Spherical law of cosines

Percentage Accurate: 73.3% → 94.0%

Time: 17.8s

Alternatives: 25

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: 73.3% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 94.0%

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

3. Applied rewrites 94.0%

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

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. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 94.0%

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

8. Applied rewrites 94.0%

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

Alternative 2: 94.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 94.0%

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

3. Applied rewrites 94.0%

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

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: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -0.0032:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\frac{\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_0}{t\_1}, \frac{t\_2}{t\_3}, 1\right) \cdot \left(t\_3 \cdot t\_1\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 200:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{max}\left(\phi_1, \phi_2\right), t\_1, t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_1 \cdot t\_3 + \left(t\_0 \cdot t\_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (sin (fmin phi1 phi2)))
        (t_2 (cos (fmax phi1 phi2)))
        (t_3 (sin (fmax phi1 phi2))))
   (if (<= (fmax phi1 phi2) -0.0032)
     (*
      (acos
       (*
        (fma (/ (* (cos (- lambda2 lambda1)) t_0) t_1) (/ t_2 t_3) 1.0)
        (* t_3 t_1)))
      R)
     (if (<= (fmax phi1 phi2) 200.0)
       (*
        (acos
         (fma
          (fmax phi1 phi2)
          t_1
          (*
           t_0
           (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
        R)
       (*
        (acos (+ (* t_1 t_3) (* (* t_0 t_2) (cos (- lambda1 lambda2)))))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2));
	double t_1 = sin(fmin(phi1, phi2));
	double t_2 = cos(fmax(phi1, phi2));
	double t_3 = sin(fmax(phi1, phi2));
	double tmp;
	if (fmax(phi1, phi2) <= -0.0032) {
		tmp = acos((fma(((cos((lambda2 - lambda1)) * t_0) / t_1), (t_2 / t_3), 1.0) * (t_3 * t_1))) * R;
	} else if (fmax(phi1, phi2) <= 200.0) {
		tmp = acos(fma(fmax(phi1, phi2), t_1, (t_0 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))))) * R;
	} else {
		tmp = acos(((t_1 * t_3) + ((t_0 * t_2) * cos((lambda1 - lambda2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	t_1 = sin(fmin(phi1, phi2))
	t_2 = cos(fmax(phi1, phi2))
	t_3 = sin(fmax(phi1, phi2))
	tmp = 0.0
	if (fmax(phi1, phi2) <= -0.0032)
		tmp = Float64(acos(Float64(fma(Float64(Float64(cos(Float64(lambda2 - lambda1)) * t_0) / t_1), Float64(t_2 / t_3), 1.0) * Float64(t_3 * t_1))) * R);
	elseif (fmax(phi1, phi2) <= 200.0)
		tmp = Float64(acos(fma(fmax(phi1, phi2), t_1, Float64(t_0 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))))) * R);
	else
		tmp = Float64(acos(Float64(Float64(t_1 * t_3) + Float64(Float64(t_0 * t_2) * cos(Float64(lambda1 - lambda2))))) * 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[Sin[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[Max[phi1, phi2], $MachinePrecision], -0.0032], N[(N[ArcCos[N[(N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t$95$2 / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 200.0], N[(N[ArcCos[N[(N[Max[phi1, phi2], $MachinePrecision] * t$95$1 + 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]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(t$95$1 * t$95$3), $MachinePrecision] + N[(N[(t$95$0 * 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 phi2 < -0.0032000000000000002

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

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

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

   3. Applied rewrites 65.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 65.7%

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

   5. Applied rewrites 65.7%

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

   if -0.0032000000000000002 < phi2 < 200

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   if 200 < phi2

   1. Initial program 73.3%

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

Alternative 4: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -5.8 \cdot 10^{-16}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t\_0 \cdot t\_1, t\_2 \cdot t\_3\right)\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 200:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{max}\left(\phi_1, \phi_2\right), t\_3, t\_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_3 \cdot t\_2 + \left(t\_1 \cdot t\_0\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (sin (fmax phi1 phi2)))
        (t_3 (sin (fmin phi1 phi2))))
   (if (<= (fmax phi1 phi2) -5.8e-16)
     (*
      (-
       (* PI 0.5)
       (asin (fma (cos (- lambda2 lambda1)) (* t_0 t_1) (* t_2 t_3))))
      R)
     (if (<= (fmax phi1 phi2) 200.0)
       (*
        (acos
         (fma
          (fmax phi1 phi2)
          t_3
          (*
           t_1
           (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
        R)
       (*
        (acos (+ (* t_3 t_2) (* (* t_1 t_0) (cos (- lambda1 lambda2)))))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = sin(fmax(phi1, phi2));
	double t_3 = sin(fmin(phi1, phi2));
	double tmp;
	if (fmax(phi1, phi2) <= -5.8e-16) {
		tmp = ((((double) M_PI) * 0.5) - asin(fma(cos((lambda2 - lambda1)), (t_0 * t_1), (t_2 * t_3)))) * R;
	} else if (fmax(phi1, phi2) <= 200.0) {
		tmp = acos(fma(fmax(phi1, phi2), t_3, (t_1 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))))) * R;
	} else {
		tmp = acos(((t_3 * t_2) + ((t_1 * t_0) * cos((lambda1 - lambda2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = sin(fmax(phi1, phi2))
	t_3 = sin(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(phi1, phi2) <= -5.8e-16)
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(fma(cos(Float64(lambda2 - lambda1)), Float64(t_0 * t_1), Float64(t_2 * t_3)))) * R);
	elseif (fmax(phi1, phi2) <= 200.0)
		tmp = Float64(acos(fma(fmax(phi1, phi2), t_3, Float64(t_1 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))))) * R);
	else
		tmp = Float64(acos(Float64(Float64(t_3 * t_2) + Float64(Float64(t_1 * t_0) * cos(Float64(lambda1 - lambda2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], -5.8e-16], N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 200.0], N[(N[ArcCos[N[(N[Max[phi1, phi2], $MachinePrecision] * t$95$3 + N[(t$95$1 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(t$95$3 * t$95$2), $MachinePrecision] + N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.7999999999999996e-16

   1. Initial program 73.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 73.2%

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

      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 73.2%

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

   3. Applied rewrites 73.2%

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

   if -5.7999999999999996e-16 < phi2 < 200

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   if 200 < phi2

   1. Initial program 73.3%

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

Alternative 5: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -55:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t\_0 \cdot t\_1, t\_2 \cdot t\_3\right)\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 680:\\ \;\;\;\;\cos^{-1} \left(t\_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_3 \cdot t\_2 + \left(t\_1 \cdot t\_0\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (sin (fmax phi1 phi2)))
        (t_3 (sin (fmin phi1 phi2))))
   (if (<= (fmax phi1 phi2) -55.0)
     (*
      (-
       (* PI 0.5)
       (asin (fma (cos (- lambda2 lambda1)) (* t_0 t_1) (* t_2 t_3))))
      R)
     (if (<= (fmax phi1 phi2) 680.0)
       (*
        (acos
         (*
          t_1
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (*
        (acos (+ (* t_3 t_2) (* (* t_1 t_0) (cos (- lambda1 lambda2)))))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = sin(fmax(phi1, phi2));
	double t_3 = sin(fmin(phi1, phi2));
	double tmp;
	if (fmax(phi1, phi2) <= -55.0) {
		tmp = ((((double) M_PI) * 0.5) - asin(fma(cos((lambda2 - lambda1)), (t_0 * t_1), (t_2 * t_3)))) * R;
	} else if (fmax(phi1, phi2) <= 680.0) {
		tmp = acos((t_1 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = acos(((t_3 * t_2) + ((t_1 * t_0) * cos((lambda1 - lambda2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = sin(fmax(phi1, phi2))
	t_3 = sin(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(phi1, phi2) <= -55.0)
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(fma(cos(Float64(lambda2 - lambda1)), Float64(t_0 * t_1), Float64(t_2 * t_3)))) * R);
	elseif (fmax(phi1, phi2) <= 680.0)
		tmp = Float64(acos(Float64(t_1 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(acos(Float64(Float64(t_3 * t_2) + Float64(Float64(t_1 * t_0) * cos(Float64(lambda1 - lambda2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], -55.0], N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 680.0], N[(N[ArcCos[N[(t$95$1 * 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$3 * t$95$2), $MachinePrecision] + N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -55

   1. Initial program 73.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 73.2%

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

      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 73.2%

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

   3. Applied rewrites 73.2%

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

   if -55 < phi2 < 680

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 53.0%

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

   6. Applied rewrites 53.0%

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

   if 680 < phi2

   1. Initial program 73.3%

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

Alternative 6: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -55:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_3, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_0\right) \cdot t\_1\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 680:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_3 \cdot t\_2 + \left(t\_0 \cdot t\_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (cos (fmax phi1 phi2)))
        (t_2 (sin (fmax phi1 phi2)))
        (t_3 (sin (fmin phi1 phi2))))
   (if (<= (fmax phi1 phi2) -55.0)
     (* (acos (fma t_2 t_3 (* (* (cos (- lambda2 lambda1)) t_0) t_1))) R)
     (if (<= (fmax phi1 phi2) 680.0)
       (*
        (acos
         (*
          t_0
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (*
        (acos (+ (* t_3 t_2) (* (* t_0 t_1) (cos (- lambda1 lambda2)))))
        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 = sin(fmax(phi1, phi2));
	double t_3 = sin(fmin(phi1, phi2));
	double tmp;
	if (fmax(phi1, phi2) <= -55.0) {
		tmp = acos(fma(t_2, t_3, ((cos((lambda2 - lambda1)) * t_0) * t_1))) * R;
	} else if (fmax(phi1, phi2) <= 680.0) {
		tmp = acos((t_0 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = acos(((t_3 * t_2) + ((t_0 * t_1) * cos((lambda1 - lambda2))))) * 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 = sin(fmax(phi1, phi2))
	t_3 = sin(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(phi1, phi2) <= -55.0)
		tmp = Float64(acos(fma(t_2, t_3, Float64(Float64(cos(Float64(lambda2 - lambda1)) * t_0) * t_1))) * R);
	elseif (fmax(phi1, phi2) <= 680.0)
		tmp = Float64(acos(Float64(t_0 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(acos(Float64(Float64(t_3 * t_2) + Float64(Float64(t_0 * t_1) * cos(Float64(lambda1 - lambda2))))) * 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[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], -55.0], N[(N[ArcCos[N[(t$95$2 * t$95$3 + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 680.0], N[(N[ArcCos[N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(t$95$3 * t$95$2), $MachinePrecision] + N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -55

   1. Initial program 73.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 73.3%

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

      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 73.3%

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

      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 73.3%

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

   3. Applied rewrites 73.3%

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

   if -55 < phi2 < 680

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 53.0%

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

   6. Applied rewrites 53.0%

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

   if 680 < phi2

   1. Initial program 73.3%

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

Alternative 7: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(\sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right), \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right), \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_0\right) \cdot \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -55:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 680:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1
         (*
          (acos
           (fma
            (sin (fmax phi1 phi2))
            (sin (fmin phi1 phi2))
            (* (* (cos (- lambda2 lambda1)) t_0) (cos (fmax phi1 phi2)))))
          R)))
   (if (<= (fmax phi1 phi2) -55.0)
     t_1
     (if (<= (fmax phi1 phi2) 680.0)
       (*
        (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(fmin(phi1, phi2));
	double t_1 = acos(fma(sin(fmax(phi1, phi2)), sin(fmin(phi1, phi2)), ((cos((lambda2 - lambda1)) * t_0) * cos(fmax(phi1, phi2))))) * R;
	double tmp;
	if (fmax(phi1, phi2) <= -55.0) {
		tmp = t_1;
	} else if (fmax(phi1, phi2) <= 680.0) {
		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(fmin(phi1, phi2))
	t_1 = Float64(acos(fma(sin(fmax(phi1, phi2)), sin(fmin(phi1, phi2)), Float64(Float64(cos(Float64(lambda2 - lambda1)) * t_0) * cos(fmax(phi1, phi2))))) * R)
	tmp = 0.0
	if (fmax(phi1, phi2) <= -55.0)
		tmp = t_1;
	elseif (fmax(phi1, phi2) <= 680.0)
		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[Min[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] * t$95$0), $MachinePrecision] * N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], -55.0], t$95$1, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 680.0], 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 phi2 < -55 or 680 < phi2

   1. Initial program 73.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 73.3%

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

      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 73.3%

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

      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 73.3%

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

   3. Applied rewrites 73.3%

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

   if -55 < phi2 < 680

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 53.0%

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

   6. Applied rewrites 53.0%

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

Alternative 8: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_4 := \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\\ t_5 := \cos^{-1} \left(t\_3 \cdot \mathsf{fma}\left(t\_1, t\_4, \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\right) \cdot R\\ t_6 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -1.6 \cdot 10^{+201}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -7.4 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_0 \cdot t\_3, t\_6 \cdot t\_2\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq 1.65 \cdot 10^{-67}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_4 \cdot t\_0, t\_3, t\_2 \cdot t\_6\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (cos (fmin lambda1 lambda2)))
        (t_2 (sin (fmax phi1 phi2)))
        (t_3 (cos (fmax phi1 phi2)))
        (t_4 (cos (fmax lambda1 lambda2)))
        (t_5
         (*
          (acos
           (*
            t_3
            (fma
             t_1
             t_4
             (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2))))))
          R))
        (t_6 (sin (fmin phi1 phi2))))
   (if (<= (fmin lambda1 lambda2) -1.6e+201)
     t_5
     (if (<= (fmin lambda1 lambda2) -7.4e-8)
       (* (acos (fma t_1 (* t_0 t_3) (* t_6 t_2))) R)
       (if (<= (fmin lambda1 lambda2) 1.65e-67)
         (* (acos (fma (* t_4 t_0) t_3 (* t_2 t_6))) R)
         t_5)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2));
	double t_1 = cos(fmin(lambda1, lambda2));
	double t_2 = sin(fmax(phi1, phi2));
	double t_3 = cos(fmax(phi1, phi2));
	double t_4 = cos(fmax(lambda1, lambda2));
	double t_5 = acos((t_3 * fma(t_1, t_4, (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R;
	double t_6 = sin(fmin(phi1, phi2));
	double tmp;
	if (fmin(lambda1, lambda2) <= -1.6e+201) {
		tmp = t_5;
	} else if (fmin(lambda1, lambda2) <= -7.4e-8) {
		tmp = acos(fma(t_1, (t_0 * t_3), (t_6 * t_2))) * R;
	} else if (fmin(lambda1, lambda2) <= 1.65e-67) {
		tmp = acos(fma((t_4 * t_0), t_3, (t_2 * t_6))) * R;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	t_1 = cos(fmin(lambda1, lambda2))
	t_2 = sin(fmax(phi1, phi2))
	t_3 = cos(fmax(phi1, phi2))
	t_4 = cos(fmax(lambda1, lambda2))
	t_5 = Float64(acos(Float64(t_3 * fma(t_1, t_4, Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R)
	t_6 = sin(fmin(phi1, phi2))
	tmp = 0.0
	if (fmin(lambda1, lambda2) <= -1.6e+201)
		tmp = t_5;
	elseif (fmin(lambda1, lambda2) <= -7.4e-8)
		tmp = Float64(acos(fma(t_1, Float64(t_0 * t_3), Float64(t_6 * t_2))) * R);
	elseif (fmin(lambda1, lambda2) <= 1.65e-67)
		tmp = Float64(acos(fma(Float64(t_4 * t_0), t_3, Float64(t_2 * t_6))) * R);
	else
		tmp = t_5;
	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[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[ArcCos[N[(t$95$3 * N[(t$95$1 * t$95$4 + N[(N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$6 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -1.6e+201], t$95$5, If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -7.4e-8], N[(N[ArcCos[N[(t$95$1 * N[(t$95$0 * t$95$3), $MachinePrecision] + N[(t$95$6 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], 1.65e-67], N[(N[ArcCos[N[(N[(t$95$4 * t$95$0), $MachinePrecision] * t$95$3 + N[(t$95$2 * t$95$6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$5]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.6e201 or 1.6500000000000001e-67 < lambda1

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   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 52.7%

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

   6. Applied rewrites 52.7%

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

   if -1.6e201 < lambda1 < -7.4000000000000001e-8

   1. Initial program 73.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.8%

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

   4. Applied rewrites 52.8%

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

   if -7.4000000000000001e-8 < lambda1 < 1.6500000000000001e-67

   1. Initial program 73.3%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-neg.f64 53.3%

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 53.3%

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

      9. lift-cos.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 53.3%

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

   6. Applied rewrites 53.3%

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

Alternative 9: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_4 := \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\\ t_5 := \cos^{-1} \left(t\_3 \cdot \mathsf{fma}\left(t\_1, t\_4, \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\right) \cdot R\\ t_6 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -1.6 \cdot 10^{+201}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -7.4 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_0 \cdot t\_3, t\_6 \cdot t\_2\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq 1.65 \cdot 10^{-67}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_6, \left(t\_4 \cdot t\_0\right) \cdot t\_3\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (cos (fmin lambda1 lambda2)))
        (t_2 (sin (fmax phi1 phi2)))
        (t_3 (cos (fmax phi1 phi2)))
        (t_4 (cos (fmax lambda1 lambda2)))
        (t_5
         (*
          (acos
           (*
            t_3
            (fma
             t_1
             t_4
             (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2))))))
          R))
        (t_6 (sin (fmin phi1 phi2))))
   (if (<= (fmin lambda1 lambda2) -1.6e+201)
     t_5
     (if (<= (fmin lambda1 lambda2) -7.4e-8)
       (* (acos (fma t_1 (* t_0 t_3) (* t_6 t_2))) R)
       (if (<= (fmin lambda1 lambda2) 1.65e-67)
         (* (acos (fma t_2 t_6 (* (* t_4 t_0) t_3))) R)
         t_5)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2));
	double t_1 = cos(fmin(lambda1, lambda2));
	double t_2 = sin(fmax(phi1, phi2));
	double t_3 = cos(fmax(phi1, phi2));
	double t_4 = cos(fmax(lambda1, lambda2));
	double t_5 = acos((t_3 * fma(t_1, t_4, (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R;
	double t_6 = sin(fmin(phi1, phi2));
	double tmp;
	if (fmin(lambda1, lambda2) <= -1.6e+201) {
		tmp = t_5;
	} else if (fmin(lambda1, lambda2) <= -7.4e-8) {
		tmp = acos(fma(t_1, (t_0 * t_3), (t_6 * t_2))) * R;
	} else if (fmin(lambda1, lambda2) <= 1.65e-67) {
		tmp = acos(fma(t_2, t_6, ((t_4 * t_0) * t_3))) * R;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	t_1 = cos(fmin(lambda1, lambda2))
	t_2 = sin(fmax(phi1, phi2))
	t_3 = cos(fmax(phi1, phi2))
	t_4 = cos(fmax(lambda1, lambda2))
	t_5 = Float64(acos(Float64(t_3 * fma(t_1, t_4, Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R)
	t_6 = sin(fmin(phi1, phi2))
	tmp = 0.0
	if (fmin(lambda1, lambda2) <= -1.6e+201)
		tmp = t_5;
	elseif (fmin(lambda1, lambda2) <= -7.4e-8)
		tmp = Float64(acos(fma(t_1, Float64(t_0 * t_3), Float64(t_6 * t_2))) * R);
	elseif (fmin(lambda1, lambda2) <= 1.65e-67)
		tmp = Float64(acos(fma(t_2, t_6, Float64(Float64(t_4 * t_0) * t_3))) * R);
	else
		tmp = t_5;
	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[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[ArcCos[N[(t$95$3 * N[(t$95$1 * t$95$4 + N[(N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$6 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -1.6e+201], t$95$5, If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -7.4e-8], N[(N[ArcCos[N[(t$95$1 * N[(t$95$0 * t$95$3), $MachinePrecision] + N[(t$95$6 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], 1.65e-67], N[(N[ArcCos[N[(t$95$2 * t$95$6 + N[(N[(t$95$4 * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$5]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.6e201 or 1.6500000000000001e-67 < lambda1

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   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 52.7%

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

   6. Applied rewrites 52.7%

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

   if -1.6e201 < lambda1 < -7.4000000000000001e-8

   1. Initial program 73.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.8%

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

   4. Applied rewrites 52.8%

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

   if -7.4000000000000001e-8 < lambda1 < 1.6500000000000001e-67

   1. Initial program 73.3%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-neg.f64 53.3%

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 53.3%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 53.3%

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

      11. lift-cos.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 53.3%

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

   6. Applied rewrites 53.3%

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

Alternative 10: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_3 := \mathsf{fma}\left(t\_2, \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -55:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_0 \cdot t\_1, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 16000000:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot t\_3\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_1 \cdot t\_3\right) \cdot R\\ \end{array} \]

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) -55.0)
     (*
      (acos
       (fma t_2 (* t_0 t_1) (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))))
      R)
     (if (<= (fmax phi1 phi2) 16000000.0)
       (* (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) <= -55.0) {
		tmp = acos(fma(t_2, (t_0 * t_1), (sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R;
	} else if (fmax(phi1, phi2) <= 16000000.0) {
		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) <= -55.0)
		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) <= 16000000.0)
		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], -55.0], 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], 16000000.0], 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 < -55

   1. Initial program 73.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.8%

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

   4. Applied rewrites 52.8%

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

   if -55 < phi2 < 1.6e7

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 53.0%

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

   6. Applied rewrites 53.0%

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

   if 1.6e7 < phi2

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   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 52.7%

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

   6. Applied rewrites 52.7%

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

Alternative 11: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos^{-1} \left(t\_2 + t\_1 \cdot \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right) \cdot R\\ t_4 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_5 := \cos^{-1} \left(\mathsf{fma}\left(t\_4, t\_0 \cdot t\_1, t\_2\right)\right) \cdot R\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -55:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 16000000:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(t\_4, \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 3 \cdot 10^{+103}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 1.52 \cdot 10^{+262}:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (cos (fmax phi1 phi2)))
        (t_2 (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2))))
        (t_3
         (*
          (acos
           (+
            t_2
            (* t_1 (cos (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))))))
          R))
        (t_4 (cos (fmin lambda1 lambda2)))
        (t_5 (* (acos (fma t_4 (* t_0 t_1) t_2)) R)))
   (if (<= (fmax phi1 phi2) -55.0)
     t_5
     (if (<= (fmax phi1 phi2) 16000000.0)
       (*
        (acos
         (*
          t_0
          (fma
           t_4
           (cos (fmax lambda1 lambda2))
           (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2))))))
        R)
       (if (<= (fmax phi1 phi2) 3e+103)
         t_3
         (if (<= (fmax phi1 phi2) 1.52e+262) t_5 t_3))))))

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 = sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2));
	double t_3 = acos((t_2 + (t_1 * cos((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R;
	double t_4 = cos(fmin(lambda1, lambda2));
	double t_5 = acos(fma(t_4, (t_0 * t_1), t_2)) * R;
	double tmp;
	if (fmax(phi1, phi2) <= -55.0) {
		tmp = t_5;
	} else if (fmax(phi1, phi2) <= 16000000.0) {
		tmp = acos((t_0 * fma(t_4, cos(fmax(lambda1, lambda2)), (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R;
	} else if (fmax(phi1, phi2) <= 3e+103) {
		tmp = t_3;
	} else if (fmax(phi1, phi2) <= 1.52e+262) {
		tmp = t_5;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	t_1 = cos(fmax(phi1, phi2))
	t_2 = Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2)))
	t_3 = Float64(acos(Float64(t_2 + Float64(t_1 * cos(Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R)
	t_4 = cos(fmin(lambda1, lambda2))
	t_5 = Float64(acos(fma(t_4, Float64(t_0 * t_1), t_2)) * R)
	tmp = 0.0
	if (fmax(phi1, phi2) <= -55.0)
		tmp = t_5;
	elseif (fmax(phi1, phi2) <= 16000000.0)
		tmp = Float64(acos(Float64(t_0 * fma(t_4, cos(fmax(lambda1, lambda2)), Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R);
	elseif (fmax(phi1, phi2) <= 3e+103)
		tmp = t_3;
	elseif (fmax(phi1, phi2) <= 1.52e+262)
		tmp = t_5;
	else
		tmp = t_3;
	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[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcCos[N[(t$95$2 + N[(t$95$1 * N[Cos[N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[ArcCos[N[(t$95$4 * N[(t$95$0 * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], -55.0], t$95$5, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 16000000.0], N[(N[ArcCos[N[(t$95$0 * N[(t$95$4 * 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], If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 3e+103], t$95$3, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 1.52e+262], t$95$5, t$95$3]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -55 or 3e103 < phi2 < 1.5200000000000001e262

   1. Initial program 73.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.8%

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

   4. Applied rewrites 52.8%

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

   if -55 < phi2 < 1.6e7

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. 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 53.0%

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

   6. Applied rewrites 53.0%

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

   if 1.6e7 < phi2 < 3e103 or 1.5200000000000001e262 < phi2

   1. Initial program 73.3%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 42.0%

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

   4. Applied rewrites 42.0%

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

Alternative 12: 62.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 0.0023:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right), t\_0 \cdot t\_1, t\_2\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 10^{+69}:\\ \;\;\;\;\cos^{-1} \left(t\_2 + t\_1 \cdot \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \pi - \sin^{-1} \left(\cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot t\_0\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (cos (fmax phi1 phi2)))
        (t_2 (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))))
   (if (<= (fmax lambda1 lambda2) 0.0023)
     (* (acos (fma (cos (fmin lambda1 lambda2)) (* t_0 t_1) t_2)) R)
     (if (<= (fmax lambda1 lambda2) 1e+69)
       (*
        (acos
         (+
          t_2
          (* t_1 (cos (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))))))
        R)
       (*
        (-
         (* 0.5 PI)
         (asin
          (* (cos (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))) t_0)))
        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 = sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2));
	double tmp;
	if (fmax(lambda1, lambda2) <= 0.0023) {
		tmp = acos(fma(cos(fmin(lambda1, lambda2)), (t_0 * t_1), t_2)) * R;
	} else if (fmax(lambda1, lambda2) <= 1e+69) {
		tmp = acos((t_2 + (t_1 * cos((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R;
	} else {
		tmp = ((0.5 * ((double) M_PI)) - asin((cos((fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * t_0))) * 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 = Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2)))
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 0.0023)
		tmp = Float64(acos(fma(cos(fmin(lambda1, lambda2)), Float64(t_0 * t_1), t_2)) * R);
	elseif (fmax(lambda1, lambda2) <= 1e+69)
		tmp = Float64(acos(Float64(t_2 + Float64(t_1 * cos(Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R);
	else
		tmp = Float64(Float64(Float64(0.5 * pi) - asin(Float64(cos(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * t_0))) * 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[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 0.0023], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 1e+69], N[(N[ArcCos[N[(t$95$2 + N[(t$95$1 * N[Cos[N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 0.0023

   1. Initial program 73.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.8%

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

   4. Applied rewrites 52.8%

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

   if 0.0023 < lambda2 < 1.0000000000000001e69

   1. Initial program 73.3%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 42.0%

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

   4. Applied rewrites 42.0%

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

   if 1.0000000000000001e69 < lambda2

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-asin.f64 42.8%

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

      10. lift-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

   6. Applied rewrites 42.8%

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

Alternative 13: 59.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot t\_3\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 2.05:\\ \;\;\;\;\cos^{-1} \left(\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \left(1 + -0.16666666666666666 \cdot {\left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)}^{2}\right)\right) \cdot t\_0 + \left(t\_2 \cdot t\_1\right) \cdot t\_3\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_1, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmax phi1 phi2)))
        (t_1 (cos (fmax phi1 phi2)))
        (t_2 (cos (fmin phi1 phi2)))
        (t_3 (cos (- lambda1 lambda2))))
   (if (<= (fmin phi1 phi2) -7.5e-8)
     (* (acos (* t_2 t_3)) R)
     (if (<= (fmin phi1 phi2) 2.05)
       (*
        (acos
         (+
          (*
           (*
            (fmin phi1 phi2)
            (+ 1.0 (* -0.16666666666666666 (pow (fmin phi1 phi2) 2.0))))
           t_0)
          (* (* t_2 t_1) t_3)))
        R)
       (* (acos (fma t_2 t_1 (* (sin (fmin phi1 phi2)) t_0))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmax(phi1, phi2));
	double t_1 = cos(fmax(phi1, phi2));
	double t_2 = cos(fmin(phi1, phi2));
	double t_3 = cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -7.5e-8) {
		tmp = acos((t_2 * t_3)) * R;
	} else if (fmin(phi1, phi2) <= 2.05) {
		tmp = acos((((fmin(phi1, phi2) * (1.0 + (-0.16666666666666666 * pow(fmin(phi1, phi2), 2.0)))) * t_0) + ((t_2 * t_1) * t_3))) * R;
	} else {
		tmp = acos(fma(t_2, t_1, (sin(fmin(phi1, phi2)) * t_0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmax(phi1, phi2))
	t_1 = cos(fmax(phi1, phi2))
	t_2 = cos(fmin(phi1, phi2))
	t_3 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -7.5e-8)
		tmp = Float64(acos(Float64(t_2 * t_3)) * R);
	elseif (fmin(phi1, phi2) <= 2.05)
		tmp = Float64(acos(Float64(Float64(Float64(fmin(phi1, phi2) * Float64(1.0 + Float64(-0.16666666666666666 * (fmin(phi1, phi2) ^ 2.0)))) * t_0) + Float64(Float64(t_2 * t_1) * t_3))) * R);
	else
		tmp = Float64(acos(fma(t_2, t_1, Float64(sin(fmin(phi1, phi2)) * t_0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -7.5e-8], N[(N[ArcCos[N[(t$95$2 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 2.05], N[(N[ArcCos[N[(N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Min[phi1, phi2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(t$95$2 * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 * t$95$1 + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -7.4999999999999997e-8

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   if -7.4999999999999997e-8 < phi1 < 2.0499999999999998

   1. Initial program 73.3%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 37.0%

         \[\leadsto \cos^{-1} \left(\left(\phi_1 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_1}^{\color{blue}{2}}\right)\right) \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 37.0%

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(\phi_1 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_1}^{2}\right)\right)} \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 2.0499999999999998 < phi1

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   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 52.8%

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

   9. Applied rewrites 52.8%

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

   10. Taylor expanded in lambda1 around 0

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

       1. lower-cos.f64 31.8%

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

   12. Applied rewrites 31.8%

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

Alternative 14: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.84:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot t\_3\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 1.6:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \left(\left(1 + -0.5 \cdot {\left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)}^{2}\right) \cdot t\_0\right) \cdot t\_3\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_0, t\_1\right)\right) \cdot R\\ \end{array} \]

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 phi1 phi2)))
        (t_3 (cos (- lambda1 lambda2))))
   (if (<= (fmin phi1 phi2) -0.84)
     (* (acos (* t_2 t_3)) R)
     (if (<= (fmin phi1 phi2) 1.6)
       (*
        (acos
         (+ t_1 (* (* (+ 1.0 (* -0.5 (pow (fmin phi1 phi2) 2.0))) t_0) t_3)))
        R)
       (* (acos (fma t_2 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(phi1, phi2));
	double t_3 = cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -0.84) {
		tmp = acos((t_2 * t_3)) * R;
	} else if (fmin(phi1, phi2) <= 1.6) {
		tmp = acos((t_1 + (((1.0 + (-0.5 * pow(fmin(phi1, phi2), 2.0))) * t_0) * t_3))) * R;
	} else {
		tmp = acos(fma(t_2, 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(phi1, phi2))
	t_3 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.84)
		tmp = Float64(acos(Float64(t_2 * t_3)) * R);
	elseif (fmin(phi1, phi2) <= 1.6)
		tmp = Float64(acos(Float64(t_1 + Float64(Float64(Float64(1.0 + Float64(-0.5 * (fmin(phi1, phi2) ^ 2.0))) * t_0) * t_3))) * R);
	else
		tmp = Float64(acos(fma(t_2, 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[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.84], N[(N[ArcCos[N[(t$95$2 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 1.6], N[(N[ArcCos[N[(t$95$1 + N[(N[(N[(1.0 + N[(-0.5 * N[Power[N[Min[phi1, phi2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.83999999999999997

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   if -0.83999999999999997 < phi1 < 1.6000000000000001

   1. Initial program 73.3%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 33.6%

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

   4. Applied rewrites 33.6%

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

   if 1.6000000000000001 < phi1

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   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 52.8%

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

   9. Applied rewrites 52.8%

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

   10. Taylor expanded in lambda1 around 0

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

       1. lower-cos.f64 31.8%

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

   12. Applied rewrites 31.8%

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

Alternative 15: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.31:\\ \;\;\;\;\cos^{-1} \left(t\_1 \cdot t\_2\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 1.12:\\ \;\;\;\;\cos^{-1} \left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot t\_0 + \left(t\_1 \cdot t\_3\right) \cdot t\_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_3, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmax phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (cos (- lambda1 lambda2)))
        (t_3 (cos (fmax phi1 phi2))))
   (if (<= (fmin phi1 phi2) -0.31)
     (* (acos (* t_1 t_2)) R)
     (if (<= (fmin phi1 phi2) 1.12)
       (* (acos (+ (* (fmin phi1 phi2) t_0) (* (* t_1 t_3) t_2))) R)
       (* (acos (fma t_1 t_3 (* (sin (fmin phi1 phi2)) t_0))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmax(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = cos((lambda1 - lambda2));
	double t_3 = cos(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -0.31) {
		tmp = acos((t_1 * t_2)) * R;
	} else if (fmin(phi1, phi2) <= 1.12) {
		tmp = acos(((fmin(phi1, phi2) * t_0) + ((t_1 * t_3) * t_2))) * R;
	} else {
		tmp = acos(fma(t_1, t_3, (sin(fmin(phi1, phi2)) * t_0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmax(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = cos(Float64(lambda1 - lambda2))
	t_3 = cos(fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.31)
		tmp = Float64(acos(Float64(t_1 * t_2)) * R);
	elseif (fmin(phi1, phi2) <= 1.12)
		tmp = Float64(acos(Float64(Float64(fmin(phi1, phi2) * t_0) + Float64(Float64(t_1 * t_3) * t_2))) * R);
	else
		tmp = Float64(acos(fma(t_1, t_3, Float64(sin(fmin(phi1, phi2)) * t_0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.31], N[(N[ArcCos[N[(t$95$1 * t$95$2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 1.12], N[(N[ArcCos[N[(N[(N[Min[phi1, phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(t$95$1 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 * t$95$3 + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.31

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   if -0.31 < phi1 < 1.1200000000000001

   1. Initial program 73.3%

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

   2. Taylor expanded in phi1 around 0

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

   4. Applied rewrites 43.2%

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

   if 1.1200000000000001 < phi1

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   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 52.8%

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

   9. Applied rewrites 52.8%

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

   10. Taylor expanded in lambda1 around 0

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

       1. lower-cos.f64 31.8%

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

   12. Applied rewrites 31.8%

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

Alternative 16: 59.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot t\_3\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 1.2:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right), t\_0, t\_1 \cdot t\_3\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_1, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmax phi1 phi2)))
        (t_1 (cos (fmax phi1 phi2)))
        (t_2 (cos (fmin phi1 phi2)))
        (t_3 (cos (- lambda1 lambda2))))
   (if (<= (fmin phi1 phi2) -7.5e-8)
     (* (acos (* t_2 t_3)) R)
     (if (<= (fmin phi1 phi2) 1.2)
       (* (acos (fma (fmin phi1 phi2) t_0 (* t_1 t_3))) R)
       (* (acos (fma t_2 t_1 (* (sin (fmin phi1 phi2)) t_0))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmax(phi1, phi2));
	double t_1 = cos(fmax(phi1, phi2));
	double t_2 = cos(fmin(phi1, phi2));
	double t_3 = cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -7.5e-8) {
		tmp = acos((t_2 * t_3)) * R;
	} else if (fmin(phi1, phi2) <= 1.2) {
		tmp = acos(fma(fmin(phi1, phi2), t_0, (t_1 * t_3))) * R;
	} else {
		tmp = acos(fma(t_2, t_1, (sin(fmin(phi1, phi2)) * t_0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmax(phi1, phi2))
	t_1 = cos(fmax(phi1, phi2))
	t_2 = cos(fmin(phi1, phi2))
	t_3 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -7.5e-8)
		tmp = Float64(acos(Float64(t_2 * t_3)) * R);
	elseif (fmin(phi1, phi2) <= 1.2)
		tmp = Float64(acos(fma(fmin(phi1, phi2), t_0, Float64(t_1 * t_3))) * R);
	else
		tmp = Float64(acos(fma(t_2, t_1, Float64(sin(fmin(phi1, phi2)) * t_0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -7.5e-8], N[(N[ArcCos[N[(t$95$2 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 1.2], N[(N[ArcCos[N[(N[Min[phi1, phi2], $MachinePrecision] * t$95$0 + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 * t$95$1 + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -7.4999999999999997e-8

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   if -7.4999999999999997e-8 < phi1 < 1.2

   1. Initial program 73.3%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 35.4%

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

   4. Applied rewrites 35.4%

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

   if 1.2 < phi1

   1. Initial program 73.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 94.0%

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

   3. Applied rewrites 94.0%

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

   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 52.8%

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

   9. Applied rewrites 52.8%

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

   10. Taylor expanded in lambda1 around 0

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

       1. lower-cos.f64 31.8%

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

   12. Applied rewrites 31.8%

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

Alternative 17: 57.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (fmin phi1 phi2) -7.5e-8)
     (* (acos (* (cos (fmin phi1 phi2)) t_0)) R)
     (* (acos (* (cos (fmax phi1 phi2)) t_0)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -7.5e-8) {
		tmp = acos((cos(fmin(phi1, phi2)) * t_0)) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * t_0)) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    if (fmin(phi1, phi2) <= (-7.5d-8)) then
        tmp = acos((cos(fmin(phi1, phi2)) * t_0)) * r
    else
        tmp = acos((cos(fmax(phi1, phi2)) * t_0)) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -7.5e-8) {
		tmp = Math.acos((Math.cos(fmin(phi1, phi2)) * t_0)) * R;
	} else {
		tmp = Math.acos((Math.cos(fmax(phi1, phi2)) * t_0)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if fmin(phi1, phi2) <= -7.5e-8:
		tmp = math.acos((math.cos(fmin(phi1, phi2)) * t_0)) * R
	else:
		tmp = math.acos((math.cos(fmax(phi1, phi2)) * t_0)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -7.5e-8)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * t_0)) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * t_0)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (min(phi1, phi2) <= -7.5e-8)
		tmp = acos((cos(min(phi1, phi2)) * t_0)) * R;
	else
		tmp = acos((cos(max(phi1, phi2)) * t_0)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -7.5e-8], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.4999999999999997e-8

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   if -7.4999999999999997e-8 < phi1

   1. Initial program 73.3%

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

   2. 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 42.4%

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

   4. Applied rewrites 42.4%

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

Alternative 18: 42.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -7.4 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \left(-\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin lambda1 lambda2) -7.4e-8)
   (* (acos (* (cos (fmin lambda1 lambda2)) (cos phi1))) R)
   (* (acos (* (cos phi1) (cos (- (fmax lambda1 lambda2))))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(lambda1, lambda2) <= -7.4e-8) {
		tmp = acos((cos(fmin(lambda1, lambda2)) * cos(phi1))) * R;
	} else {
		tmp = acos((cos(phi1) * 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) :: tmp
    if (fmin(lambda1, lambda2) <= (-7.4d-8)) then
        tmp = acos((cos(fmin(lambda1, lambda2)) * cos(phi1))) * r
    else
        tmp = acos((cos(phi1) * 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 tmp;
	if (fmin(lambda1, lambda2) <= -7.4e-8) {
		tmp = Math.acos((Math.cos(fmin(lambda1, lambda2)) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((Math.cos(phi1) * Math.cos(-fmax(lambda1, lambda2)))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmin(lambda1, lambda2) <= -7.4e-8:
		tmp = math.acos((math.cos(fmin(lambda1, lambda2)) * math.cos(phi1))) * R
	else:
		tmp = math.acos((math.cos(phi1) * math.cos(-fmax(lambda1, lambda2)))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(lambda1, lambda2) <= -7.4e-8)
		tmp = Float64(acos(Float64(cos(fmin(lambda1, lambda2)) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi1) * cos(Float64(-fmax(lambda1, lambda2))))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (min(lambda1, lambda2) <= -7.4e-8)
		tmp = acos((cos(min(lambda1, lambda2)) * cos(phi1))) * R;
	else
		tmp = acos((cos(phi1) * cos(-max(lambda1, lambda2)))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -7.4e-8], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[(-N[Max[lambda1, lambda2], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -7.4000000000000001e-8

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   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 31.0%

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

   7. Applied rewrites 31.0%

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

   if -7.4000000000000001e-8 < lambda1

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-neg.f64 31.5%

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

   7. Applied rewrites 31.5%

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

Alternative 19: 42.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 42.9%

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

4. Applied rewrites 42.9%

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

Alternative 20: 36.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.32:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(-0.5 \cdot \mathsf{min}\left(\phi_1, \phi_2\right), \mathsf{min}\left(\phi_1, \phi_2\right), 1\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -0.32)
   (* (acos (* (cos lambda1) (cos (fmin phi1 phi2)))) R)
   (*
    (-
     (* PI 0.5)
     (asin
      (*
       (cos (- lambda2 lambda1))
       (fma (* -0.5 (fmin phi1 phi2)) (fmin phi1 phi2) 1.0))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -0.32) {
		tmp = acos((cos(lambda1) * cos(fmin(phi1, phi2)))) * R;
	} else {
		tmp = ((((double) M_PI) * 0.5) - asin((cos((lambda2 - lambda1)) * fma((-0.5 * fmin(phi1, phi2)), fmin(phi1, phi2), 1.0)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.32)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(fmin(phi1, phi2)))) * R);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(Float64(cos(Float64(lambda2 - lambda1)) * fma(Float64(-0.5 * fmin(phi1, phi2)), fmin(phi1, phi2), 1.0)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.32], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.5 * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.32000000000000001

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   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 31.0%

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

   7. Applied rewrites 31.0%

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

   if -0.32000000000000001 < phi1

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   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 18.3%

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

   7. Applied rewrites 18.3%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-asin.f64 18.3%

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

   9. Applied rewrites 18.3%

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

Alternative 21: 27.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1550000:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) + \frac{1}{\mathsf{max}\left(\lambda_1, \lambda_2\right)}\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \mathsf{fma}\left(-0.5 \cdot \mathsf{min}\left(\phi_1, \phi_2\right), \mathsf{min}\left(\phi_1, \phi_2\right), 1\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -1550000.0)
   (*
    (acos
     (*
      (cos (fmin phi1 phi2))
      (*
       (fmax lambda1 lambda2)
       (+ (fmin lambda1 lambda2) (/ 1.0 (fmax lambda1 lambda2))))))
    R)
   (*
    (-
     (* PI 0.5)
     (asin
      (*
       (cos (- (fmax lambda1 lambda2) (fmin lambda1 lambda2)))
       (fma (* -0.5 (fmin phi1 phi2)) (fmin phi1 phi2) 1.0))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -1550000.0) {
		tmp = acos((cos(fmin(phi1, phi2)) * (fmax(lambda1, lambda2) * (fmin(lambda1, lambda2) + (1.0 / fmax(lambda1, lambda2)))))) * R;
	} else {
		tmp = ((((double) M_PI) * 0.5) - asin((cos((fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * fma((-0.5 * fmin(phi1, phi2)), fmin(phi1, phi2), 1.0)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1550000.0)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * Float64(fmax(lambda1, lambda2) * Float64(fmin(lambda1, lambda2) + Float64(1.0 / fmax(lambda1, lambda2)))))) * R);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(Float64(cos(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * fma(Float64(-0.5 * fmin(phi1, phi2)), fmin(phi1, phi2), 1.0)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1550000.0], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[(N[Max[lambda1, lambda2], $MachinePrecision] * N[(N[Min[lambda1, lambda2], $MachinePrecision] + N[(1.0 / N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.5 * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.55e6

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   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 25.0%

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

   7. Applied rewrites 25.0%

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

   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 11.6%

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

   10. Applied rewrites 11.6%

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

   11. Taylor expanded in lambda2 around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 11.7%

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

   13. Applied rewrites 11.7%

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

   if -1.55e6 < phi1

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   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 18.3%

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

   7. Applied rewrites 18.3%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-asin.f64 18.3%

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

   9. Applied rewrites 18.3%

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

Alternative 22: 27.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1550000:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) + \frac{1}{\mathsf{max}\left(\lambda_1, \lambda_2\right)}\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \mathsf{fma}\left(-0.5 \cdot \mathsf{min}\left(\phi_1, \phi_2\right), \mathsf{min}\left(\phi_1, \phi_2\right), 1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -1550000.0)
   (*
    (acos
     (*
      (cos (fmin phi1 phi2))
      (*
       (fmax lambda1 lambda2)
       (+ (fmin lambda1 lambda2) (/ 1.0 (fmax lambda1 lambda2))))))
    R)
   (*
    (acos
     (*
      (cos (- (fmax lambda1 lambda2) (fmin lambda1 lambda2)))
      (fma (* -0.5 (fmin phi1 phi2)) (fmin phi1 phi2) 1.0)))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -1550000.0) {
		tmp = acos((cos(fmin(phi1, phi2)) * (fmax(lambda1, lambda2) * (fmin(lambda1, lambda2) + (1.0 / fmax(lambda1, lambda2)))))) * R;
	} else {
		tmp = acos((cos((fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * fma((-0.5 * fmin(phi1, phi2)), fmin(phi1, phi2), 1.0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1550000.0)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * Float64(fmax(lambda1, lambda2) * Float64(fmin(lambda1, lambda2) + Float64(1.0 / fmax(lambda1, lambda2)))))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * fma(Float64(-0.5 * fmin(phi1, phi2)), fmin(phi1, phi2), 1.0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1550000.0], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[(N[Max[lambda1, lambda2], $MachinePrecision] * N[(N[Min[lambda1, lambda2], $MachinePrecision] + N[(1.0 / N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]), $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[(-0.5 * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.55e6

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   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 25.0%

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

   7. Applied rewrites 25.0%

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

   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 11.6%

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

   10. Applied rewrites 11.6%

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

   11. Taylor expanded in lambda2 around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 11.7%

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

   13. Applied rewrites 11.7%

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

   if -1.55e6 < phi1

   1. Initial program 73.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   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 18.3%

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

   7. Applied rewrites 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 18.3%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 18.3%

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

   9. Applied rewrites 18.3%

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

Alternative 23: 11.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos((cos(phi1) * (fmin(lambda1, lambda2) * (fmax(lambda1, lambda2) + (1.0d0 / fmin(lambda1, lambda2)))))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos((Math.cos(phi1) * (fmin(lambda1, lambda2) * (fmax(lambda1, lambda2) + (1.0 / fmin(lambda1, lambda2)))))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos((math.cos(phi1) * (fmin(lambda1, lambda2) * (fmax(lambda1, lambda2) + (1.0 / fmin(lambda1, lambda2)))))) * R

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos((cos(phi1) * (min(lambda1, lambda2) * (max(lambda1, lambda2) + (1.0 / min(lambda1, lambda2)))))) * R;
end

Wolfram

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

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 42.9%

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

4. Applied rewrites 42.9%

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

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 25.0%

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

7. Applied rewrites 25.0%

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

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 11.6%

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

10. Applied rewrites 11.6%

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

11. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-/.f64 11.7%

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

13. Applied rewrites 11.7%

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

Alternative 24: 11.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 42.9%

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

4. Applied rewrites 42.9%

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

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 25.0%

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

7. Applied rewrites 25.0%

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

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 11.6%

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

10. Applied rewrites 11.6%

    \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \left(1 + \lambda_1 \cdot \color{blue}{\lambda_2}\right)\right) \cdot R \]
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 11.6%

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

12. Applied rewrites 11.6%

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

Alternative 25: 1.5% accurate, 4.4× speedup

Math

\[\cos^{-1} \left(\left(1 + -0.5 \cdot {\left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)}^{2}\right) \cdot \left(1 + \lambda_1 \cdot \lambda_2\right)\right) \cdot R \]

FPCore

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(((1.0 + (-0.5 * math.pow(fmin(phi1, phi2), 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 * (fmin(phi1, phi2) ^ 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 * (min(phi1, phi2) ^ 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[N[Min[phi1, phi2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(lambda1 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 42.9%

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

4. Applied rewrites 42.9%

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

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 25.0%

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

7. Applied rewrites 25.0%

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

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 11.6%

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

10. Applied rewrites 11.6%

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

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 1.6%

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

13. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2025187 
(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))