
          R)))
   (if (<= lambda1 -1.5e+250)
     t_0
     (if (<= lambda1 -1.16e-6)
       (*
        (acos
         (fma
          (cos lambda1)
          (* (cos phi2) (cos phi1))
          (* (sin phi2) (sin phi1))))
        R)
       (if (<= lambda1 0.054)
         (*
          (acos
           (fma
            (cos lambda2)
            (* (cos phi1) (cos phi2))
            (* (sin phi1) (sin phi2))))
          R)
         t_0)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) - (-sin(lambda1) * sin(lambda2))))) * R;
	double tmp;
	if (lambda1 <= -1.5e+250) {
		tmp = t_0;
	} else if (lambda1 <= -1.16e-6) {
		tmp = acos(fma(cos(lambda1), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))) * R;
	} else if (lambda1 <= 0.054) {
		tmp = acos(fma(cos(lambda2), (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) - Float64(Float64(-sin(lambda1)) * sin(lambda2))))) * R)
	tmp = 0.0
	if (lambda1 <= -1.5e+250)
		tmp = t_0;
	elseif (lambda1 <= -1.16e-6)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))) * R);
	elseif (lambda1 <= 0.054)
		tmp = Float64(acos(fma(cos(lambda2), Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[((-N[Sin[lambda1], $MachinePrecision]) * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[lambda1, -1.5e+250], t$95$0, If[LessEqual[lambda1, -1.16e-6], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda1, 0.054], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.49999999999999988e250 or 0.0539999999999999994 < lambda1

   1. Initial program 59.2%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

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

      12. cos-neg-rev N/A

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

      13. lower-*.f64 N/A

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. sin-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 99.2

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

   3. Applied rewrites 99.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sub-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-sin.f64 59.3

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

   8. Applied rewrites 59.3%

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

   if -1.49999999999999988e250 < lambda1 < -1.1599999999999999e-6

   1. Initial program 60.2%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 59.7

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

   4. Applied rewrites 59.7%

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

   if -1.1599999999999999e-6 < lambda1 < 0.0539999999999999994

   1. Initial program 88.5%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. cos-sum N/A

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

      6. cos-neg N/A

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

      7. cos-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      12. cos-neg-rev N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \color{blue}{\cos \lambda_1} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      13. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      14. cos-neg N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      15. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      16. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      17. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1}\right)\right) \cdot R \]

      18. sin-neg N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)} \cdot \sin \lambda_1\right)\right) \cdot R \]

      19. lower-neg.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \color{blue}{\left(-\sin \lambda_2\right)} \cdot \sin \lambda_1\right)\right) \cdot R \]

      20. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\color{blue}{\sin \lambda_2}\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      21. lower-sin.f64 88.8

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

   3. Applied rewrites 88.8%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)}\right) \cdot R \]

   4. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   5. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lift-*.f64 88.2

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   6. Applied rewrites 88.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.16 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.16e-6)
   (*
    (acos
     (fma (cos lambda1) (* (cos phi2) (cos phi1)) (* (sin phi2) (sin phi1))))
    R)
   (*
    (acos
     (fma (cos lambda2) (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.16e-6) {
		tmp = acos(fma(cos(lambda1), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))) * R;
	} else {
		tmp = acos(fma(cos(lambda2), (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.16e-6)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))) * R);
	else
		tmp = Float64(acos(fma(cos(lambda2), Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.16e-6], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.1599999999999999e-6

   1. Initial program 59.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \color{blue}{\phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      9. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      10. lift-sin.f64 59.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

   4. Applied rewrites 59.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if -1.1599999999999999e-6 < lambda1

   1. Initial program 78.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. sub-flip N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)}\right) \cdot R \]

      5. cos-sum N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1 - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)}\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      7. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \color{blue}{\left(-1 \cdot \lambda_1\right)} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      9. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \left(-1 \cdot \lambda_1\right) - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)}\right) \cdot R \]

      10. cos-neg N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \cos \left(-1 \cdot \lambda_1\right) - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      12. cos-neg-rev N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \color{blue}{\cos \lambda_1} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      13. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \lambda_1} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      14. cos-neg N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      15. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      16. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} - \sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      17. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \sin \lambda_1}\right)\right) \cdot R \]

      18. sin-neg N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \color{blue}{\left(\mathsf{neg}\left(\sin \lambda_2\right)\right)} \cdot \sin \lambda_1\right)\right) \cdot R \]

      19. lower-neg.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \color{blue}{\left(-\sin \lambda_2\right)} \cdot \sin \lambda_1\right)\right) \cdot R \]

      20. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\color{blue}{\sin \lambda_2}\right) \cdot \sin \lambda_1\right)\right) \cdot R \]

      21. lower-sin.f64 92.4

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

   3. Applied rewrites 92.4%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 - \left(-\sin \lambda_2\right) \cdot \sin \lambda_1\right)}\right) \cdot R \]

   4. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   5. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lift-*.f64 64.2

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   6. Applied rewrites 64.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -1.16 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, t\_0\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, t\_0\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi2) (sin phi1))))
   (if (<= lambda1 -1.16e-6)
     (* (acos (fma (cos lambda1) (* (cos phi2) (cos phi1)) t_0)) R)
     (* (acos (fma (* (cos lambda2) (cos phi2)) (cos phi1) t_0)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi2) * sin(phi1);
	double tmp;
	if (lambda1 <= -1.16e-6) {
		tmp = acos(fma(cos(lambda1), (cos(phi2) * cos(phi1)), t_0)) * R;
	} else {
		tmp = acos(fma((cos(lambda2) * cos(phi2)), cos(phi1), t_0)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi2) * sin(phi1))
	tmp = 0.0
	if (lambda1 <= -1.16e-6)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(phi2) * cos(phi1)), t_0)) * R);
	else
		tmp = Float64(acos(fma(Float64(cos(lambda2) * cos(phi2)), cos(phi1), t_0)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.16e-6], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.1599999999999999e-6

   1. Initial program 59.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \color{blue}{\phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      9. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      10. lift-sin.f64 59.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

   4. Applied rewrites 59.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if -1.1599999999999999e-6 < lambda1

   1. Initial program 78.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1 + \color{blue}{\sin \phi_1} \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2, \cos \color{blue}{\phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2, \cos \color{blue}{\phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      12. lift-sin.f64 64.2

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

   4. Applied rewrites 64.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.3e-5)
   (*
    (acos
     (fma (cos lambda1) (* (cos phi2) (cos phi1)) (* (sin phi2) (sin phi1))))
    R)
   (* (acos (* (cos (- lambda2 lambda1)) (cos phi1))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.3e-5) {
		tmp = acos(fma(cos(lambda1), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))) * R;
	} else {
		tmp = acos((cos((lambda2 - lambda1)) * cos(phi1))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.3e-5)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.3e-5], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.29999999999999992e-5

   1. Initial program 78.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \color{blue}{\phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      9. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      10. lift-sin.f64 65.5

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

   4. Applied rewrites 65.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if 1.29999999999999992e-5 < lambda2

   1. Initial program 57.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lift-cos.f64 39.4

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   4. Applied rewrites 39.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 53.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.4 \cdot 10^{+42}:\\ \;\;\;\;\cos^{-1} \left(t\_2 + t\_0 \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 3.35:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.0001984126984126984, 0.008333333333333333\right), \phi_2 \cdot \phi_2, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2\right) + t\_0 \cdot t\_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_2 + \cos \phi_2 \cdot t\_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -4.4e+42)
     (* (acos (+ t_2 (* t_0 (- 1.0 (* 0.5 (* lambda2 lambda2)))))) R)
     (if (<= phi2 3.35)
       (*
        (acos
         (+
          (*
           (sin phi1)
           (*
            (fma
             (fma
              (fma (* phi2 phi2) -0.0001984126984126984 0.008333333333333333)
              (* phi2 phi2)
              -0.16666666666666666)
             (* phi2 phi2)
             1.0)
            phi2))
          (* t_0 t_1)))
        R)
       (* (acos (+ t_2 (* (cos phi2) t_1))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -4.4e+42) {
		tmp = acos((t_2 + (t_0 * (1.0 - (0.5 * (lambda2 * lambda2)))))) * R;
	} else if (phi2 <= 3.35) {
		tmp = acos(((sin(phi1) * (fma(fma(fma((phi2 * phi2), -0.0001984126984126984, 0.008333333333333333), (phi2 * phi2), -0.16666666666666666), (phi2 * phi2), 1.0) * phi2)) + (t_0 * t_1))) * R;
	} else {
		tmp = acos((t_2 + (cos(phi2) * t_1))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -4.4e+42)
		tmp = Float64(acos(Float64(t_2 + Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(lambda2 * lambda2)))))) * R);
	elseif (phi2 <= 3.35)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * Float64(fma(fma(fma(Float64(phi2 * phi2), -0.0001984126984126984, 0.008333333333333333), Float64(phi2 * phi2), -0.16666666666666666), Float64(phi2 * phi2), 1.0) * phi2)) + Float64(t_0 * t_1))) * R);
	else
		tmp = Float64(acos(Float64(t_2 + Float64(cos(phi2) * t_1))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.4e+42], N[(N[ArcCos[N[(t$95$2 + N[(t$95$0 * N[(1.0 - N[(0.5 * N[(lambda2 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 3.35], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.4000000000000003e42

   1. Initial program 77.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 + \lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right)\right)}\right) \cdot R \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right) + \color{blue}{\cos \lambda_1}\right)\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \color{blue}{\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - \color{blue}{-1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      5. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      6. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      9. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(\mathsf{neg}\left(\sin \lambda_1\right)\right), \cos \lambda_1\right)\right) \cdot R \]

      10. lower-neg.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      12. lower-cos.f64 48.5

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

   4. Applied rewrites 48.5%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)}\right) \cdot R \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\lambda_2}^{2}}\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      2. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      3. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{2}\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{\color{blue}{2}}\right)\right) \cdot R \]

      5. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

      6. lower-*.f64 28.9

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

   7. Applied rewrites 28.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \color{blue}{0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)}\right)\right) \cdot R \]

   if -4.4000000000000003e42 < phi2 < 3.35000000000000009

   1. Initial program 69.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\phi_2}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\phi_2}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\phi_2}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 65.6%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.0001984126984126984, 0.008333333333333333\right), \phi_2 \cdot \phi_2, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 3.35000000000000009 < phi2

   1. Initial program 79.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. lift-cos.f64 47.5

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 47.5%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.4 \cdot 10^{+42}:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 0.345:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right), \phi_2 \cdot \phi_2, -0.5\right), \phi_2 \cdot \phi_2, 1\right)\right) \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \cos \phi_2 \cdot t\_0\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -4.4e+42)
     (*
      (acos
       (+
        t_1
        (* (* (cos phi1) (cos phi2)) (- 1.0 (* 0.5 (* lambda2 lambda2))))))
      R)
     (if (<= phi2 0.345)
       (*
        (acos
         (+
          t_1
          (*
           (*
            (cos phi1)
            (fma
             (fma
              (fma (* phi2 phi2) -0.001388888888888889 0.041666666666666664)
              (* phi2 phi2)
              -0.5)
             (* phi2 phi2)
             1.0))
           t_0)))
        R)
       (* (acos (+ t_1 (* (cos phi2) t_0))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -4.4e+42) {
		tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * (1.0 - (0.5 * (lambda2 * lambda2)))))) * R;
	} else if (phi2 <= 0.345) {
		tmp = acos((t_1 + ((cos(phi1) * fma(fma(fma((phi2 * phi2), -0.001388888888888889, 0.041666666666666664), (phi2 * phi2), -0.5), (phi2 * phi2), 1.0)) * t_0))) * R;
	} else {
		tmp = acos((t_1 + (cos(phi2) * t_0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -4.4e+42)
		tmp = Float64(acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * Float64(1.0 - Float64(0.5 * Float64(lambda2 * lambda2)))))) * R);
	elseif (phi2 <= 0.345)
		tmp = Float64(acos(Float64(t_1 + Float64(Float64(cos(phi1) * fma(fma(fma(Float64(phi2 * phi2), -0.001388888888888889, 0.041666666666666664), Float64(phi2 * phi2), -0.5), Float64(phi2 * phi2), 1.0)) * t_0))) * R);
	else
		tmp = Float64(acos(Float64(t_1 + Float64(cos(phi2) * t_0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.4e+42], N[(N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(lambda2 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 0.345], N[(N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + -0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.4000000000000003e42

   1. Initial program 77.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 + \lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right)\right)}\right) \cdot R \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right) + \color{blue}{\cos \lambda_1}\right)\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \color{blue}{\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - \color{blue}{-1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      5. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      6. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      9. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(\mathsf{neg}\left(\sin \lambda_1\right)\right), \cos \lambda_1\right)\right) \cdot R \]

      10. lower-neg.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      12. lower-cos.f64 48.5

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

   4. Applied rewrites 48.5%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)}\right) \cdot R \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\lambda_2}^{2}}\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      2. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      3. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{2}\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{\color{blue}{2}}\right)\right) \cdot R \]

      5. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

      6. lower-*.f64 28.9

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

   7. Applied rewrites 28.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \color{blue}{0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)}\right)\right) \cdot R \]

   if -4.4000000000000003e42 < phi2 < 0.34499999999999997

   1. Initial program 69.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}\right)\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left({\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left(\left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. sub-flip N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{\phi_2}}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) \cdot {\phi_2}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}\right) \cdot {\phi_2}^{2} + \frac{-1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_2}^{2}, {\phi_2}^{2}, \frac{-1}{2}\right), {\color{blue}{\phi_2}}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      8. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720} \cdot {\phi_2}^{2} + \frac{1}{24}, {\phi_2}^{2}, \frac{-1}{2}\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      9. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{-1}{720} + \frac{1}{24}, {\phi_2}^{2}, \frac{-1}{2}\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      10. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{720}, \frac{1}{24}\right), {\phi_2}^{2}, \frac{-1}{2}\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right), {\phi_2}^{2}, \frac{-1}{2}\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right), {\phi_2}^{2}, \frac{-1}{2}\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      13. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right), \phi_2 \cdot \phi_2, \frac{-1}{2}\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      14. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right), \phi_2 \cdot \phi_2, \frac{-1}{2}\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      15. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{720}, \frac{1}{24}\right), \phi_2 \cdot \phi_2, \frac{-1}{2}\right), \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      16. lower-*.f64 64.2

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right), \phi_2 \cdot \phi_2, -0.5\right), \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 64.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.001388888888888889, 0.041666666666666664\right), \phi_2 \cdot \phi_2, -0.5\right), \phi_2 \cdot \phi_2, 1\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 0.34499999999999997 < phi2

   1. Initial program 79.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. lift-cos.f64 47.6

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 47.6%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.4 \cdot 10^{+42}:\\ \;\;\;\;\cos^{-1} \left(t\_2 + t\_0 \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 3.75:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2\right) + t\_0 \cdot t\_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_2 + \cos \phi_2 \cdot t\_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -4.4e+42)
     (* (acos (+ t_2 (* t_0 (- 1.0 (* 0.5 (* lambda2 lambda2)))))) R)
     (if (<= phi2 3.75)
       (*
        (acos
         (+
          (*
           (sin phi1)
           (*
            (fma
             (fma (* phi2 phi2) 0.008333333333333333 -0.16666666666666666)
             (* phi2 phi2)
             1.0)
            phi2))
          (* t_0 t_1)))
        R)
       (* (acos (+ t_2 (* (cos phi2) t_1))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -4.4e+42) {
		tmp = acos((t_2 + (t_0 * (1.0 - (0.5 * (lambda2 * lambda2)))))) * R;
	} else if (phi2 <= 3.75) {
		tmp = acos(((sin(phi1) * (fma(fma((phi2 * phi2), 0.008333333333333333, -0.16666666666666666), (phi2 * phi2), 1.0) * phi2)) + (t_0 * t_1))) * R;
	} else {
		tmp = acos((t_2 + (cos(phi2) * t_1))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -4.4e+42)
		tmp = Float64(acos(Float64(t_2 + Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(lambda2 * lambda2)))))) * R);
	elseif (phi2 <= 3.75)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * Float64(fma(fma(Float64(phi2 * phi2), 0.008333333333333333, -0.16666666666666666), Float64(phi2 * phi2), 1.0) * phi2)) + Float64(t_0 * t_1))) * R);
	else
		tmp = Float64(acos(Float64(t_2 + Float64(cos(phi2) * t_1))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.4e+42], N[(N[ArcCos[N[(t$95$2 + N[(t$95$0 * N[(1.0 - N[(0.5 * N[(lambda2 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 3.75], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.4000000000000003e42

   1. Initial program 77.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 + \lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right)\right)}\right) \cdot R \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right) + \color{blue}{\cos \lambda_1}\right)\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \color{blue}{\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - \color{blue}{-1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      5. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      6. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      9. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(\mathsf{neg}\left(\sin \lambda_1\right)\right), \cos \lambda_1\right)\right) \cdot R \]

      10. lower-neg.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      12. lower-cos.f64 48.5

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

   4. Applied rewrites 48.5%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)}\right) \cdot R \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\lambda_2}^{2}}\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      2. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      3. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{2}\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{\color{blue}{2}}\right)\right) \cdot R \]

      5. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

      6. lower-*.f64 28.9

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

   7. Applied rewrites 28.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \color{blue}{0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)}\right)\right) \cdot R \]

   if -4.4000000000000003e42 < phi2 < 3.75

   1. Initial program 69.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{\phi_2}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{\phi_2}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left({\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right) + 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left(\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right) \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. sub-flip N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {\phi_2}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\phi_2}^{2}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\phi_2}^{2}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      8. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {\phi_2}^{2}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      9. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({\phi_2}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      10. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), {\phi_2}^{2}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      12. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{120}, \frac{-1}{6}\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      13. lower-*.f64 66.0

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 66.0%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.008333333333333333, -0.16666666666666666\right), \phi_2 \cdot \phi_2, 1\right) \cdot \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 3.75 < phi2

   1. Initial program 79.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. lift-cos.f64 47.5

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 47.5%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.4 \cdot 10^{+42}:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 0.24:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.041666666666666664, -0.5\right), \phi_2 \cdot \phi_2, 1\right)\right) \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \cos \phi_2 \cdot t\_0\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -4.4e+42)
     (*
      (acos
       (+
        t_1
        (* (* (cos phi1) (cos phi2)) (- 1.0 (* 0.5 (* lambda2 lambda2))))))
      R)
     (if (<= phi2 0.24)
       (*
        (acos
         (+
          t_1
          (*
           (*
            (cos phi1)
            (fma
             (fma (* phi2 phi2) 0.041666666666666664 -0.5)
             (* phi2 phi2)
             1.0))
           t_0)))
        R)
       (* (acos (+ t_1 (* (cos phi2) t_0))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -4.4e+42) {
		tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * (1.0 - (0.5 * (lambda2 * lambda2)))))) * R;
	} else if (phi2 <= 0.24) {
		tmp = acos((t_1 + ((cos(phi1) * fma(fma((phi2 * phi2), 0.041666666666666664, -0.5), (phi2 * phi2), 1.0)) * t_0))) * R;
	} else {
		tmp = acos((t_1 + (cos(phi2) * t_0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -4.4e+42)
		tmp = Float64(acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * Float64(1.0 - Float64(0.5 * Float64(lambda2 * lambda2)))))) * R);
	elseif (phi2 <= 0.24)
		tmp = Float64(acos(Float64(t_1 + Float64(Float64(cos(phi1) * fma(fma(Float64(phi2 * phi2), 0.041666666666666664, -0.5), Float64(phi2 * phi2), 1.0)) * t_0))) * R);
	else
		tmp = Float64(acos(Float64(t_1 + Float64(cos(phi2) * t_0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.4e+42], N[(N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(lambda2 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 0.24], N[(N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[(phi2 * phi2), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.4000000000000003e42

   1. Initial program 77.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 + \lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right)\right)}\right) \cdot R \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right) + \color{blue}{\cos \lambda_1}\right)\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \color{blue}{\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - \color{blue}{-1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      5. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      6. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      9. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(\mathsf{neg}\left(\sin \lambda_1\right)\right), \cos \lambda_1\right)\right) \cdot R \]

      10. lower-neg.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      12. lower-cos.f64 48.5

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

   4. Applied rewrites 48.5%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)}\right) \cdot R \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\lambda_2}^{2}}\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      2. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      3. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{2}\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{\color{blue}{2}}\right)\right) \cdot R \]

      5. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

      6. lower-*.f64 28.9

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

   7. Applied rewrites 28.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \color{blue}{0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)}\right)\right) \cdot R \]

   if -4.4000000000000003e42 < phi2 < 0.23999999999999999

   1. Initial program 69.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left({\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \left(\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right) \cdot {\phi_2}^{2} + 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}, \color{blue}{{\phi_2}^{2}}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. sub-flip N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\phi_2}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{\phi_2}}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left({\phi_2}^{2} \cdot \frac{1}{24} + \frac{-1}{2}, {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left({\phi_2}^{2}, \frac{1}{24}, \frac{-1}{2}\right), {\color{blue}{\phi_2}}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      8. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{24}, \frac{-1}{2}\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{24}, \frac{-1}{2}\right), {\phi_2}^{2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      10. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{1}{24}, \frac{-1}{2}\right), \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-*.f64 64.2

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.041666666666666664, -0.5\right), \phi_2 \cdot \color{blue}{\phi_2}, 1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 64.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, 0.041666666666666664, -0.5\right), \phi_2 \cdot \phi_2, 1\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 0.23999999999999999 < phi2

   1. Initial program 79.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. lift-cos.f64 47.6

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 47.6%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 52.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -4.4 \cdot 10^{+42}:\\ \;\;\;\;\cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 0.24:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right)\right)\right) \cdot t\_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_0 + \cos \phi_2 \cdot t\_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= phi2 -4.4e+42)
     (*
      (acos
       (+
        t_0
        (* (* (cos phi1) (cos phi2)) (- 1.0 (* 0.5 (* lambda2 lambda2))))))
      R)
     (if (<= phi2 0.24)
       (*
        (acos
         (+
          (* (sin phi1) (* (fma (* phi2 phi2) -0.16666666666666666 1.0) phi2))
          (*
           (*
            (cos phi1)
            (+
             1.0
             (* (* phi2 phi2) (fma 0.041666666666666664 (* phi2 phi2) -0.5))))
           t_1)))
        R)
       (* (acos (+ t_0 (* (cos phi2) t_1))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -4.4e+42) {
		tmp = acos((t_0 + ((cos(phi1) * cos(phi2)) * (1.0 - (0.5 * (lambda2 * lambda2)))))) * R;
	} else if (phi2 <= 0.24) {
		tmp = acos(((sin(phi1) * (fma((phi2 * phi2), -0.16666666666666666, 1.0) * phi2)) + ((cos(phi1) * (1.0 + ((phi2 * phi2) * fma(0.041666666666666664, (phi2 * phi2), -0.5)))) * t_1))) * R;
	} else {
		tmp = acos((t_0 + (cos(phi2) * t_1))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= -4.4e+42)
		tmp = Float64(acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * Float64(1.0 - Float64(0.5 * Float64(lambda2 * lambda2)))))) * R);
	elseif (phi2 <= 0.24)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * Float64(fma(Float64(phi2 * phi2), -0.16666666666666666, 1.0) * phi2)) + Float64(Float64(cos(phi1) * Float64(1.0 + Float64(Float64(phi2 * phi2) * fma(0.041666666666666664, Float64(phi2 * phi2), -0.5)))) * t_1))) * R);
	else
		tmp = Float64(acos(Float64(t_0 + Float64(cos(phi2) * t_1))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -4.4e+42], N[(N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(lambda2 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 0.24], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[(1.0 + N[(N[(phi2 * phi2), $MachinePrecision] * N[(0.041666666666666664 * N[(phi2 * phi2), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.4000000000000003e42

   1. Initial program 77.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 + \lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right)\right)}\right) \cdot R \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\lambda_2 \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1\right) + \color{blue}{\cos \lambda_1}\right)\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \color{blue}{\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - -1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right) - \color{blue}{-1 \cdot \sin \lambda_1}, \cos \lambda_1\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      5. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\lambda_2 \cdot \cos \lambda_1\right) \cdot \frac{-1}{2} - \color{blue}{-1} \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      6. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - -1 \cdot \sin \lambda_1, \cos \lambda_1\right)\right) \cdot R \]

      9. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(\mathsf{neg}\left(\sin \lambda_1\right)\right), \cos \lambda_1\right)\right) \cdot R \]

      10. lower-neg.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot \frac{-1}{2} - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

      12. lower-cos.f64 48.5

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)\right) \cdot R \]

   4. Applied rewrites 48.5%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\lambda_2, \left(\cos \lambda_1 \cdot \lambda_2\right) \cdot -0.5 - \left(-\sin \lambda_1\right), \cos \lambda_1\right)}\right) \cdot R \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\lambda_2}^{2}}\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      2. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{{\lambda_2}^{2}}\right)\right) \cdot R \]

      3. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{2}\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot {\lambda_2}^{\color{blue}{2}}\right)\right) \cdot R \]

      5. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \frac{1}{2} \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

      6. lower-*.f64 28.9

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - 0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right) \cdot R \]

   7. Applied rewrites 28.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(1 - \color{blue}{0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)}\right)\right) \cdot R \]

   if -4.4000000000000003e42 < phi2 < 0.23999999999999999

   1. Initial program 69.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\phi_2 \cdot \left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \color{blue}{\phi_2}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right) \cdot \color{blue}{\phi_2}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left(\frac{-1}{6} \cdot {\phi_2}^{2} + 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\left({\phi_2}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left({\phi_2}^{2}, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lower-*.f64 66.2

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 66.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + \color{blue}{{\phi_2}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + {\phi_2}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {\phi_2}^{2} - \frac{1}{2}\right)}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. pow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {\phi_2}^{2}} - \frac{1}{2}\right)\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {\phi_2}^{2}} - \frac{1}{2}\right)\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. sub-flip N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \left(\frac{1}{24} \cdot {\phi_2}^{2} + \frac{-1}{2}\right)\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\frac{1}{24}, \color{blue}{{\phi_2}^{2}}, \frac{-1}{2}\right)\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      8. pow2 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{6}, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(\frac{1}{24}, \phi_2 \cdot \color{blue}{\phi_2}, \frac{-1}{2}\right)\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      9. lift-*.f64 64.1

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \color{blue}{\phi_2}, -0.5\right)\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 64.1%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \left(\mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.16666666666666666, 1\right) \cdot \phi_2\right) + \left(\cos \phi_1 \cdot \color{blue}{\left(1 + \left(\phi_2 \cdot \phi_2\right) \cdot \mathsf{fma}\left(0.041666666666666664, \phi_2 \cdot \phi_2, -0.5\right)\right)}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 0.23999999999999999 < phi2

   1. Initial program 79.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. lift-cos.f64 47.6

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 47.6%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 50.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.35e-6)
   (* (acos (* (cos (- lambda2 lambda1)) (cos phi1))) R)
   (*
    (acos
     (+ (* (sin phi1) (sin phi2)) (* (cos phi2) (cos (- lambda1 lambda2)))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.35e-6) {
		tmp = acos((cos((lambda2 - lambda1)) * cos(phi1))) * R;
	} else {
		tmp = acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * cos((lambda1 - lambda2))))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 <= 1.35d-6) then
        tmp = acos((cos((lambda2 - lambda1)) * cos(phi1))) * r
    else
        tmp = acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * cos((lambda1 - lambda2))))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.35e-6) {
		tmp = Math.acos((Math.cos((lambda2 - lambda1)) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2))))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.35e-6:
		tmp = math.acos((math.cos((lambda2 - lambda1)) * math.cos(phi1))) * R
	else:
		tmp = math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi2) * math.cos((lambda1 - lambda2))))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.35e-6)
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.35e-6)
		tmp = acos((cos((lambda2 - lambda1)) * cos(phi1))) * R;
	else
		tmp = acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * cos((lambda1 - lambda2))))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.35e-6], N[(N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.34999999999999999e-6

   1. Initial program 71.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lift-cos.f64 51.3

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   4. Applied rewrites 51.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if 1.34999999999999999e-6 < phi2

   1. Initial program 79.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   3. Step-by-step derivation

      1. lift-cos.f64 47.5

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 47.5%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 50.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 1.35e-6)
     (* (acos (* t_0 (cos phi1))) R)
     (* (acos (* t_0 (cos phi2))) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 1.35e-6) {
		tmp = acos((t_0 * cos(phi1))) * R;
	} else {
		tmp = acos((t_0 * cos(phi2))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 1.35d-6) then
        tmp = acos((t_0 * cos(phi1))) * r
    else
        tmp = acos((t_0 * cos(phi2))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 1.35e-6) {
		tmp = Math.acos((t_0 * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((t_0 * Math.cos(phi2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 1.35e-6:
		tmp = math.acos((t_0 * math.cos(phi1))) * R
	else:
		tmp = math.acos((t_0 * math.cos(phi2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 1.35e-6)
		tmp = Float64(acos(Float64(t_0 * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(t_0 * cos(phi2))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 1.35e-6)
		tmp = acos((t_0 * cos(phi1))) * R;
	else
		tmp = acos((t_0 * cos(phi2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 1.35e-6], 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 phi2 < 1.34999999999999999e-6

   1. Initial program 71.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lift-cos.f64 51.3

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   4. Applied rewrites 51.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if 1.34999999999999999e-6 < phi2

   1. Initial program 79.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_2}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_2}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot R \]

      7. lift-cos.f64 48.0

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot R \]

   4. Applied rewrites 48.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 42.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.16 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.16e-6)
   (* (acos (* (cos lambda1) (cos phi1))) R)
   (* (acos (* (cos lambda2) (cos phi1))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.16e-6) {
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	} else {
		tmp = acos((cos(lambda2) * cos(phi1))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-1.16d-6)) then
        tmp = acos((cos(lambda1) * cos(phi1))) * r
    else
        tmp = acos((cos(lambda2) * cos(phi1))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.16e-6) {
		tmp = Math.acos((Math.cos(lambda1) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((Math.cos(lambda2) * Math.cos(phi1))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.16e-6:
		tmp = math.acos((math.cos(lambda1) * math.cos(phi1))) * R
	else:
		tmp = math.acos((math.cos(lambda2) * math.cos(phi1))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.16e-6)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(lambda2) * cos(phi1))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.16e-6)
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	else
		tmp = acos((cos(lambda2) * cos(phi1))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.16e-6], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.1599999999999999e-6

   1. Initial program 59.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lift-cos.f64 39.5

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   4. Applied rewrites 39.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      2. lift-cos.f64 39.3

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

   7. Applied rewrites 39.3%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   if -1.1599999999999999e-6 < lambda1

   1. Initial program 78.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lift-cos.f64 44.1

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   4. Applied rewrites 44.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R \]
   6. Step-by-step derivation

   7. Applied rewrites 35.7%

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 36.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (* (cos (- lambda2 lambda1)) (cos phi1))) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos((cos((lambda2 - lambda1)) * cos(phi1))) * R;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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((lambda2 - lambda1)) * cos(phi1))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos((Math.cos((lambda2 - lambda1)) * Math.cos(phi1))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos((math.cos((lambda2 - lambda1)) * math.cos(phi1))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos((cos((lambda2 - lambda1)) * cos(phi1))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   3. sub-negate N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   4. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   6. lower--.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   7. lift-cos.f64 42.9

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

4. Applied rewrites 42.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]
5. Add Preprocessing

Alternative 25: 34.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.04:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{2} - \sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 0.04)
   (* (acos (* (cos lambda1) (cos phi1))) R)
   (* (- (/ PI 2.0) (asin (cos (- lambda2 lambda1)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.04) {
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	} else {
		tmp = ((((double) M_PI) / 2.0) - asin(cos((lambda2 - lambda1)))) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.04) {
		tmp = Math.acos((Math.cos(lambda1) * Math.cos(phi1))) * R;
	} else {
		tmp = ((Math.PI / 2.0) - Math.asin(Math.cos((lambda2 - lambda1)))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 0.04:
		tmp = math.acos((math.cos(lambda1) * math.cos(phi1))) * R
	else:
		tmp = ((math.pi / 2.0) - math.asin(math.cos((lambda2 - lambda1)))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 0.04)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R);
	else
		tmp = Float64(Float64(Float64(pi / 2.0) - asin(cos(Float64(lambda2 - lambda1)))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 0.04)
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	else
		tmp = ((pi / 2.0) - asin(cos((lambda2 - lambda1)))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 0.04], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.0400000000000000008

   1. Initial program 78.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lift-cos.f64 44.1

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   4. Applied rewrites 44.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      2. lift-cos.f64 36.5

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

   7. Applied rewrites 36.5%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   if 0.0400000000000000008 < lambda2

   1. Initial program 58.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lift-cos.f64 39.4

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   4. Applied rewrites 39.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

      2. lift--.f64 29.3

         \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   7. Applied rewrites 29.3%

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
   8. Step-by-step derivation

      1. lift-acos.f64 N/A

         \[\leadsto \color{blue}{\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]

      2. acos-asin N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} - \sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\pi}}{2} - \sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R \]

      6. lower-asin.f64 29.2

         \[\leadsto \left(\frac{\pi}{2} - \color{blue}{\sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   9. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 26.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \cos \lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \lambda_2 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -3.8e-8)
   (* (acos (cos lambda1)) R)
   (* (acos (cos lambda2)) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.8e-8) {
		tmp = acos(cos(lambda1)) * R;
	} else {
		tmp = acos(cos(lambda2)) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-3.8d-8)) then
        tmp = acos(cos(lambda1)) * r
    else
        tmp = acos(cos(lambda2)) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.8e-8) {
		tmp = Math.acos(Math.cos(lambda1)) * R;
	} else {
		tmp = Math.acos(Math.cos(lambda2)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -3.8e-8:
		tmp = math.acos(math.cos(lambda1)) * R
	else:
		tmp = math.acos(math.cos(lambda2)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -3.8e-8)
		tmp = Float64(acos(cos(lambda1)) * R);
	else
		tmp = Float64(acos(cos(lambda2)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -3.8e-8)
		tmp = acos(cos(lambda1)) * R;
	else
		tmp = acos(cos(lambda2)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -3.8e-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 lambda1 < -3.80000000000000028e-8

   1. Initial program 59.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lift-cos.f64 39.5

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   4. Applied rewrites 39.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

      2. lift--.f64 29.4

         \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   7. Applied rewrites 29.4%

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot R \]
   9. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

      2. lift-cos.f64 29.3

         \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

   10. Applied rewrites 29.3%

       \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

   if -3.80000000000000028e-8 < lambda1

   1. Initial program 78.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

      3. sub-negate N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      4. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lift-cos.f64 44.2

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   4. Applied rewrites 44.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

      2. lift--.f64 25.6

         \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   7. Applied rewrites 25.6%

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \cos \lambda_2 \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 19.2%

       \[\leadsto \cos^{-1} \cos \lambda_2 \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 21.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (cos (- lambda2 lambda1))) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(cos((lambda2 - lambda1))) * R;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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((lambda2 - lambda1))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(Math.cos((lambda2 - lambda1))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(math.cos((lambda2 - lambda1))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(cos(Float64(lambda2 - lambda1))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(cos((lambda2 - lambda1))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   3. sub-negate N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   4. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   6. lower--.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   7. lift-cos.f64 42.9

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

4. Applied rewrites 42.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   2. lift--.f64 26.6

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

7. Applied rewrites 26.6%

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
8. Add Preprocessing

Alternative 28: 17.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \cos \lambda_1 \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (cos lambda1)) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(cos(lambda1)) * R;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(Math.cos(lambda1)) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(math.cos(lambda1)) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(cos(lambda1)) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(cos(lambda1)) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   3. sub-negate N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   4. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   6. lower--.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   7. lift-cos.f64 42.9

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

4. Applied rewrites 42.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   2. lift--.f64 26.6

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

7. Applied rewrites 26.6%

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot R \]
9. Step-by-step derivation

   1. cos-neg-rev N/A

      \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

   2. lift-cos.f64 17.8

      \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

10. Applied rewrites 17.8%

    \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]
11. Add Preprocessing

Alternative 29: 4.3% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} 1 \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* (acos 1.0) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(1.0) * R;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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(1.0d0) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(1.0) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(1.0) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(1.0) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(1.0) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[1.0], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   3. sub-negate N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   4. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   6. lower--.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

   7. lift-cos.f64 42.9

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

4. Applied rewrites 42.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

   2. lift--.f64 26.6

      \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

7. Applied rewrites 26.6%

   \[\leadsto \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot R \]
9. Step-by-step derivation

   1. cos-neg-rev N/A

      \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

   2. lift-cos.f64 17.8

      \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

10. Applied rewrites 17.8%

    \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

11. Taylor expanded in lambda1 around 0

    \[\leadsto \cos^{-1} 1 \cdot R \]
12. Step-by-step derivation

13. Applied rewrites 4.3%

    \[\leadsto \cos^{-1} 1 \cdot R \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025128 
(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))