Spherical law of cosines

Percentage Accurate: 72.5% → 94.1%

Time: 17.9s

Alternatives: 26

Speedup: 1.0×

Specification

Math

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

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: 72.5% accurate, 1.0× speedup

Math

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

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: 94.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-sin.f64 N/A

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

   21. lower-sin.f64 94.1

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

3. Applied rewrites 94.1%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 94.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 94.1

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

7. Applied rewrites 94.1%

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

Alternative 2: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right), \mathsf{fma}\left(t\_0, \sin \lambda_2 \cdot \sin \lambda_1, \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot t\_0\right), \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2))))
   (*
    (acos
     (fma
      (cos (fmax phi1 phi2))
      (fma
       t_0
       (* (sin lambda2) (sin lambda1))
       (* (* (cos lambda2) (cos lambda1)) t_0))
      (* (sin (fmax phi1 phi2)) (sin (fmin phi1 phi2)))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2));
	return acos(fma(cos(fmax(phi1, phi2)), fma(t_0, (sin(lambda2) * sin(lambda1)), ((cos(lambda2) * cos(lambda1)) * t_0)), (sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))))) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	return Float64(acos(fma(cos(fmax(phi1, phi2)), fma(t_0, Float64(sin(lambda2) * sin(lambda1)), Float64(Float64(cos(lambda2) * cos(lambda1)) * t_0)), Float64(sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))))) * R)
end

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-sin.f64 N/A

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

   21. lower-sin.f64 94.1

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

3. Applied rewrites 94.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

5. Applied rewrites 94.1%

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

Alternative 3: 94.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-sin.f64 N/A

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

   21. lower-sin.f64 94.1

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

3. Applied rewrites 94.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

5. Applied rewrites 94.1%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. *-commutative N/A

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

   5. distribute-rgt-out-- N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 94.1

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 94.1

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

7. Applied rewrites 94.1%

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

Alternative 4: 94.1% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 72.5%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 94.1

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

3. Applied rewrites 94.1%

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

Alternative 5: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\phi_1 \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right) \cdot \cos \lambda_2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.0078:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\left(1 + \frac{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2}{t\_0}\right) \cdot t\_0\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi2) (sin phi1))))
   (if (<= phi1 -3.4e-9)
     (*
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (* (* (* (cos phi1) (cos phi2)) (cos lambda1)) (cos lambda2))))
      R)
     (if (<= phi1 0.0078)
       (*
        (acos
         (fma
          phi1
          (sin phi2)
          (*
           (cos phi2)
           (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))))
        R)
       (*
        (acos
         (*
          (+
           1.0
           (/ (* (* (cos (- lambda2 lambda1)) (cos phi1)) (cos phi2)) t_0))
          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 (phi1 <= -3.4e-9) {
		tmp = acos(((sin(phi1) * sin(phi2)) + (((cos(phi1) * cos(phi2)) * cos(lambda1)) * cos(lambda2)))) * R;
	} else if (phi1 <= 0.0078) {
		tmp = acos(fma(phi1, sin(phi2), (cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))))) * R;
	} else {
		tmp = acos(((1.0 + (((cos((lambda2 - lambda1)) * cos(phi1)) * cos(phi2)) / t_0)) * t_0)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi2) * sin(phi1))
	tmp = 0.0
	if (phi1 <= -3.4e-9)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)) * cos(lambda2)))) * R);
	elseif (phi1 <= 0.0078)
		tmp = Float64(acos(fma(phi1, sin(phi2), Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))))) * R);
	else
		tmp = Float64(acos(Float64(Float64(1.0 + Float64(Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1)) * cos(phi2)) / t_0)) * 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[phi1, -3.4e-9], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 0.0078], N[(N[ArcCos[N[(phi1 * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(1.0 + N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.3999999999999998e-9

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 72.8

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

   8. Applied rewrites 72.8%

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

   if -3.3999999999999998e-9 < phi1 < 0.0077999999999999996

   1. Initial program 72.5%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 35.3

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

   4. Applied rewrites 35.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 46.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 46.3

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

   6. Applied rewrites 46.3%

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

   if 0.0077999999999999996 < phi1

   1. Initial program 72.5%

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

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

      3. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 65.8%

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

Alternative 6: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right) \cdot \cos \lambda_2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.0078:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \frac{\sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)}{\pi \cdot 0.5}\right) \cdot \left(\pi \cdot 0.5\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.4e-9)
   (*
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (* (cos phi1) (cos phi2)) (cos lambda1)) (cos lambda2))))
    R)
   (if (<= phi1 0.0078)
     (*
      (acos
       (fma
        phi1
        (sin phi2)
        (*
         (cos phi2)
         (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))))
      R)
     (*
      (*
       (-
        1.0
        (/
         (asin
          (fma
           (cos (- lambda2 lambda1))
           (* (cos phi2) (cos phi1))
           (* (sin phi2) (sin phi1))))
         (* PI 0.5)))
       (* PI 0.5))
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.4e-9) {
		tmp = acos(((sin(phi1) * sin(phi2)) + (((cos(phi1) * cos(phi2)) * cos(lambda1)) * cos(lambda2)))) * R;
	} else if (phi1 <= 0.0078) {
		tmp = acos(fma(phi1, sin(phi2), (cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))))) * R;
	} else {
		tmp = ((1.0 - (asin(fma(cos((lambda2 - lambda1)), (cos(phi2) * cos(phi1)), (sin(phi2) * sin(phi1)))) / (((double) M_PI) * 0.5))) * (((double) M_PI) * 0.5)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.4e-9)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)) * cos(lambda2)))) * R);
	elseif (phi1 <= 0.0078)
		tmp = Float64(acos(fma(phi1, sin(phi2), Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))))) * R);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(asin(fma(cos(Float64(lambda2 - lambda1)), Float64(cos(phi2) * cos(phi1)), Float64(sin(phi2) * sin(phi1)))) / Float64(pi * 0.5))) * Float64(pi * 0.5)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.4e-9], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 0.0078], N[(N[ArcCos[N[(phi1 * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(1.0 - N[(N[ArcSin[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] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.3999999999999998e-9

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 72.8

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

   8. Applied rewrites 72.8%

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

   if -3.3999999999999998e-9 < phi1 < 0.0077999999999999996

   1. Initial program 72.5%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 35.3

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

   4. Applied rewrites 35.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 46.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 46.3

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

   6. Applied rewrites 46.3%

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

   if 0.0077999999999999996 < phi1

   1. Initial program 72.5%

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

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

      3. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 72.3%

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

Alternative 7: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -3.45 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(t\_3 \cdot t\_0 + \left(\left(t\_1 \cdot t\_2\right) \cdot \cos \lambda_1\right) \cdot \cos \lambda_2\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 0.0078:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \frac{\sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t\_2 \cdot t\_1, t\_0 \cdot t\_3\right)\right)}{\pi \cdot 0.5}\right) \cdot \left(\pi \cdot 0.5\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmax phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (cos (fmax phi1 phi2)))
        (t_3 (sin (fmin phi1 phi2))))
   (if (<= (fmin phi1 phi2) -3.45e-6)
     (*
      (acos (+ (* t_3 t_0) (* (* (* t_1 t_2) (cos lambda1)) (cos lambda2))))
      R)
     (if (<= (fmin phi1 phi2) 0.0078)
       (*
        (acos
         (*
          t_2
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (*
        (*
         (-
          1.0
          (/
           (asin (fma (cos (- lambda2 lambda1)) (* t_2 t_1) (* t_0 t_3)))
           (* PI 0.5)))
         (* PI 0.5))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmax(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = cos(fmax(phi1, phi2));
	double t_3 = sin(fmin(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -3.45e-6) {
		tmp = acos(((t_3 * t_0) + (((t_1 * t_2) * cos(lambda1)) * cos(lambda2)))) * R;
	} else if (fmin(phi1, phi2) <= 0.0078) {
		tmp = acos((t_2 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = ((1.0 - (asin(fma(cos((lambda2 - lambda1)), (t_2 * t_1), (t_0 * t_3))) / (((double) M_PI) * 0.5))) * (((double) M_PI) * 0.5)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmax(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = cos(fmax(phi1, phi2))
	t_3 = sin(fmin(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -3.45e-6)
		tmp = Float64(acos(Float64(Float64(t_3 * t_0) + Float64(Float64(Float64(t_1 * t_2) * cos(lambda1)) * cos(lambda2)))) * R);
	elseif (fmin(phi1, phi2) <= 0.0078)
		tmp = Float64(acos(Float64(t_2 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(asin(fma(cos(Float64(lambda2 - lambda1)), Float64(t_2 * t_1), Float64(t_0 * t_3))) / Float64(pi * 0.5))) * Float64(pi * 0.5)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -3.45e-6], N[(N[ArcCos[N[(N[(t$95$3 * t$95$0), $MachinePrecision] + N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 0.0078], N[(N[ArcCos[N[(t$95$2 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(1.0 - N[(N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.45e-6

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in lambda1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 72.8

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

   8. Applied rewrites 72.8%

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

   if -3.45e-6 < phi1 < 0.0077999999999999996

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 0.0077999999999999996 < phi1

   1. Initial program 72.5%

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

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

      3. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 72.3%

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

Alternative 8: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -3.45 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_3 \cdot t\_0, t\_1, t\_2\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 0.0078:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \frac{\sin^{-1} \left(\mathsf{fma}\left(t\_1, t\_0 \cdot t\_3, t\_2\right)\right)}{\pi \cdot 0.5}\right) \cdot \left(\pi \cdot 0.5\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (- lambda2 lambda1)))
        (t_2 (* (sin (fmax phi1 phi2)) (sin (fmin phi1 phi2))))
        (t_3 (cos (fmin phi1 phi2))))
   (if (<= (fmin phi1 phi2) -3.45e-6)
     (* (acos (fma (* t_3 t_0) t_1 t_2)) R)
     (if (<= (fmin phi1 phi2) 0.0078)
       (*
        (acos
         (*
          t_0
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (*
        (* (- 1.0 (/ (asin (fma t_1 (* t_0 t_3) t_2)) (* PI 0.5))) (* PI 0.5))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos((lambda2 - lambda1));
	double t_2 = sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2));
	double t_3 = cos(fmin(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -3.45e-6) {
		tmp = acos(fma((t_3 * t_0), t_1, t_2)) * R;
	} else if (fmin(phi1, phi2) <= 0.0078) {
		tmp = acos((t_0 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = ((1.0 - (asin(fma(t_1, (t_0 * t_3), t_2)) / (((double) M_PI) * 0.5))) * (((double) M_PI) * 0.5)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(Float64(lambda2 - lambda1))
	t_2 = Float64(sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2)))
	t_3 = cos(fmin(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -3.45e-6)
		tmp = Float64(acos(fma(Float64(t_3 * t_0), t_1, t_2)) * R);
	elseif (fmin(phi1, phi2) <= 0.0078)
		tmp = Float64(acos(Float64(t_0 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(asin(fma(t_1, Float64(t_0 * t_3), t_2)) / Float64(pi * 0.5))) * Float64(pi * 0.5)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -3.45e-6], N[(N[ArcCos[N[(N[(t$95$3 * t$95$0), $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 0.0078], N[(N[ArcCos[N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(1.0 - N[(N[ArcSin[N[(t$95$1 * N[(t$95$0 * t$95$3), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.45e-6

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 72.5%

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

   if -3.45e-6 < phi1 < 0.0077999999999999996

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 0.0077999999999999996 < phi1

   1. Initial program 72.5%

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

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

      3. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 72.3%

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

Alternative 9: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.45 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, t\_0, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.0078:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(t\_0 \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -3.45e-6)
     (* (acos (fma (* (cos phi1) (cos phi2)) t_0 (* (sin phi2) (sin phi1)))) R)
     (if (<= phi1 0.0078)
       (*
        (acos
         (*
          (cos phi2)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (*
        (acos (fma (sin phi2) (sin phi1) (* (* t_0 (cos phi1)) (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 (phi1 <= -3.45e-6) {
		tmp = acos(fma((cos(phi1) * cos(phi2)), t_0, (sin(phi2) * sin(phi1)))) * R;
	} else if (phi1 <= 0.0078) {
		tmp = acos((cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = acos(fma(sin(phi2), sin(phi1), ((t_0 * cos(phi1)) * cos(phi2)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -3.45e-6)
		tmp = Float64(acos(fma(Float64(cos(phi1) * cos(phi2)), t_0, Float64(sin(phi2) * sin(phi1)))) * R);
	elseif (phi1 <= 0.0078)
		tmp = Float64(acos(Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(t_0 * cos(phi1)) * cos(phi2)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.45e-6], N[(N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 0.0078], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.45e-6

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 72.5%

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

   if -3.45e-6 < phi1 < 0.0077999999999999996

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 0.0077999999999999996 < phi1

   1. Initial program 72.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 72.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 72.5

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lower--.f64 72.5

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

   3. Applied rewrites 72.5%

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

Alternative 10: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.45 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(t\_0 \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.0078:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(t\_0 \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -3.45e-6)
     (* (acos (fma (sin phi2) (sin phi1) (* (* t_0 (cos phi2)) (cos phi1)))) R)
     (if (<= phi1 0.0078)
       (*
        (acos
         (*
          (cos phi2)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (*
        (acos (fma (sin phi2) (sin phi1) (* (* t_0 (cos phi1)) (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 (phi1 <= -3.45e-6) {
		tmp = acos(fma(sin(phi2), sin(phi1), ((t_0 * cos(phi2)) * cos(phi1)))) * R;
	} else if (phi1 <= 0.0078) {
		tmp = acos((cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = acos(fma(sin(phi2), sin(phi1), ((t_0 * cos(phi1)) * cos(phi2)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -3.45e-6)
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(t_0 * cos(phi2)) * cos(phi1)))) * R);
	elseif (phi1 <= 0.0078)
		tmp = Float64(acos(Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(t_0 * cos(phi1)) * cos(phi2)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.45e-6], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 0.0078], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.45e-6

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 94.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

   7. Applied rewrites 72.5%

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

   if -3.45e-6 < phi1 < 0.0077999999999999996

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 0.0077999999999999996 < phi1

   1. Initial program 72.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 72.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 72.5

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lower--.f64 72.5

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

   3. Applied rewrites 72.5%

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

Alternative 11: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -3.45 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 0.0078:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (sin phi2)
            (sin phi1)
            (* (* (cos (- lambda2 lambda1)) (cos phi1)) (cos phi2))))
          R)))
   (if (<= phi1 -3.45e-6)
     t_0
     (if (<= phi1 0.0078)
       (*
        (acos
         (*
          (cos phi2)
          (fma (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(phi1)) * cos(phi2)))) * R;
	double tmp;
	if (phi1 <= -3.45e-6) {
		tmp = t_0;
	} else if (phi1 <= 0.0078) {
		tmp = acos((cos(phi2) * fma(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(phi1)) * cos(phi2)))) * R)
	tmp = 0.0
	if (phi1 <= -3.45e-6)
		tmp = t_0;
	elseif (phi1 <= 0.0078)
		tmp = Float64(acos(Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), 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[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -3.45e-6], t$95$0, If[LessEqual[phi1, 0.0078], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.45e-6 or 0.0077999999999999996 < phi1

   1. Initial program 72.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 72.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 72.5

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lower--.f64 72.5

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

   3. Applied rewrites 72.5%

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

   if -3.45e-6 < phi1 < 0.0077999999999999996

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.2

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

   8. Applied rewrites 53.2%

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

Alternative 12: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_3 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_4 := \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\\ t_5 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -2.06 \cdot 10^{+15}:\\ \;\;\;\;\cos^{-1} \left(t\_3 \cdot \mathsf{fma}\left(t\_4, t\_0, t\_1 \cdot t\_2\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 3400000:\\ \;\;\;\;\cos^{-1} \left(t\_5 \cdot \mathsf{fma}\left(t\_2, t\_1, t\_0 \cdot t\_4\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_3 \cdot t\_5, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmin lambda1 lambda2)))
        (t_1 (cos (fmax lambda1 lambda2)))
        (t_2 (cos (fmin lambda1 lambda2)))
        (t_3 (cos (fmin phi1 phi2)))
        (t_4 (sin (fmax lambda1 lambda2)))
        (t_5 (cos (fmax phi1 phi2))))
   (if (<= (fmin phi1 phi2) -2.06e+15)
     (* (acos (* t_3 (fma t_4 t_0 (* t_1 t_2)))) R)
     (if (<= (fmin phi1 phi2) 3400000.0)
       (* (acos (* t_5 (fma t_2 t_1 (* t_0 t_4)))) R)
       (*
        (acos
         (fma
          t_2
          (* t_3 t_5)
          (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmin(lambda1, lambda2));
	double t_1 = cos(fmax(lambda1, lambda2));
	double t_2 = cos(fmin(lambda1, lambda2));
	double t_3 = cos(fmin(phi1, phi2));
	double t_4 = sin(fmax(lambda1, lambda2));
	double t_5 = cos(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -2.06e+15) {
		tmp = acos((t_3 * fma(t_4, t_0, (t_1 * t_2)))) * R;
	} else if (fmin(phi1, phi2) <= 3400000.0) {
		tmp = acos((t_5 * fma(t_2, t_1, (t_0 * t_4)))) * R;
	} else {
		tmp = acos(fma(t_2, (t_3 * t_5), (sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmin(lambda1, lambda2))
	t_1 = cos(fmax(lambda1, lambda2))
	t_2 = cos(fmin(lambda1, lambda2))
	t_3 = cos(fmin(phi1, phi2))
	t_4 = sin(fmax(lambda1, lambda2))
	t_5 = cos(fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -2.06e+15)
		tmp = Float64(acos(Float64(t_3 * fma(t_4, t_0, Float64(t_1 * t_2)))) * R);
	elseif (fmin(phi1, phi2) <= 3400000.0)
		tmp = Float64(acos(Float64(t_5 * fma(t_2, t_1, Float64(t_0 * t_4)))) * R);
	else
		tmp = Float64(acos(fma(t_2, Float64(t_3 * t_5), Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -2.06e+15], N[(N[ArcCos[N[(t$95$3 * N[(t$95$4 * t$95$0 + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 3400000.0], N[(N[ArcCos[N[(t$95$5 * N[(t$95$2 * t$95$1 + N[(t$95$0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 * N[(t$95$3 * t$95$5), $MachinePrecision] + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.06e15

   1. Initial program 72.5%

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

   2. 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.0

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

   4. Applied rewrites 42.0%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. cos-neg-rev N/A

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

      5. cos-diff-rev N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. remove-double-neg N/A

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

      16. lift-sin.f64 N/A

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

      17. sin-neg N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-sin.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

   6. Applied rewrites 53.1%

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

   if -2.06e15 < phi1 < 3.4e6

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 3.4e6 < phi1

   1. Initial program 72.5%

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

   2. 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. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.3

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

   4. Applied rewrites 52.3%

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

Alternative 13: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -2.06 \cdot 10^{+15}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -2.06e+15)
   (*
    (acos
     (*
      (cos (fmin phi1 phi2))
      (fma (sin lambda2) (sin lambda1) (* (cos lambda2) (cos lambda1)))))
    R)
   (*
    (acos
     (*
      (cos (fmax phi1 phi2))
      (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -2.06e+15) {
		tmp = acos((cos(fmin(phi1, phi2)) * fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1))))) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -2.06e+15)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda2) * cos(lambda1))))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -2.06e+15], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.06e15

   1. Initial program 72.5%

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

   2. 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.0

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

   4. Applied rewrites 42.0%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. cos-neg-rev N/A

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

      5. cos-diff-rev N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. remove-double-neg N/A

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

      16. lift-sin.f64 N/A

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

      17. sin-neg N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-sin.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

   6. Applied rewrites 53.1%

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

   if -2.06e15 < phi1

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.2

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

   8. Applied rewrites 53.2%

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

Alternative 14: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -2.06 \cdot 10^{+15}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -2.06e+15)
   (* (acos (* (cos (fmin phi1 phi2)) (cos (- lambda1 lambda2)))) R)
   (*
    (acos
     (*
      (cos (fmax phi1 phi2))
      (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -2.06e+15) {
		tmp = acos((cos(fmin(phi1, phi2)) * cos((lambda1 - lambda2)))) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -2.06e+15)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * cos(Float64(lambda1 - lambda2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -2.06e+15], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.06e15

   1. Initial program 72.5%

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

   2. 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.0

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

   4. Applied rewrites 42.0%

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

   if -2.06e15 < phi1

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sin.f64 N/A

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

      21. lower-sin.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 94.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \phi_1, \sin \lambda_2 \cdot \sin \lambda_1, \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_1\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \color{blue}{\lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. lower-sin.f64 53.2

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   8. Applied rewrites 53.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -0.096:\\ \;\;\;\;\cos^{-1} \left(\sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) + \left(t\_0 \cdot t\_1\right) \cdot 1\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 8.5 \cdot 10^{-162}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot t\_2\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_1 \cdot t\_2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (cos (fmax phi1 phi2)))
        (t_2 (cos (- lambda1 lambda2))))
   (if (<= (fmax phi1 phi2) -0.096)
     (*
      (acos
       (+
        (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))
        (* (* t_0 t_1) 1.0)))
      R)
     (if (<= (fmax phi1 phi2) 8.5e-162)
       (* (acos (* t_0 t_2)) R)
       (if (<= (fmax phi1 phi2) 9.5e-12)
         (*
          (acos
           (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))
          R)
         (* (acos (* t_1 t_2)) R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2));
	double t_1 = cos(fmax(phi1, phi2));
	double t_2 = cos((lambda1 - lambda2));
	double tmp;
	if (fmax(phi1, phi2) <= -0.096) {
		tmp = acos(((sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))) + ((t_0 * t_1) * 1.0))) * R;
	} else if (fmax(phi1, phi2) <= 8.5e-162) {
		tmp = acos((t_0 * t_2)) * R;
	} else if (fmax(phi1, phi2) <= 9.5e-12) {
		tmp = acos(fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))) * R;
	} else {
		tmp = acos((t_1 * t_2)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	t_1 = cos(fmax(phi1, phi2))
	t_2 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmax(phi1, phi2) <= -0.096)
		tmp = Float64(acos(Float64(Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))) + Float64(Float64(t_0 * t_1) * 1.0))) * R);
	elseif (fmax(phi1, phi2) <= 8.5e-162)
		tmp = Float64(acos(Float64(t_0 * t_2)) * R);
	elseif (fmax(phi1, phi2) <= 9.5e-12)
		tmp = Float64(acos(fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))) * R);
	else
		tmp = Float64(acos(Float64(t_1 * t_2)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], -0.096], N[(N[ArcCos[N[(N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 8.5e-162], N[(N[ArcCos[N[(t$95$0 * t$95$2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 9.5e-12], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 * t$95$2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -0.096000000000000002

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]
   3. Step-by-step derivation

      1. lower-cos.f64 52.3

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right) \cdot R \]

   4. Applied rewrites 52.3%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 1\right) \cdot R \]
   6. Step-by-step derivation

   7. Applied rewrites 31.6%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 1\right) \cdot R \]

   if -0.096000000000000002 < phi2 < 8.49999999999999955e-162

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   if 8.49999999999999955e-162 < phi2 < 9.4999999999999995e-12

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-pow.f64 18.0

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.0%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \color{blue}{\lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      6. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      9. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \color{blue}{\lambda_2}\right)\right) \cdot R \]

      10. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      11. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

      12. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

      13. distribute-rgt-in N/A

          \[\leadsto \cos^{-1} \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) + \color{blue}{\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)}\right) \cdot R \]

      14. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \lambda_1, \color{blue}{1 + \frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)\right) \cdot R \]

   9. Applied rewrites 23.9%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)}, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right)\right) \cdot R \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right) \cdot R \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       2. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       4. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       6. lower-sin.f64 32.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   12. Applied rewrites 32.0%

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   if 9.4999999999999995e-12 < phi2

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.2

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 56.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 8.5 \cdot 10^{-162}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (fmax phi1 phi2) 8.5e-162)
     (* (acos (* (cos (fmin phi1 phi2)) t_0)) R)
     (if (<= (fmax phi1 phi2) 9.5e-12)
       (*
        (acos
         (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))
        R)
       (* (acos (* (cos (fmax phi1 phi2)) t_0)) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (fmax(phi1, phi2) <= 8.5e-162) {
		tmp = acos((cos(fmin(phi1, phi2)) * t_0)) * R;
	} else if (fmax(phi1, phi2) <= 9.5e-12) {
		tmp = acos(fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * t_0)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmax(phi1, phi2) <= 8.5e-162)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * t_0)) * R);
	elseif (fmax(phi1, phi2) <= 9.5e-12)
		tmp = Float64(acos(fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, 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]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 8.5e-162], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 9.5e-12], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 8.49999999999999955e-162

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   if 8.49999999999999955e-162 < phi2 < 9.4999999999999995e-12

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-pow.f64 18.0

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.0%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \color{blue}{\lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      6. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      9. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \color{blue}{\lambda_2}\right)\right) \cdot R \]

      10. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      11. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

      12. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

      13. distribute-rgt-in N/A

          \[\leadsto \cos^{-1} \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) + \color{blue}{\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)}\right) \cdot R \]

      14. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \lambda_1, \color{blue}{1 + \frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)\right) \cdot R \]

   9. Applied rewrites 23.9%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)}, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right)\right) \cdot R \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right) \cdot R \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       2. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       4. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

       6. lower-sin.f64 32.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   12. Applied rewrites 32.0%

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   if 9.4999999999999995e-12 < phi2

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.2

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 55.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -2.06 \cdot 10^{+15}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (fmin phi1 phi2) -2.06e+15)
     (* (acos (* (cos (fmin phi1 phi2)) t_0)) R)
     (* (acos (* (cos (fmax phi1 phi2)) t_0)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -2.06e+15) {
		tmp = acos((cos(fmin(phi1, phi2)) * t_0)) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * t_0)) * 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((lambda1 - lambda2))
    if (fmin(phi1, phi2) <= (-2.06d+15)) then
        tmp = acos((cos(fmin(phi1, phi2)) * t_0)) * r
    else
        tmp = acos((cos(fmax(phi1, phi2)) * t_0)) * 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((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -2.06e+15) {
		tmp = Math.acos((Math.cos(fmin(phi1, phi2)) * t_0)) * R;
	} else {
		tmp = Math.acos((Math.cos(fmax(phi1, phi2)) * t_0)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if fmin(phi1, phi2) <= -2.06e+15:
		tmp = math.acos((math.cos(fmin(phi1, phi2)) * t_0)) * R
	else:
		tmp = math.acos((math.cos(fmax(phi1, phi2)) * t_0)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -2.06e+15)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * t_0)) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * t_0)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (min(phi1, phi2) <= -2.06e+15)
		tmp = acos((cos(min(phi1, phi2)) * t_0)) * R;
	else
		tmp = acos((cos(max(phi1, phi2)) * t_0)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -2.06e+15], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.06e15

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   if -2.06e15 < phi1

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.2

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 42.0% accurate, 2.3× speedup

Math

\[\cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (* (cos phi1) (cos (- lambda1 lambda2)))) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos((cos(phi1) * 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((cos(phi1) * 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.cos(phi1) * Math.cos((lambda1 - lambda2)))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos((math.cos(phi1) * math.cos((lambda1 - lambda2)))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(cos(phi1) * cos(Float64(lambda1 - lambda2)))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos((cos(phi1) * cos((lambda1 - lambda2)))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 72.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. 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. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   3. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. lower--.f64 42.0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

4. Applied rewrites 42.0%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
5. Add Preprocessing

Alternative 19: 41.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmax lambda1 lambda2) 2.7e-11)
   (* (acos (* (cos (fmin lambda1 lambda2)) (cos phi1))) R)
   (* (acos (* (cos (fmax lambda1 lambda2)) (cos phi1))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(lambda1, lambda2) <= 2.7e-11) {
		tmp = acos((cos(fmin(lambda1, lambda2)) * cos(phi1))) * R;
	} else {
		tmp = acos((cos(fmax(lambda1, 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 (fmax(lambda1, lambda2) <= 2.7d-11) then
        tmp = acos((cos(fmin(lambda1, lambda2)) * cos(phi1))) * r
    else
        tmp = acos((cos(fmax(lambda1, 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 (fmax(lambda1, lambda2) <= 2.7e-11) {
		tmp = Math.acos((Math.cos(fmin(lambda1, lambda2)) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((Math.cos(fmax(lambda1, lambda2)) * Math.cos(phi1))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmax(lambda1, lambda2) <= 2.7e-11:
		tmp = math.acos((math.cos(fmin(lambda1, lambda2)) * math.cos(phi1))) * R
	else:
		tmp = math.acos((math.cos(fmax(lambda1, lambda2)) * math.cos(phi1))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 2.7e-11)
		tmp = Float64(acos(Float64(cos(fmin(lambda1, lambda2)) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(lambda1, lambda2)) * cos(phi1))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (max(lambda1, lambda2) <= 2.7e-11)
		tmp = acos((cos(min(lambda1, lambda2)) * cos(phi1))) * R;
	else
		tmp = acos((cos(max(lambda1, lambda2)) * cos(phi1))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 2.7e-11], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.70000000000000005e-11

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 30.5

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

   7. Applied rewrites 30.5%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   if 2.70000000000000005e-11 < lambda2

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-pow.f64 18.0

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.0%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \color{blue}{\lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      6. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)\right) \cdot R \]

      9. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \color{blue}{\lambda_2}\right)\right) \cdot R \]

      10. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      11. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

      12. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

      13. distribute-rgt-in N/A

          \[\leadsto \cos^{-1} \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) + \color{blue}{\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)}\right) \cdot R \]

      14. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \lambda_1, \color{blue}{1 + \frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)\right) \cdot R \]

   9. Applied rewrites 23.9%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)}, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right)\right) \cdot R \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R \]

       2. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R \]

       3. lower-cos.f64 31.0

          \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R \]

   12. Applied rewrites 31.0%

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.215:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -0.215)
   (* (acos (* (cos lambda1) (cos (fmin phi1 phi2)))) R)
   (*
    (acos
     (*
      (cos (- lambda2 lambda1))
      (fma (* (fmin phi1 phi2) (fmin phi1 phi2)) -0.5 1.0)))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -0.215) {
		tmp = acos((cos(lambda1) * cos(fmin(phi1, phi2)))) * R;
	} else {
		tmp = acos((cos((lambda2 - lambda1)) * fma((fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.215)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(fmin(phi1, phi2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * fma(Float64(fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.215], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.214999999999999997

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 30.5

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

   7. Applied rewrites 30.5%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   if -0.214999999999999997 < phi1

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-pow.f64 18.0

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.0%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   8. Step-by-step derivation

   9. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 19.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 0.01:\\ \;\;\;\;\cos^{-1} \left(\mathsf{max}\left(\phi_1, \phi_2\right) \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\mathsf{max}\left(\phi_1, \phi_2\right)}\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmax phi1 phi2) 0.01)
   (*
    (acos
     (*
      (fmax phi1 phi2)
      (+ (fmin phi1 phi2) (/ (cos (- lambda1 lambda2)) (fmax phi1 phi2)))))
    R)
   (*
    (-
     (* PI 0.5)
     (asin
      (*
       (cos (- lambda2 lambda1))
       (fma (* (fmin phi1 phi2) (fmin phi1 phi2)) -0.5 1.0))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(phi1, phi2) <= 0.01) {
		tmp = acos((fmax(phi1, phi2) * (fmin(phi1, phi2) + (cos((lambda1 - lambda2)) / fmax(phi1, phi2))))) * R;
	} else {
		tmp = ((((double) M_PI) * 0.5) - asin((cos((lambda2 - lambda1)) * fma((fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(phi1, phi2) <= 0.01)
		tmp = Float64(acos(Float64(fmax(phi1, phi2) * Float64(fmin(phi1, phi2) + Float64(cos(Float64(lambda1 - lambda2)) / fmax(phi1, phi2))))) * R);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(Float64(cos(Float64(lambda2 - lambda1)) * fma(Float64(fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 0.01], N[(N[ArcCos[N[(N[Max[phi1, phi2], $MachinePrecision] * N[(N[Min[phi1, phi2], $MachinePrecision] + N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0100000000000000002

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. lower--.f64 35.3

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   4. Applied rewrites 35.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

      4. lower-*.f64 17.7

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   7. Applied rewrites 17.7%

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]

   8. Taylor expanded in phi2 around inf

      \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \color{blue}{\frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}}\right)\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}}\right)\right) \cdot R \]

      2. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

      3. lower-/.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

      5. lower--.f64 17.8

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

   10. Applied rewrites 17.8%

       \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \color{blue}{\frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}}\right)\right) \cdot R \]

   if 0.0100000000000000002 < phi2

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-pow.f64 18.0

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.0%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   8. Step-by-step derivation

      1. lift-acos.f64 N/A

         \[\leadsto \color{blue}{\cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. acos-asin N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\pi}}{2} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. mult-flip N/A

         \[\leadsto \left(\color{blue}{\pi \cdot \frac{1}{2}} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. metadata-eval N/A

         \[\leadsto \left(\pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      7. metadata-eval N/A

         \[\leadsto \left(\pi \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      9. metadata-eval N/A

         \[\leadsto \left(\pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      10. lower-asin.f64 18.0

          \[\leadsto \left(\pi \cdot 0.5 - \color{blue}{\sin^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot R \]

   9. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 19.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 450000:\\ \;\;\;\;\cos^{-1} \left(\mathsf{max}\left(\phi_1, \phi_2\right) \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\mathsf{max}\left(\phi_1, \phi_2\right)}\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmax phi1 phi2) 450000.0)
   (*
    (acos
     (*
      (fmax phi1 phi2)
      (+ (fmin phi1 phi2) (/ (cos (- lambda1 lambda2)) (fmax phi1 phi2)))))
    R)
   (*
    (acos
     (*
      (cos (- lambda2 lambda1))
      (fma (* (fmin phi1 phi2) (fmin phi1 phi2)) -0.5 1.0)))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(phi1, phi2) <= 450000.0) {
		tmp = acos((fmax(phi1, phi2) * (fmin(phi1, phi2) + (cos((lambda1 - lambda2)) / fmax(phi1, phi2))))) * R;
	} else {
		tmp = acos((cos((lambda2 - lambda1)) * fma((fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(phi1, phi2) <= 450000.0)
		tmp = Float64(acos(Float64(fmax(phi1, phi2) * Float64(fmin(phi1, phi2) + Float64(cos(Float64(lambda1 - lambda2)) / fmax(phi1, phi2))))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * fma(Float64(fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 450000.0], N[(N[ArcCos[N[(N[Max[phi1, phi2], $MachinePrecision] * N[(N[Min[phi1, phi2], $MachinePrecision] + N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.5e5

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. lower--.f64 35.3

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   4. Applied rewrites 35.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

      4. lower-*.f64 17.7

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   7. Applied rewrites 17.7%

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]

   8. Taylor expanded in phi2 around inf

      \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \color{blue}{\frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}}\right)\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}}\right)\right) \cdot R \]

      2. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

      3. lower-/.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

      5. lower--.f64 17.8

         \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

   10. Applied rewrites 17.8%

       \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \color{blue}{\frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}}\right)\right) \cdot R \]

   if 4.5e5 < phi2

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.0

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-pow.f64 18.0

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.0%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   8. Step-by-step derivation

   9. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 17.8% accurate, 3.0× speedup

Math

\[\cos^{-1} \left(\mathsf{max}\left(\phi_1, \phi_2\right) \cdot \left(\mathsf{min}\left(\phi_1, \phi_2\right) + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\mathsf{max}\left(\phi_1, \phi_2\right)}\right)\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (*
    (fmax phi1 phi2)
    (+ (fmin phi1 phi2) (/ (cos (- lambda1 lambda2)) (fmax phi1 phi2)))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos((fmax(phi1, phi2) * (fmin(phi1, phi2) + (cos((lambda1 - lambda2)) / fmax(phi1, phi2))))) * 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((fmax(phi1, phi2) * (fmin(phi1, phi2) + (cos((lambda1 - lambda2)) / fmax(phi1, phi2))))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos((fmax(phi1, phi2) * (fmin(phi1, phi2) + (Math.cos((lambda1 - lambda2)) / fmax(phi1, phi2))))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos((fmax(phi1, phi2) * (fmin(phi1, phi2) + (math.cos((lambda1 - lambda2)) / fmax(phi1, phi2))))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(fmax(phi1, phi2) * Float64(fmin(phi1, phi2) + Float64(cos(Float64(lambda1 - lambda2)) / fmax(phi1, phi2))))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos((max(phi1, phi2) * (min(phi1, phi2) + (cos((lambda1 - lambda2)) / max(phi1, phi2))))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[Max[phi1, phi2], $MachinePrecision] * N[(N[Min[phi1, phi2], $MachinePrecision] + N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 72.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   2. lower-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   3. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   4. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   6. lower--.f64 35.3

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

4. Applied rewrites 35.3%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   3. lower--.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   4. lower-*.f64 17.7

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

7. Applied rewrites 17.7%

   \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]

8. Taylor expanded in phi2 around inf

   \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \color{blue}{\frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}}\right)\right) \cdot R \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}}\right)\right) \cdot R \]

   2. lower-+.f64 N/A

      \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

   3. lower-/.f64 N/A

      \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

   4. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

   5. lower--.f64 17.8

      \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}\right)\right) \cdot R \]

10. Applied rewrites 17.8%

    \[\leadsto \cos^{-1} \left(\phi_2 \cdot \left(\phi_1 + \color{blue}{\frac{\cos \left(\lambda_1 - \lambda_2\right)}{\phi_2}}\right)\right) \cdot R \]
11. Add Preprocessing

Alternative 24: 17.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 4.2 \cdot 10^{-13}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) + \phi_1 \cdot \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(-\mathsf{max}\left(\lambda_1, \lambda_2\right)\right) + \phi_1 \cdot \phi_2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmax lambda1 lambda2) 4.2e-13)
   (* (acos (+ (cos (fmin lambda1 lambda2)) (* phi1 phi2))) R)
   (* (acos (+ (cos (- (fmax lambda1 lambda2))) (* phi1 phi2))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(lambda1, lambda2) <= 4.2e-13) {
		tmp = acos((cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * R;
	} else {
		tmp = acos((cos(-fmax(lambda1, lambda2)) + (phi1 * 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) :: tmp
    if (fmax(lambda1, lambda2) <= 4.2d-13) then
        tmp = acos((cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * r
    else
        tmp = acos((cos(-fmax(lambda1, lambda2)) + (phi1 * phi2))) * 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 (fmax(lambda1, lambda2) <= 4.2e-13) {
		tmp = Math.acos((Math.cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * R;
	} else {
		tmp = Math.acos((Math.cos(-fmax(lambda1, lambda2)) + (phi1 * phi2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmax(lambda1, lambda2) <= 4.2e-13:
		tmp = math.acos((math.cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * R
	else:
		tmp = math.acos((math.cos(-fmax(lambda1, lambda2)) + (phi1 * phi2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 4.2e-13)
		tmp = Float64(acos(Float64(cos(fmin(lambda1, lambda2)) + Float64(phi1 * phi2))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(-fmax(lambda1, lambda2))) + Float64(phi1 * phi2))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (max(lambda1, lambda2) <= 4.2e-13)
		tmp = acos((cos(min(lambda1, lambda2)) + (phi1 * phi2))) * R;
	else
		tmp = acos((cos(-max(lambda1, lambda2)) + (phi1 * phi2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 4.2e-13], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[(-N[Max[lambda1, lambda2], $MachinePrecision])], $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 4.19999999999999977e-13

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. lower--.f64 35.3

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   4. Applied rewrites 35.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

      4. lower-*.f64 17.7

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   7. Applied rewrites 17.7%

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \phi_2\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \phi_2\right) \cdot R \]

      3. lower-*.f64 10.9

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \phi_2\right) \cdot R \]

   10. Applied rewrites 10.9%

       \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

   if 4.19999999999999977e-13 < lambda2

   1. Initial program 72.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. lower--.f64 35.3

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   4. Applied rewrites 35.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

      4. lower-*.f64 17.7

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   7. Applied rewrites 17.7%

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \phi_1 \cdot \phi_2\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

      2. lower-neg.f64 11.4

         \[\leadsto \cos^{-1} \left(\cos \left(-\lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   10. Applied rewrites 11.4%

       \[\leadsto \cos^{-1} \left(\cos \left(-\lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 17.6% accurate, 3.8× speedup

Math

\[\cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \left(\lambda_2 - \lambda_1\right)\right)\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (fma phi2 phi1 (cos (- lambda2 lambda1)))) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(fma(phi2, phi1, cos((lambda2 - lambda1)))) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(fma(phi2, phi1, cos(Float64(lambda2 - lambda1)))) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(phi2 * phi1 + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 72.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   2. lower-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   3. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   4. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   6. lower--.f64 35.3

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

4. Applied rewrites 35.3%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   3. lower--.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   4. lower-*.f64 17.7

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

7. Applied rewrites 17.7%

   \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

   2. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   5. lower-fma.f64 17.7

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   6. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   7. cos-neg-rev N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) \cdot R \]

   8. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) \cdot R \]

   9. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) \cdot R \]

   10. sub-negate-rev N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \left(\lambda_2 - \lambda_1\right)\right)\right) \cdot R \]

   11. lower--.f64 17.7

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \left(\lambda_2 - \lambda_1\right)\right)\right) \cdot R \]

9. Applied rewrites 17.7%

   \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \left(\lambda_2 - \lambda_1\right)\right)\right) \cdot R \]
10. Add Preprocessing

Alternative 26: 11.3% accurate, 3.7× speedup

Math

\[\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (+ (cos (fmin lambda1 lambda2)) (* phi1 phi2))) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos((cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * 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(fmin(lambda1, lambda2)) + (phi1 * phi2))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos((Math.cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos((math.cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(cos(fmin(lambda1, lambda2)) + Float64(phi1 * phi2))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos((cos(min(lambda1, lambda2)) + (phi1 * phi2))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 72.5%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   2. lower-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   3. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   4. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   6. lower--.f64 35.3

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

4. Applied rewrites 35.3%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   3. lower--.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

   4. lower-*.f64 17.7

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right) \cdot R \]

7. Applied rewrites 17.7%

   \[\leadsto \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_1 \cdot \phi_2}\right) \cdot R \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \phi_2\right) \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \phi_2\right) \cdot R \]

   3. lower-*.f64 10.9

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \phi_2\right) \cdot R \]

10. Applied rewrites 10.9%

    \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025173 
(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))