Spherical law of cosines

Percentage Accurate: 73.5% → 94.0%

Time: 13.8s

Alternatives: 22

Speedup: 0.9×

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: 73.5% 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: 94.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-*.f64 N/A

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

   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-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

3. Applied rewrites 94.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.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. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

5. Applied rewrites 94.0%

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

Alternative 2: 94.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-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 94.0

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

3. Applied rewrites 94.0%

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

4. Taylor expanded in lambda1 around inf

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

   1. lower-fma.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-sin.f64 94.0

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

6. Applied rewrites 94.0%

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

Alternative 3: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_4 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, 1, \left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{2}{\pi}, \cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_3 \cdot t\_4, t\_0 \cdot t\_1\right)\right), -1\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 0.07:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot t\_2, t\_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t\_3, t\_2 \cdot t\_4, t\_1 \cdot t\_0\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmin phi1 phi2)))
        (t_1 (sin (fmax phi1 phi2)))
        (t_2 (cos (fmax phi1 phi2)))
        (t_3 (cos (- lambda2 lambda1)))
        (t_4 (cos (fmin phi1 phi2))))
   (if (<= (fmin phi1 phi2) -1000000.0)
     (*
      (fma
       (* PI 0.5)
       1.0
       (*
        (* PI 0.5)
        (fma (/ 2.0 PI) (acos (fma t_2 (* t_3 t_4) (* t_0 t_1))) -1.0)))
      R)
     (if (<= (fmin phi1 phi2) 0.07)
       (*
        (acos
         (fma
          (cos lambda1)
          (* (cos lambda2) t_2)
          (* t_2 (* (sin lambda1) (sin lambda2)))))
        R)
       (* (- (* PI 0.5) (asin (fma t_3 (* t_2 t_4) (* t_1 t_0)))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmin(phi1, phi2));
	double t_1 = sin(fmax(phi1, phi2));
	double t_2 = cos(fmax(phi1, phi2));
	double t_3 = cos((lambda2 - lambda1));
	double t_4 = cos(fmin(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -1000000.0) {
		tmp = fma((((double) M_PI) * 0.5), 1.0, ((((double) M_PI) * 0.5) * fma((2.0 / ((double) M_PI)), acos(fma(t_2, (t_3 * t_4), (t_0 * t_1))), -1.0))) * R;
	} else if (fmin(phi1, phi2) <= 0.07) {
		tmp = acos(fma(cos(lambda1), (cos(lambda2) * t_2), (t_2 * (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = ((((double) M_PI) * 0.5) - asin(fma(t_3, (t_2 * t_4), (t_1 * t_0)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmin(phi1, phi2))
	t_1 = sin(fmax(phi1, phi2))
	t_2 = cos(fmax(phi1, phi2))
	t_3 = cos(Float64(lambda2 - lambda1))
	t_4 = cos(fmin(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1000000.0)
		tmp = Float64(fma(Float64(pi * 0.5), 1.0, Float64(Float64(pi * 0.5) * fma(Float64(2.0 / pi), acos(fma(t_2, Float64(t_3 * t_4), Float64(t_0 * t_1))), -1.0))) * R);
	elseif (fmin(phi1, phi2) <= 0.07)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(lambda2) * t_2), Float64(t_2 * Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(fma(t_3, Float64(t_2 * t_4), Float64(t_1 * t_0)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1000000.0], N[(N[(N[(Pi * 0.5), $MachinePrecision] * 1.0 + N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(2.0 / Pi), $MachinePrecision] * N[ArcCos[N[(t$95$2 * N[(t$95$3 * t$95$4), $MachinePrecision] + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 0.07], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$2 * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(t$95$3 * N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1e6

   1. Initial program 73.5%

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

      1. lift-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 73.3%

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

   5. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift-PI.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-flip N/A

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

   7. Applied rewrites 73.4%

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

   if -1e6 < phi1 < 0.070000000000000007

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      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-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-sin.f64 N/A

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 52.9

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

   6. Applied rewrites 52.9%

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

   if 0.070000000000000007 < phi1

   1. Initial program 73.5%

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

      1. lift-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. lower--.f64 N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-asin.f64 73.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 73.4

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

   3. Applied rewrites 73.4%

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

Alternative 4: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_4 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, 1, \left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{2}{\pi}, \cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_3 \cdot t\_4, t\_0 \cdot t\_1\right)\right), -1\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 0.07:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t\_3, t\_2 \cdot t\_4, t\_1 \cdot t\_0\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmin phi1 phi2)))
        (t_1 (sin (fmax phi1 phi2)))
        (t_2 (cos (fmax phi1 phi2)))
        (t_3 (cos (- lambda2 lambda1)))
        (t_4 (cos (fmin phi1 phi2))))
   (if (<= (fmin phi1 phi2) -1000000.0)
     (*
      (fma
       (* PI 0.5)
       1.0
       (*
        (* PI 0.5)
        (fma (/ 2.0 PI) (acos (fma t_2 (* t_3 t_4) (* t_0 t_1))) -1.0)))
      R)
     (if (<= (fmin phi1 phi2) 0.07)
       (*
        (acos
         (*
          t_2
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (* (- (* PI 0.5) (asin (fma t_3 (* t_2 t_4) (* t_1 t_0)))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmin(phi1, phi2));
	double t_1 = sin(fmax(phi1, phi2));
	double t_2 = cos(fmax(phi1, phi2));
	double t_3 = cos((lambda2 - lambda1));
	double t_4 = cos(fmin(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -1000000.0) {
		tmp = fma((((double) M_PI) * 0.5), 1.0, ((((double) M_PI) * 0.5) * fma((2.0 / ((double) M_PI)), acos(fma(t_2, (t_3 * t_4), (t_0 * t_1))), -1.0))) * R;
	} else if (fmin(phi1, phi2) <= 0.07) {
		tmp = acos((t_2 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = ((((double) M_PI) * 0.5) - asin(fma(t_3, (t_2 * t_4), (t_1 * t_0)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmin(phi1, phi2))
	t_1 = sin(fmax(phi1, phi2))
	t_2 = cos(fmax(phi1, phi2))
	t_3 = cos(Float64(lambda2 - lambda1))
	t_4 = cos(fmin(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1000000.0)
		tmp = Float64(fma(Float64(pi * 0.5), 1.0, Float64(Float64(pi * 0.5) * fma(Float64(2.0 / pi), acos(fma(t_2, Float64(t_3 * t_4), Float64(t_0 * t_1))), -1.0))) * R);
	elseif (fmin(phi1, phi2) <= 0.07)
		tmp = Float64(acos(Float64(t_2 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(fma(t_3, Float64(t_2 * t_4), Float64(t_1 * t_0)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1000000.0], N[(N[(N[(Pi * 0.5), $MachinePrecision] * 1.0 + N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(2.0 / Pi), $MachinePrecision] * N[ArcCos[N[(t$95$2 * N[(t$95$3 * t$95$4), $MachinePrecision] + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 0.07], 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[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(t$95$3 * N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1e6

   1. Initial program 73.5%

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

      1. lift-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 73.3%

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

   5. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift-PI.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-flip N/A

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

   7. Applied rewrites 73.4%

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

   if -1e6 < phi1 < 0.070000000000000007

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 52.9

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

   6. Applied rewrites 52.9%

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

   if 0.070000000000000007 < phi1

   1. Initial program 73.5%

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

      1. lift-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. lower--.f64 N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-asin.f64 73.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 73.4

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

   3. Applied rewrites 73.4%

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

Alternative 5: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_4 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1000000:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_4, t\_0, \left(t\_1 \cdot t\_2\right) \cdot t\_3\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 0.07:\\ \;\;\;\;\cos^{-1} \left(t\_3 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t\_1, t\_3 \cdot t\_2, t\_4 \cdot t\_0\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmin phi1 phi2)))
        (t_1 (cos (- lambda2 lambda1)))
        (t_2 (cos (fmin phi1 phi2)))
        (t_3 (cos (fmax phi1 phi2)))
        (t_4 (sin (fmax phi1 phi2))))
   (if (<= (fmin phi1 phi2) -1000000.0)
     (* (acos (fma t_4 t_0 (* (* t_1 t_2) t_3))) R)
     (if (<= (fmin phi1 phi2) 0.07)
       (*
        (acos
         (*
          t_3
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (* (- (* PI 0.5) (asin (fma t_1 (* t_3 t_2) (* t_4 t_0)))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmin(phi1, phi2));
	double t_1 = cos((lambda2 - lambda1));
	double t_2 = cos(fmin(phi1, phi2));
	double t_3 = cos(fmax(phi1, phi2));
	double t_4 = sin(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -1000000.0) {
		tmp = acos(fma(t_4, t_0, ((t_1 * t_2) * t_3))) * R;
	} else if (fmin(phi1, phi2) <= 0.07) {
		tmp = acos((t_3 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = ((((double) M_PI) * 0.5) - asin(fma(t_1, (t_3 * t_2), (t_4 * t_0)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmin(phi1, phi2))
	t_1 = cos(Float64(lambda2 - lambda1))
	t_2 = cos(fmin(phi1, phi2))
	t_3 = cos(fmax(phi1, phi2))
	t_4 = sin(fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1000000.0)
		tmp = Float64(acos(fma(t_4, t_0, Float64(Float64(t_1 * t_2) * t_3))) * R);
	elseif (fmin(phi1, phi2) <= 0.07)
		tmp = Float64(acos(Float64(t_3 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(fma(t_1, Float64(t_3 * t_2), Float64(t_4 * t_0)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1000000.0], N[(N[ArcCos[N[(t$95$4 * t$95$0 + N[(N[(t$95$1 * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 0.07], N[(N[ArcCos[N[(t$95$3 * 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[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(t$95$1 * N[(t$95$3 * t$95$2), $MachinePrecision] + N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1e6

   1. Initial program 73.5%

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

      1. lift-+.f64 N/A

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

      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 73.5

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

      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 73.5

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

      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 73.5

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

   3. Applied rewrites 73.5%

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

   if -1e6 < phi1 < 0.070000000000000007

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 52.9

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

   6. Applied rewrites 52.9%

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

   if 0.070000000000000007 < phi1

   1. Initial program 73.5%

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

      1. lift-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. lower--.f64 N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-asin.f64 73.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 73.4

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

   3. Applied rewrites 73.4%

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

Alternative 6: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1000000:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_3, t\_0, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_1\right) \cdot t\_2\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 0.07:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot t\_3 + \left(t\_1 \cdot t\_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmin phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (cos (fmax phi1 phi2)))
        (t_3 (sin (fmax phi1 phi2))))
   (if (<= (fmin phi1 phi2) -1000000.0)
     (* (acos (fma t_3 t_0 (* (* (cos (- lambda2 lambda1)) t_1) t_2))) R)
     (if (<= (fmin phi1 phi2) 0.07)
       (*
        (acos
         (*
          t_2
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (*
        (acos (+ (* t_0 t_3) (* (* t_1 t_2) (cos (- lambda1 lambda2)))))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmin(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = cos(fmax(phi1, phi2));
	double t_3 = sin(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -1000000.0) {
		tmp = acos(fma(t_3, t_0, ((cos((lambda2 - lambda1)) * t_1) * t_2))) * R;
	} else if (fmin(phi1, phi2) <= 0.07) {
		tmp = acos((t_2 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = acos(((t_0 * t_3) + ((t_1 * t_2) * cos((lambda1 - lambda2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmin(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = cos(fmax(phi1, phi2))
	t_3 = sin(fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1000000.0)
		tmp = Float64(acos(fma(t_3, t_0, Float64(Float64(cos(Float64(lambda2 - lambda1)) * t_1) * t_2))) * R);
	elseif (fmin(phi1, phi2) <= 0.07)
		tmp = Float64(acos(Float64(t_2 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(acos(Float64(Float64(t_0 * t_3) + Float64(Float64(t_1 * t_2) * cos(Float64(lambda1 - lambda2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Min[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[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1000000.0], N[(N[ArcCos[N[(t$95$3 * t$95$0 + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 0.07], 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[ArcCos[N[(N[(t$95$0 * t$95$3), $MachinePrecision] + N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1e6

   1. Initial program 73.5%

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

      1. lift-+.f64 N/A

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

      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 73.5

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

      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 73.5

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

      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 73.5

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

   3. Applied rewrites 73.5%

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

   if -1e6 < phi1 < 0.070000000000000007

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 52.9

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

   6. Applied rewrites 52.9%

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

   if 0.070000000000000007 < phi1

   1. Initial program 73.5%

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

Alternative 7: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(\sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right), \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right), \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right) \cdot t\_0\right)\right) \cdot R\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 0.07:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1
         (*
          (acos
           (fma
            (sin (fmax phi1 phi2))
            (sin (fmin phi1 phi2))
            (* (* (cos (- lambda2 lambda1)) (cos (fmin phi1 phi2))) t_0)))
          R)))
   (if (<= (fmin phi1 phi2) -1000000.0)
     t_1
     (if (<= (fmin phi1 phi2) 0.07)
       (*
        (acos
         (*
          t_0
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = acos(fma(sin(fmax(phi1, phi2)), sin(fmin(phi1, phi2)), ((cos((lambda2 - lambda1)) * cos(fmin(phi1, phi2))) * t_0))) * R;
	double tmp;
	if (fmin(phi1, phi2) <= -1000000.0) {
		tmp = t_1;
	} else if (fmin(phi1, phi2) <= 0.07) {
		tmp = acos((t_0 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = Float64(acos(fma(sin(fmax(phi1, phi2)), sin(fmin(phi1, phi2)), Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(fmin(phi1, phi2))) * t_0))) * R)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1000000.0)
		tmp = t_1;
	elseif (fmin(phi1, phi2) <= 0.07)
		tmp = Float64(acos(Float64(t_0 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = t_1;
	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[(N[ArcCos[N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1000000.0], t$95$1, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 0.07], 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], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1e6 or 0.070000000000000007 < phi1

   1. Initial program 73.5%

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

      1. lift-+.f64 N/A

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

      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 73.5

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

      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 73.5

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

      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 73.5

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

   3. Applied rewrites 73.5%

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

   if -1e6 < phi1 < 0.070000000000000007

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 52.9

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

   6. Applied rewrites 52.9%

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

Alternative 8: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_4 := \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\\ t_5 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq -1.4 \cdot 10^{-15}:\\ \;\;\;\;\cos^{-1} \left(t\_5 \cdot \mathsf{fma}\left(\sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right), t\_3 \cdot t\_4\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 63:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_5, t\_3 \cdot t\_0, t\_1 \cdot t\_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_1, \left(t\_4 \cdot t\_5\right) \cdot t\_0\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (sin (fmin phi1 phi2)))
        (t_2 (sin (fmax phi1 phi2)))
        (t_3 (cos (fmin lambda1 lambda2)))
        (t_4 (cos (fmax lambda1 lambda2)))
        (t_5 (cos (fmin phi1 phi2))))
   (if (<= (fmax lambda1 lambda2) -1.4e-15)
     (*
      (acos
       (*
        t_5
        (fma
         (sin (fmax lambda1 lambda2))
         (sin (fmin lambda1 lambda2))
         (* t_3 t_4))))
      R)
     (if (<= (fmax lambda1 lambda2) 63.0)
       (* (acos (fma t_5 (* t_3 t_0) (* t_1 t_2))) R)
       (* (acos (fma t_2 t_1 (* (* t_4 t_5) t_0))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = sin(fmin(phi1, phi2));
	double t_2 = sin(fmax(phi1, phi2));
	double t_3 = cos(fmin(lambda1, lambda2));
	double t_4 = cos(fmax(lambda1, lambda2));
	double t_5 = cos(fmin(phi1, phi2));
	double tmp;
	if (fmax(lambda1, lambda2) <= -1.4e-15) {
		tmp = acos((t_5 * fma(sin(fmax(lambda1, lambda2)), sin(fmin(lambda1, lambda2)), (t_3 * t_4)))) * R;
	} else if (fmax(lambda1, lambda2) <= 63.0) {
		tmp = acos(fma(t_5, (t_3 * t_0), (t_1 * t_2))) * R;
	} else {
		tmp = acos(fma(t_2, t_1, ((t_4 * t_5) * t_0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = sin(fmin(phi1, phi2))
	t_2 = sin(fmax(phi1, phi2))
	t_3 = cos(fmin(lambda1, lambda2))
	t_4 = cos(fmax(lambda1, lambda2))
	t_5 = cos(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= -1.4e-15)
		tmp = Float64(acos(Float64(t_5 * fma(sin(fmax(lambda1, lambda2)), sin(fmin(lambda1, lambda2)), Float64(t_3 * t_4)))) * R);
	elseif (fmax(lambda1, lambda2) <= 63.0)
		tmp = Float64(acos(fma(t_5, Float64(t_3 * t_0), Float64(t_1 * t_2))) * R);
	else
		tmp = Float64(acos(fma(t_2, t_1, Float64(Float64(t_4 * t_5) * t_0))) * 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[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], -1.4e-15], N[(N[ArcCos[N[(t$95$5 * N[(N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 63.0], N[(N[ArcCos[N[(t$95$5 * N[(t$95$3 * t$95$0), $MachinePrecision] + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 * t$95$1 + N[(N[(t$95$4 * t$95$5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.40000000000000007e-15

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      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. lower-fma.f64 53.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 53.5

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

   6. Applied rewrites 53.5%

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

   if -1.40000000000000007e-15 < lambda2 < 63

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in lambda2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 53.1

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

   9. Applied rewrites 53.1%

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

   if 63 < lambda2

   1. Initial program 73.5%

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

      1. lift-+.f64 N/A

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

      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 73.5

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

      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 73.5

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

      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 73.5

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

   3. Applied rewrites 73.5%

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

   4. Taylor expanded in lambda1 around 0

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

   6. Applied rewrites 52.9%

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

Alternative 9: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_3 := \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\\ t_4 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq -1.4 \cdot 10^{-15}:\\ \;\;\;\;\cos^{-1} \left(t\_4 \cdot \mathsf{fma}\left(\sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right), t\_2 \cdot t\_3\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 63:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_4, t\_2 \cdot t\_0, t\_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_3, t\_4 \cdot t\_0, t\_1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2))))
        (t_2 (cos (fmin lambda1 lambda2)))
        (t_3 (cos (fmax lambda1 lambda2)))
        (t_4 (cos (fmin phi1 phi2))))
   (if (<= (fmax lambda1 lambda2) -1.4e-15)
     (*
      (acos
       (*
        t_4
        (fma
         (sin (fmax lambda1 lambda2))
         (sin (fmin lambda1 lambda2))
         (* t_2 t_3))))
      R)
     (if (<= (fmax lambda1 lambda2) 63.0)
       (* (acos (fma t_4 (* t_2 t_0) t_1)) R)
       (* (acos (fma t_3 (* t_4 t_0) t_1)) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2));
	double t_2 = cos(fmin(lambda1, lambda2));
	double t_3 = cos(fmax(lambda1, lambda2));
	double t_4 = cos(fmin(phi1, phi2));
	double tmp;
	if (fmax(lambda1, lambda2) <= -1.4e-15) {
		tmp = acos((t_4 * fma(sin(fmax(lambda1, lambda2)), sin(fmin(lambda1, lambda2)), (t_2 * t_3)))) * R;
	} else if (fmax(lambda1, lambda2) <= 63.0) {
		tmp = acos(fma(t_4, (t_2 * t_0), t_1)) * R;
	} else {
		tmp = acos(fma(t_3, (t_4 * t_0), t_1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2)))
	t_2 = cos(fmin(lambda1, lambda2))
	t_3 = cos(fmax(lambda1, lambda2))
	t_4 = cos(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= -1.4e-15)
		tmp = Float64(acos(Float64(t_4 * fma(sin(fmax(lambda1, lambda2)), sin(fmin(lambda1, lambda2)), Float64(t_2 * t_3)))) * R);
	elseif (fmax(lambda1, lambda2) <= 63.0)
		tmp = Float64(acos(fma(t_4, Float64(t_2 * t_0), t_1)) * R);
	else
		tmp = Float64(acos(fma(t_3, Float64(t_4 * t_0), t_1)) * 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[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], -1.4e-15], N[(N[ArcCos[N[(t$95$4 * N[(N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 63.0], N[(N[ArcCos[N[(t$95$4 * N[(t$95$2 * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$3 * N[(t$95$4 * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.40000000000000007e-15

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      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. lower-fma.f64 53.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 53.5

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

   6. Applied rewrites 53.5%

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

   if -1.40000000000000007e-15 < lambda2 < 63

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in lambda2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 53.1

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

   9. Applied rewrites 53.1%

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

   if 63 < lambda2

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      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-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-sin.f64 N/A

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in lambda1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.9

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

   6. Applied rewrites 52.9%

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

Alternative 10: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_1 := \mathsf{fma}\left(t\_0, \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -21000000:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_3, t\_0 \cdot t\_2, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(t\_3 \cdot t\_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot t\_1\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin lambda1 lambda2)))
        (t_1
         (fma
          t_0
          (cos (fmax lambda1 lambda2))
          (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2)))))
        (t_2 (cos (fmax phi1 phi2)))
        (t_3 (cos (fmin phi1 phi2))))
   (if (<= (fmax phi1 phi2) -21000000.0)
     (*
      (acos
       (fma t_3 (* t_0 t_2) (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))))
      R)
     (if (<= (fmax phi1 phi2) 2.4e-17)
       (* (acos (* t_3 t_1)) R)
       (* (acos (* t_2 t_1)) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(lambda1, lambda2));
	double t_1 = fma(t_0, cos(fmax(lambda1, lambda2)), (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))));
	double t_2 = cos(fmax(phi1, phi2));
	double t_3 = cos(fmin(phi1, phi2));
	double tmp;
	if (fmax(phi1, phi2) <= -21000000.0) {
		tmp = acos(fma(t_3, (t_0 * t_2), (sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R;
	} else if (fmax(phi1, phi2) <= 2.4e-17) {
		tmp = acos((t_3 * t_1)) * R;
	} else {
		tmp = acos((t_2 * t_1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(lambda1, lambda2))
	t_1 = fma(t_0, cos(fmax(lambda1, lambda2)), Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))))
	t_2 = cos(fmax(phi1, phi2))
	t_3 = cos(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(phi1, phi2) <= -21000000.0)
		tmp = Float64(acos(fma(t_3, Float64(t_0 * t_2), Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R);
	elseif (fmax(phi1, phi2) <= 2.4e-17)
		tmp = Float64(acos(Float64(t_3 * t_1)) * R);
	else
		tmp = Float64(acos(Float64(t_2 * t_1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], -21000000.0], N[(N[ArcCos[N[(t$95$3 * N[(t$95$0 * t$95$2), $MachinePrecision] + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 2.4e-17], N[(N[ArcCos[N[(t$95$3 * t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.1e7

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in lambda2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 53.1

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

   9. Applied rewrites 53.1%

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

   if -2.1e7 < phi2 < 2.39999999999999986e-17

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.5

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

   9. Applied rewrites 53.5%

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

   if 2.39999999999999986e-17 < phi2

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 52.9

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

   6. Applied rewrites 52.9%

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

Alternative 11: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_3 := \mathsf{fma}\left(t\_2, \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -21000000:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_0 \cdot t\_1, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot t\_3\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_1 \cdot t\_3\right) \cdot R\\ \end{array} \]

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 (fmin lambda1 lambda2)))
        (t_3
         (fma
          t_2
          (cos (fmax lambda1 lambda2))
          (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2))))))
   (if (<= (fmax phi1 phi2) -21000000.0)
     (*
      (acos
       (fma t_2 (* t_0 t_1) (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))))
      R)
     (if (<= (fmax phi1 phi2) 2.4e-17)
       (* (acos (* t_0 t_3)) R)
       (* (acos (* t_1 t_3)) 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(fmin(lambda1, lambda2));
	double t_3 = fma(t_2, cos(fmax(lambda1, lambda2)), (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))));
	double tmp;
	if (fmax(phi1, phi2) <= -21000000.0) {
		tmp = acos(fma(t_2, (t_0 * t_1), (sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R;
	} else if (fmax(phi1, phi2) <= 2.4e-17) {
		tmp = acos((t_0 * t_3)) * R;
	} else {
		tmp = acos((t_1 * t_3)) * 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(fmin(lambda1, lambda2))
	t_3 = fma(t_2, cos(fmax(lambda1, lambda2)), Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))))
	tmp = 0.0
	if (fmax(phi1, phi2) <= -21000000.0)
		tmp = Float64(acos(fma(t_2, Float64(t_0 * t_1), Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R);
	elseif (fmax(phi1, phi2) <= 2.4e-17)
		tmp = Float64(acos(Float64(t_0 * t_3)) * R);
	else
		tmp = Float64(acos(Float64(t_1 * t_3)) * 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[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], -21000000.0], N[(N[ArcCos[N[(t$95$2 * N[(t$95$0 * t$95$1), $MachinePrecision] + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 2.4e-17], N[(N[ArcCos[N[(t$95$0 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.1e7

   1. Initial program 73.5%

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

   2. Taylor expanded in 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 53.1

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

   4. Applied rewrites 53.1%

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

   if -2.1e7 < phi2 < 2.39999999999999986e-17

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.5

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

   9. Applied rewrites 53.5%

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

   if 2.39999999999999986e-17 < phi2

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 52.9

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

   6. Applied rewrites 52.9%

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

Alternative 12: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(t\_0, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -21000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 0.00064:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(t\_0, \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin lambda1 lambda2)))
        (t_1
         (*
          (acos (fma t_0 (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
          R)))
   (if (<= phi2 -21000000.0)
     t_1
     (if (<= phi2 0.00064)
       (*
        (acos
         (*
          (cos phi1)
          (fma
           t_0
           (cos (fmax lambda1 lambda2))
           (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2))))))
        R)
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(lambda1, lambda2));
	double t_1 = acos(fma(t_0, (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))) * R;
	double tmp;
	if (phi2 <= -21000000.0) {
		tmp = t_1;
	} else if (phi2 <= 0.00064) {
		tmp = acos((cos(phi1) * fma(t_0, cos(fmax(lambda1, lambda2)), (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(lambda1, lambda2))
	t_1 = Float64(acos(fma(t_0, Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) * R)
	tmp = 0.0
	if (phi2 <= -21000000.0)
		tmp = t_1;
	elseif (phi2 <= 0.00064)
		tmp = Float64(acos(Float64(cos(phi1) * fma(t_0, cos(fmax(lambda1, lambda2)), Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -21000000.0], t$95$1, If[LessEqual[phi2, 0.00064], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(t$95$0 * N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.1e7 or 6.40000000000000052e-4 < phi2

   1. Initial program 73.5%

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

   2. Taylor expanded in 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 53.1

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

   4. Applied rewrites 53.1%

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

   if -2.1e7 < phi2 < 6.40000000000000052e-4

   1. Initial program 73.5%

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

      1. lift-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 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.5

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

   9. Applied rewrites 53.5%

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

Alternative 13: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(t\_0, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -21000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 0.00064:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right), t\_0 \cdot \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin lambda1 lambda2)))
        (t_1
         (*
          (acos (fma t_0 (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
          R)))
   (if (<= phi2 -21000000.0)
     t_1
     (if (<= phi2 0.00064)
       (*
        (acos
         (*
          (cos phi1)
          (fma
           (sin (fmax lambda1 lambda2))
           (sin (fmin lambda1 lambda2))
           (* t_0 (cos (fmax lambda1 lambda2))))))
        R)
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(lambda1, lambda2));
	double t_1 = acos(fma(t_0, (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))) * R;
	double tmp;
	if (phi2 <= -21000000.0) {
		tmp = t_1;
	} else if (phi2 <= 0.00064) {
		tmp = acos((cos(phi1) * fma(sin(fmax(lambda1, lambda2)), sin(fmin(lambda1, lambda2)), (t_0 * cos(fmax(lambda1, lambda2)))))) * R;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(lambda1, lambda2))
	t_1 = Float64(acos(fma(t_0, Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) * R)
	tmp = 0.0
	if (phi2 <= -21000000.0)
		tmp = t_1;
	elseif (phi2 <= 0.00064)
		tmp = Float64(acos(Float64(cos(phi1) * fma(sin(fmax(lambda1, lambda2)), sin(fmin(lambda1, lambda2)), Float64(t_0 * cos(fmax(lambda1, lambda2)))))) * R);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -21000000.0], t$95$1, If[LessEqual[phi2, 0.00064], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.1e7 or 6.40000000000000052e-4 < phi2

   1. Initial program 73.5%

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

   2. Taylor expanded in 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 53.1

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

   4. Applied rewrites 53.1%

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

   if -2.1e7 < phi2 < 6.40000000000000052e-4

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      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. lower-fma.f64 53.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 53.5

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

   6. Applied rewrites 53.5%

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

Alternative 14: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 3.5 \cdot 10^{-19}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmax phi1 phi2) 3.5e-19)
   (*
    (acos
     (*
      (cos (fmin phi1 phi2))
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))))
    R)
   (* (acos (* (cos (fmax phi1 phi2)) (cos (- lambda1 lambda2)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(phi1, phi2) <= 3.5e-19) {
		tmp = acos((cos(fmin(phi1, phi2)) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * cos((lambda1 - lambda2)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(phi1, phi2) <= 3.5e-19)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * cos(Float64(lambda1 - lambda2)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 3.5e-19], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.50000000000000015e-19

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      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. lower-fma.f64 53.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 53.5

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

   6. Applied rewrites 53.5%

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

   if 3.50000000000000015e-19 < phi2

   1. Initial program 73.5%

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

   2. Taylor expanded in 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 42.5

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

   4. Applied rewrites 42.5%

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

Alternative 15: 58.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (fmax phi1 phi2) 2.4e-17)
     (* (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 (fmax(phi1, phi2) <= 2.4e-17) {
		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 (fmax(phi1, phi2) <= 2.4d-17) 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 (fmax(phi1, phi2) <= 2.4e-17) {
		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 fmax(phi1, phi2) <= 2.4e-17:
		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 (fmax(phi1, phi2) <= 2.4e-17)
		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 (max(phi1, phi2) <= 2.4e-17)
		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[Max[phi1, phi2], $MachinePrecision], 2.4e-17], 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 phi2 < 2.39999999999999986e-17

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

   if 2.39999999999999986e-17 < phi2

   1. Initial program 73.5%

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

   2. Taylor expanded in 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 42.5

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

   4. Applied rewrites 42.5%

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

Alternative 16: 43.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2))))
   (if (<= (fmin lambda1 lambda2) -1.2e-5)
     (* (acos (* (cos (fmin lambda1 lambda2)) t_0)) R)
     (* (acos (* t_0 (cos (fmax lambda1 lambda2)))) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2));
	double tmp;
	if (fmin(lambda1, lambda2) <= -1.2e-5) {
		tmp = acos((cos(fmin(lambda1, lambda2)) * t_0)) * R;
	} else {
		tmp = acos((t_0 * cos(fmax(lambda1, lambda2)))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(fmin(phi1, phi2))
    if (fmin(lambda1, lambda2) <= (-1.2d-5)) then
        tmp = acos((cos(fmin(lambda1, lambda2)) * t_0)) * r
    else
        tmp = acos((t_0 * cos(fmax(lambda1, lambda2)))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(fmin(phi1, phi2));
	double tmp;
	if (fmin(lambda1, lambda2) <= -1.2e-5) {
		tmp = Math.acos((Math.cos(fmin(lambda1, lambda2)) * t_0)) * R;
	} else {
		tmp = Math.acos((t_0 * Math.cos(fmax(lambda1, lambda2)))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(fmin(phi1, phi2))
	tmp = 0
	if fmin(lambda1, lambda2) <= -1.2e-5:
		tmp = math.acos((math.cos(fmin(lambda1, lambda2)) * t_0)) * R
	else:
		tmp = math.acos((t_0 * math.cos(fmax(lambda1, lambda2)))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	tmp = 0.0
	if (fmin(lambda1, lambda2) <= -1.2e-5)
		tmp = Float64(acos(Float64(cos(fmin(lambda1, lambda2)) * t_0)) * R);
	else
		tmp = Float64(acos(Float64(t_0 * cos(fmax(lambda1, lambda2)))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(min(phi1, phi2));
	tmp = 0.0;
	if (min(lambda1, lambda2) <= -1.2e-5)
		tmp = acos((cos(min(lambda1, lambda2)) * t_0)) * R;
	else
		tmp = acos((t_0 * cos(max(lambda1, lambda2)))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -1.2e-5], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$0 * N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.2e-5

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

   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 31.3

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

   7. Applied rewrites 31.3%

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

   if -1.2e-5 < lambda1

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   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. cos-neg-rev N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-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. cos-mult N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 42.8

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 42.8

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

   6. Applied rewrites 42.8%

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

   7. Taylor expanded in lambda1 around 0

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

      1. lower-cos.f64 31.7

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

   9. Applied rewrites 31.7%

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

Alternative 17: 42.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

2. 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 43.3

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

4. Applied rewrites 43.3%

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

Alternative 18: 36.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.3:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -0.3)
   (* (acos (* (cos (fmin lambda1 lambda2)) (cos (fmin phi1 phi2)))) R)
   (*
    (acos
     (*
      (cos (- (fmax lambda1 lambda2) (fmin lambda1 lambda2)))
      (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) <= -0.3) {
		tmp = acos((cos(fmin(lambda1, lambda2)) * cos(fmin(phi1, phi2)))) * R;
	} else {
		tmp = acos((cos((fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * 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) <= -0.3)
		tmp = Float64(acos(Float64(cos(fmin(lambda1, lambda2)) * cos(fmin(phi1, phi2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * 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], -0.3], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $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 < -0.299999999999999989

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

   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 31.3

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

   7. Applied rewrites 31.3%

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

   if -0.299999999999999989 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

   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 18.7

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.7%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   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. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 18.7

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right)}\right) \cdot R \]

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 18.7

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 18.7

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

   9. Applied rewrites 18.7%

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

Alternative 19: 27.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -2.55 \cdot 10^{+14}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\lambda_1, \lambda_2, 1\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -2.55e+14)
   (*
    (- (* PI 0.5) (asin (* (fma lambda1 lambda2 1.0) (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) <= -2.55e+14) {
		tmp = ((((double) M_PI) * 0.5) - asin((fma(lambda1, lambda2, 1.0) * 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) <= -2.55e+14)
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(Float64(fma(lambda1, lambda2, 1.0) * 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], -2.55e+14], N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[(lambda1 * lambda2 + 1.0), $MachinePrecision] * N[Cos[N[Min[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 phi1 < -2.55e14

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-neg.f64 25.1

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

   7. Applied rewrites 25.1%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 11.5

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

   10. Applied rewrites 11.5%

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

       1. lift-acos.f64 N/A

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

       2. acos-asin N/A

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

   12. Applied rewrites 11.5%

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

   if -2.55e14 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

   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 18.7

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.7%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   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. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 18.7

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right)}\right) \cdot R \]

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 18.7

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 18.7

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

   9. Applied rewrites 18.7%

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

Alternative 20: 27.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -2.55 \cdot 10^{+14}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \mathsf{fma}\left(\lambda_2, \lambda_1, 1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -2.55e+14)
   (* (acos (* (cos (fmin phi1 phi2)) (fma lambda2 lambda1 1.0))) 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) <= -2.55e+14) {
		tmp = acos((cos(fmin(phi1, phi2)) * fma(lambda2, lambda1, 1.0))) * 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) <= -2.55e+14)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * fma(lambda2, lambda1, 1.0))) * 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], -2.55e+14], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[(lambda2 * lambda1 + 1.0), $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 < -2.55e14

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-neg.f64 25.1

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

   7. Applied rewrites 25.1%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 11.5

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

   10. Applied rewrites 11.5%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 11.5

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

   12. Applied rewrites 11.5%

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

   if -2.55e14 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. 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 43.3

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

   4. Applied rewrites 43.3%

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

   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 18.7

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.7%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   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. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-*.f64 18.7

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_1}^{2}\right)}\right) \cdot R \]

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 18.7

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 18.7

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

   9. Applied rewrites 18.7%

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

Alternative 21: 11.2% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (* (cos (fmin phi1 phi2)) (fma lambda2 lambda1 1.0))) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos((cos(fmin(phi1, phi2)) * fma(lambda2, lambda1, 1.0))) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(cos(fmin(phi1, phi2)) * fma(lambda2, lambda1, 1.0))) * R)
end

Wolfram

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

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in phi2 around 0

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

   1. 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 43.3

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

4. Applied rewrites 43.3%

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

5. Taylor expanded in lambda1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-neg.f64 25.1

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

7. Applied rewrites 25.1%

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

8. Taylor expanded in lambda2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 11.5

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

10. Applied rewrites 11.5%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 11.5

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

12. Applied rewrites 11.5%

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

Alternative 22: 1.8% accurate, 4.7× speedup

Math

\[\cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \left(1 + \lambda_1 \cdot \lambda_2\right)\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (* (+ 1.0 (* -0.5 (pow phi1 2.0))) (+ 1.0 (* lambda1 lambda2)))) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((1.0 + (-0.5 * pow(phi1, 2.0))) * (1.0 + (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(((1.0d0 + ((-0.5d0) * (phi1 ** 2.0d0))) * (1.0d0 + (lambda1 * lambda2)))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(((1.0 + (-0.5 * Math.pow(phi1, 2.0))) * (1.0 + (lambda1 * lambda2)))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(((1.0 + (-0.5 * math.pow(phi1, 2.0))) * (1.0 + (lambda1 * lambda2)))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) * Float64(1.0 + Float64(lambda1 * lambda2)))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(((1.0 + (-0.5 * (phi1 ^ 2.0))) * (1.0 + (lambda1 * lambda2)))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(lambda1 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

2. 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 43.3

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

4. Applied rewrites 43.3%

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

5. Taylor expanded in lambda1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-neg.f64 25.1

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

7. Applied rewrites 25.1%

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

8. Taylor expanded in lambda2 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 11.5

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

10. Applied rewrites 11.5%

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

11. Taylor expanded in phi1 around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-pow.f64 1.8

       \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \left(1 + \lambda_1 \cdot \lambda_2\right)\right) \cdot R \]

13. Applied rewrites 1.8%

    \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \left(\color{blue}{1} + \lambda_1 \cdot \lambda_2\right)\right) \cdot R \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025171 
(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))