Spherical law of cosines

Percentage Accurate: 74.2% → 94.4%

Time: 37.1s

Alternatives: 22

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

   1. lift-var N/A

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

   2. lift-var N/A

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

   3. lift--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lift-var N/A

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

   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. lift-var N/A

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

   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. lift-var N/A

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

   13. lower-cos.f64 N/A

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

   14. lift-var 93.7

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

4. Applied rewrites 93.7%

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

6. Applied rewrites 93.7%

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

Alternative 2: 94.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

   1. lift-var N/A

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

   2. lift-var N/A

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

   3. lift--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lift-var N/A

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

   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. lift-var N/A

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

   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. lift-var N/A

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

   13. lower-cos.f64 N/A

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

   14. lift-var 93.7

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

4. Applied rewrites 93.7%

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

Alternative 3: 94.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

   1. lift-var N/A

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

   2. lift-var N/A

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

   3. lift--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lift-var N/A

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

   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. lift-var N/A

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

   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. lift-var N/A

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

   13. lower-cos.f64 N/A

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

   14. lift-var 93.7

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

4. Applied rewrites 93.7%

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

6. Applied rewrites 93.7%

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

7. Taylor expanded in lambda1 around inf

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

9. Applied rewrites 93.7%

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

Alternative 4: 94.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

   1. lift-var N/A

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

   2. lift-var N/A

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

   3. lift--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lift-var N/A

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

   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. lift-var N/A

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

   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. lift-var N/A

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

   13. lower-cos.f64 N/A

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

   14. lift-var 93.7

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

4. Applied rewrites 93.7%

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

5. Taylor expanded in lambda1 around inf

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

7. Applied rewrites 93.7%

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

Alternative 5: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 3.3 \cdot 10^{-11}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -3.6e-6) (not (<= phi2 3.3e-11)))
   (*
    (acos
     (fma
      (sin phi2)
      (sin phi1)
      (* (* (cos (- lambda2 lambda1)) (cos phi1)) (cos phi2))))
    R)
   (*
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (*
       (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))
       (cos phi1))))
    R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -3.6e-6) || !(phi2 <= 3.3e-11)) {
		tmp = acos(fma(sin(phi2), sin(phi1), ((cos((lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R;
	} else {
		tmp = acos(((sin(phi1) * sin(phi2)) + (fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))) * cos(phi1)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -3.6e-6) || !(phi2 <= 3.3e-11))
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R);
	else
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))) * cos(phi1)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -3.6e-6], N[Not[LessEqual[phi2, 3.3e-11]], $MachinePrecision]], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.59999999999999984e-6 or 3.3000000000000002e-11 < phi2

   1. Initial program 81.0%

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

   4. Applied rewrites 81.1%

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

   if -3.59999999999999984e-6 < phi2 < 3.3000000000000002e-11

   1. Initial program 66.3%

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

      1. lift-var N/A

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

      2. lift-var N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-var N/A

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

      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. lift-var N/A

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

      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. lift-var N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-var 88.8

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in phi1 around 0

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 88.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 3.3 \cdot 10^{-11}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 3.3 \cdot 10^{-11}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -3.6e-6) (not (<= phi2 3.3e-11)))
   (*
    (acos
     (fma
      (sin phi2)
      (sin phi1)
      (* (* (cos (- lambda2 lambda1)) (cos phi1)) (cos phi2))))
    R)
   (*
    (acos
     (fma
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))
      (cos phi1)
      (* (sin phi1) phi2)))
    R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -3.6e-6) || !(phi2 <= 3.3e-11)) {
		tmp = acos(fma(sin(phi2), sin(phi1), ((cos((lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R;
	} else {
		tmp = acos(fma(fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))), cos(phi1), (sin(phi1) * phi2))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -3.6e-6) || !(phi2 <= 3.3e-11))
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R);
	else
		tmp = Float64(acos(fma(fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))), cos(phi1), Float64(sin(phi1) * phi2))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -3.6e-6], N[Not[LessEqual[phi2, 3.3e-11]], $MachinePrecision]], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.59999999999999984e-6 or 3.3000000000000002e-11 < phi2

   1. Initial program 81.0%

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

   4. Applied rewrites 81.1%

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

   if -3.59999999999999984e-6 < phi2 < 3.3000000000000002e-11

   1. Initial program 66.3%

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

      1. lift-var N/A

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

      2. lift-var N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-var N/A

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

      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. lift-var N/A

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

      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. lift-var N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-var 88.8

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

   4. Applied rewrites 88.8%

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in phi2 around 0

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

   9. Applied rewrites 88.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 3.3 \cdot 10^{-11}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 3.3 \cdot 10^{-11}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -3.6e-6) (not (<= phi2 3.3e-11)))
   (*
    (acos
     (fma
      (sin phi2)
      (sin phi1)
      (* (* (cos (- lambda2 lambda1)) (cos phi1)) (cos phi2))))
    R)
   (*
    (acos
     (fma
      (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))
      (cos phi1)
      (* (sin phi1) phi2)))
    R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -3.6e-6) || !(phi2 <= 3.3e-11)) {
		tmp = acos(fma(sin(phi2), sin(phi1), ((cos((lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R;
	} else {
		tmp = acos(fma(fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))), cos(phi1), (sin(phi1) * phi2))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -3.6e-6) || !(phi2 <= 3.3e-11))
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R);
	else
		tmp = Float64(acos(fma(fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))), cos(phi1), Float64(sin(phi1) * phi2))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -3.6e-6], N[Not[LessEqual[phi2, 3.3e-11]], $MachinePrecision]], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.59999999999999984e-6 or 3.3000000000000002e-11 < phi2

   1. Initial program 81.0%

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

   4. Applied rewrites 81.1%

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

   if -3.59999999999999984e-6 < phi2 < 3.3000000000000002e-11

   1. Initial program 66.3%

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

      1. lift-var N/A

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

      2. lift-var N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-var N/A

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

      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. lift-var N/A

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

      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. lift-var N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-var 88.8

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 88.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 3.3 \cdot 10^{-11}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -2.2 \cdot 10^{+133}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq -0.00236 \lor \neg \left(\lambda_2 \leq 2.8 \cdot 10^{-19}\right):\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))))
   (if (<= lambda2 -2.2e+133)
     (* (acos (* (cos phi1) t_0)) R)
     (if (or (<= lambda2 -0.00236) (not (<= lambda2 2.8e-19)))
       (* (acos (* t_0 (cos phi2))) R)
       (*
        (acos
         (fma
          (* (cos lambda1) (cos phi2))
          (cos phi1)
          (* (sin phi2) (sin phi1))))
        R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)));
	double tmp;
	if (lambda2 <= -2.2e+133) {
		tmp = acos((cos(phi1) * t_0)) * R;
	} else if ((lambda2 <= -0.00236) || !(lambda2 <= 2.8e-19)) {
		tmp = acos((t_0 * cos(phi2))) * R;
	} else {
		tmp = acos(fma((cos(lambda1) * cos(phi2)), cos(phi1), (sin(phi2) * sin(phi1)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))
	tmp = 0.0
	if (lambda2 <= -2.2e+133)
		tmp = Float64(acos(Float64(cos(phi1) * t_0)) * R);
	elseif ((lambda2 <= -0.00236) || !(lambda2 <= 2.8e-19))
		tmp = Float64(acos(Float64(t_0 * cos(phi2))) * R);
	else
		tmp = Float64(acos(fma(Float64(cos(lambda1) * cos(phi2)), cos(phi1), Float64(sin(phi2) * sin(phi1)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -2.2e+133], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[Or[LessEqual[lambda2, -0.00236], N[Not[LessEqual[lambda2, 2.8e-19]], $MachinePrecision]], N[(N[ArcCos[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -2.2e133

   1. Initial program 53.2%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 37.2%

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

   7. Applied rewrites 66.8%

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

   if -2.2e133 < lambda2 < -0.0023600000000000001 or 2.80000000000000003e-19 < lambda2

   1. Initial program 57.1%

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

      1. lift-var N/A

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

      2. lift-var N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-var N/A

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

      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. lift-var N/A

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

      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. lift-var N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-var 99.0

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in phi1 around 0

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

   9. Applied rewrites 57.8%

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

   if -0.0023600000000000001 < lambda2 < 2.80000000000000003e-19

   1. Initial program 88.6%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 88.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.2 \cdot 10^{+133}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq -0.00236 \lor \neg \left(\lambda_2 \leq 2.8 \cdot 10^{-19}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.2 \cdot 10^{+133}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq -0.00236 \lor \neg \left(\lambda_2 \leq 2.8 \cdot 10^{-19}\right):\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -2.2e+133)
   (*
    (acos
     (*
      (cos phi1)
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))))
    R)
   (if (or (<= lambda2 -0.00236) (not (<= lambda2 2.8e-19)))
     (*
      (acos
       (*
        (cos phi2)
        (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
      R)
     (*
      (acos
       (fma (* (cos lambda1) (cos phi2)) (cos phi1) (* (sin phi2) (sin phi1))))
      R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -2.2e+133) {
		tmp = acos((cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))) * R;
	} else if ((lambda2 <= -0.00236) || !(lambda2 <= 2.8e-19)) {
		tmp = acos((cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = acos(fma((cos(lambda1) * cos(phi2)), cos(phi1), (sin(phi2) * sin(phi1)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -2.2e+133)
		tmp = Float64(acos(Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))) * R);
	elseif ((lambda2 <= -0.00236) || !(lambda2 <= 2.8e-19))
		tmp = Float64(acos(Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(acos(fma(Float64(cos(lambda1) * cos(phi2)), cos(phi1), Float64(sin(phi2) * sin(phi1)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -2.2e+133], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[Or[LessEqual[lambda2, -0.00236], N[Not[LessEqual[lambda2, 2.8e-19]], $MachinePrecision]], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -2.2e133

   1. Initial program 53.2%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 37.2%

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

   7. Applied rewrites 66.8%

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

   if -2.2e133 < lambda2 < -0.0023600000000000001 or 2.80000000000000003e-19 < lambda2

   1. Initial program 57.1%

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

      1. lift-var N/A

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

      2. lift-var N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-var N/A

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

      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. lift-var N/A

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

      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. lift-var N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-var 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in phi1 around 0

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

   7. Applied rewrites 57.7%

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

   if -0.0023600000000000001 < lambda2 < 2.80000000000000003e-19

   1. Initial program 88.6%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 88.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.2 \cdot 10^{+133}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq -0.00236 \lor \neg \left(\lambda_2 \leq 2.8 \cdot 10^{-19}\right):\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3 \cdot 10^{-8} \lor \neg \left(\phi_2 \leq 2.65 \cdot 10^{-11}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -3e-8) (not (<= phi2 2.65e-11)))
   (*
    (acos
     (fma
      (sin phi2)
      (sin phi1)
      (* (* (cos (- lambda2 lambda1)) (cos phi1)) (cos phi2))))
    R)
   (*
    (acos
     (*
      (cos phi1)
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))))
    R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -3e-8) || !(phi2 <= 2.65e-11)) {
		tmp = acos(fma(sin(phi2), sin(phi1), ((cos((lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R;
	} else {
		tmp = acos((cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -3e-8) || !(phi2 <= 2.65e-11))
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -3e-8], N[Not[LessEqual[phi2, 2.65e-11]], $MachinePrecision]], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.99999999999999973e-8 or 2.6499999999999999e-11 < phi2

   1. Initial program 81.0%

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

   4. Applied rewrites 81.1%

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

   if -2.99999999999999973e-8 < phi2 < 2.6499999999999999e-11

   1. Initial program 66.3%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 66.2%

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

   7. Applied rewrites 88.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3 \cdot 10^{-8} \lor \neg \left(\phi_2 \leq 2.65 \cdot 10^{-11}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.9 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 1.06 \cdot 10^{-8}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -2.9e-6) (not (<= phi2 1.06e-8)))
   (*
    (acos
     (fma (* (cos lambda1) (cos phi2)) (cos phi1) (* (sin phi2) (sin phi1))))
    R)
   (*
    (acos
     (*
      (cos phi1)
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))))
    R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -2.9e-6) || !(phi2 <= 1.06e-8)) {
		tmp = acos(fma((cos(lambda1) * cos(phi2)), cos(phi1), (sin(phi2) * sin(phi1)))) * R;
	} else {
		tmp = acos((cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -2.9e-6) || !(phi2 <= 1.06e-8))
		tmp = Float64(acos(fma(Float64(cos(lambda1) * cos(phi2)), cos(phi1), Float64(sin(phi2) * sin(phi1)))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -2.9e-6], N[Not[LessEqual[phi2, 1.06e-8]], $MachinePrecision]], N[(N[ArcCos[N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.9000000000000002e-6 or 1.06000000000000006e-8 < phi2

   1. Initial program 80.9%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 65.3%

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

   if -2.9000000000000002e-6 < phi2 < 1.06000000000000006e-8

   1. Initial program 66.5%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 88.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.9 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 1.06 \cdot 10^{-8}\right):\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_2 \leq -2.15 \cdot 10^{-12}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \phi_2, \cos \phi_1, t\_0\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \lambda_2, \cos \phi_2, t\_0\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi2) (sin phi1))))
   (if (<= lambda2 -2.15e-12)
     (*
      (acos
       (*
        (cos phi1)
        (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))))
      R)
     (if (<= lambda2 3.5e-6)
       (* (acos (fma (* (cos lambda1) (cos phi2)) (cos phi1) t_0)) R)
       (* (acos (fma (* (cos phi1) (cos lambda2)) (cos phi2) t_0)) R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi2) * sin(phi1);
	double tmp;
	if (lambda2 <= -2.15e-12) {
		tmp = acos((cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))) * R;
	} else if (lambda2 <= 3.5e-6) {
		tmp = acos(fma((cos(lambda1) * cos(phi2)), cos(phi1), t_0)) * R;
	} else {
		tmp = acos(fma((cos(phi1) * cos(lambda2)), cos(phi2), t_0)) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi2) * sin(phi1))
	tmp = 0.0
	if (lambda2 <= -2.15e-12)
		tmp = Float64(acos(Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))) * R);
	elseif (lambda2 <= 3.5e-6)
		tmp = Float64(acos(fma(Float64(cos(lambda1) * cos(phi2)), cos(phi1), t_0)) * R);
	else
		tmp = Float64(acos(fma(Float64(cos(phi1) * cos(lambda2)), cos(phi2), t_0)) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -2.15e-12], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 3.5e-6], N[(N[ArcCos[N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -2.14999999999999993e-12

   1. Initial program 54.6%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 37.3%

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

   7. Applied rewrites 64.0%

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

   if -2.14999999999999993e-12 < lambda2 < 3.49999999999999995e-6

   1. Initial program 89.0%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 89.0%

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

   if 3.49999999999999995e-6 < lambda2

   1. Initial program 58.2%

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

   3. Taylor expanded in lambda1 around 0

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

   5. Applied rewrites 58.3%

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

Alternative 13: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0009:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.0009)
   (*
    (acos
     (*
      (cos phi1)
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))))
    R)
   (* (acos (* (cos phi2) (cos (- lambda2 lambda1)))) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0009) {
		tmp = acos((cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))) * R;
	} else {
		tmp = acos((cos(phi2) * cos((lambda2 - lambda1)))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.0009)
		tmp = Float64(acos(Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.0009], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.9999999999999998e-4

   1. Initial program 70.6%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 65.9%

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

   if 8.9999999999999998e-4 < phi2

   1. Initial program 82.6%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 41.9%

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

Alternative 14: 58.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 0.00115:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot t\_0\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 0.00115)
     (* (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) t_0))) R)
     (* (acos (* (cos phi2) t_0)) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 0.00115) {
		tmp = acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0))) * R;
	} else {
		tmp = acos((cos(phi2) * t_0)) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 0.00115d0) then
        tmp = acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0))) * r
    else
        tmp = acos((cos(phi2) * t_0)) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 0.00115) {
		tmp = Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * t_0))) * R;
	} else {
		tmp = Math.acos((Math.cos(phi2) * t_0)) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 0.00115:
		tmp = math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * t_0))) * R
	else:
		tmp = math.acos((math.cos(phi2) * t_0)) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 0.00115)
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * t_0))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi2) * t_0)) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 0.00115)
		tmp = acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0))) * R;
	else
		tmp = acos((cos(phi2) * t_0)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 0.00115], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.00115

   1. Initial program 70.6%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 50.8%

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

   if 0.00115 < phi2

   1. Initial program 82.6%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 41.9%

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

Alternative 15: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.9 \cdot 10^{-8} \lor \neg \left(\lambda_1 \leq 1.65 \cdot 10^{-10}\right):\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda1 -4.9e-8) (not (<= lambda1 1.65e-10)))
   (* (acos (* (cos phi1) (cos lambda1))) R)
   (* (acos (* (cos phi1) (cos lambda2))) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 <= -4.9e-8) || !(lambda1 <= 1.65e-10)) {
		tmp = acos((cos(phi1) * cos(lambda1))) * R;
	} else {
		tmp = acos((cos(phi1) * cos(lambda2))) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda1 <= (-4.9d-8)) .or. (.not. (lambda1 <= 1.65d-10))) then
        tmp = acos((cos(phi1) * cos(lambda1))) * r
    else
        tmp = acos((cos(phi1) * cos(lambda2))) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 <= -4.9e-8) || !(lambda1 <= 1.65e-10)) {
		tmp = Math.acos((Math.cos(phi1) * Math.cos(lambda1))) * R;
	} else {
		tmp = Math.acos((Math.cos(phi1) * Math.cos(lambda2))) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 <= -4.9e-8) or not (lambda1 <= 1.65e-10):
		tmp = math.acos((math.cos(phi1) * math.cos(lambda1))) * R
	else:
		tmp = math.acos((math.cos(phi1) * math.cos(lambda2))) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda1 <= -4.9e-8) || !(lambda1 <= 1.65e-10))
		tmp = Float64(acos(Float64(cos(phi1) * cos(lambda1))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi1) * cos(lambda2))) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 <= -4.9e-8) || ~((lambda1 <= 1.65e-10)))
		tmp = acos((cos(phi1) * cos(lambda1))) * R;
	else
		tmp = acos((cos(phi1) * cos(lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda1, -4.9e-8], N[Not[LessEqual[lambda1, 1.65e-10]], $MachinePrecision]], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.9000000000000002e-8 or 1.65e-10 < lambda1

   1. Initial program 63.0%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 38.9%

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

   if -4.9000000000000002e-8 < lambda1 < 1.65e-10

   1. Initial program 86.8%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 48.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.9 \cdot 10^{-8} \lor \neg \left(\lambda_1 \leq 1.65 \cdot 10^{-10}\right):\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 0.00066:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot t\_0\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 0.00066)
     (* (acos (* (cos phi1) t_0)) R)
     (* (acos (* (cos phi2) t_0)) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 0.00066) {
		tmp = acos((cos(phi1) * t_0)) * R;
	} else {
		tmp = acos((cos(phi2) * t_0)) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 0.00066d0) then
        tmp = acos((cos(phi1) * t_0)) * r
    else
        tmp = acos((cos(phi2) * t_0)) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 0.00066) {
		tmp = Math.acos((Math.cos(phi1) * t_0)) * R;
	} else {
		tmp = Math.acos((Math.cos(phi2) * t_0)) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 0.00066:
		tmp = math.acos((math.cos(phi1) * t_0)) * R
	else:
		tmp = math.acos((math.cos(phi2) * t_0)) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 0.00066)
		tmp = Float64(acos(Float64(cos(phi1) * t_0)) * R);
	else
		tmp = Float64(acos(Float64(cos(phi2) * t_0)) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 0.00066)
		tmp = acos((cos(phi1) * t_0)) * R;
	else
		tmp = acos((cos(phi2) * t_0)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 0.00066], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.6e-4

   1. Initial program 70.6%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 50.9%

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

   if 6.6e-4 < phi2

   1. Initial program 82.6%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 41.9%

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

Alternative 17: 27.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.999:\\ \;\;\;\;\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos (- lambda1 lambda2)) 0.999)
   (* (acos (cos (- lambda2 lambda1))) R)
   (* (acos (* (sin phi1) phi2)) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos((lambda1 - lambda2)) <= 0.999) {
		tmp = acos(cos((lambda2 - lambda1))) * R;
	} else {
		tmp = acos((sin(phi1) * phi2)) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (cos((lambda1 - lambda2)) <= 0.999d0) then
        tmp = acos(cos((lambda2 - lambda1))) * r
    else
        tmp = acos((sin(phi1) * phi2)) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.cos((lambda1 - lambda2)) <= 0.999) {
		tmp = Math.acos(Math.cos((lambda2 - lambda1))) * R;
	} else {
		tmp = Math.acos((Math.sin(phi1) * phi2)) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.cos((lambda1 - lambda2)) <= 0.999:
		tmp = math.acos(math.cos((lambda2 - lambda1))) * R
	else:
		tmp = math.acos((math.sin(phi1) * phi2)) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(Float64(lambda1 - lambda2)) <= 0.999)
		tmp = Float64(acos(cos(Float64(lambda2 - lambda1))) * R);
	else
		tmp = Float64(acos(Float64(sin(phi1) * phi2)) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (cos((lambda1 - lambda2)) <= 0.999)
		tmp = acos(cos((lambda2 - lambda1))) * R;
	else
		tmp = acos((sin(phi1) * phi2)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision], 0.999], N[(N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.998999999999999999

   1. Initial program 72.8%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 35.0%

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

   if 0.998999999999999999 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 75.7%

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

      1. lift-var N/A

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

      2. lift-var N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-var N/A

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

      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. lift-var N/A

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

      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. lift-var N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-var 78.1

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

   4. Applied rewrites 78.1%

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

   6. Applied rewrites 78.1%

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

   7. Taylor expanded in phi2 around 0

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

   9. Applied rewrites 35.8%

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

   10. Taylor expanded in phi2 around inf

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

   12. Applied rewrites 12.0%

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

Alternative 18: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5.1e-8)
   (* (acos (* (cos phi1) (cos lambda1))) R)
   (* (acos (cos (- lambda2 lambda1))) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.1e-8) {
		tmp = acos((cos(phi1) * cos(lambda1))) * R;
	} else {
		tmp = acos(cos((lambda2 - lambda1))) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (phi1 <= (-5.1d-8)) then
        tmp = acos((cos(phi1) * cos(lambda1))) * r
    else
        tmp = acos(cos((lambda2 - lambda1))) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.1e-8) {
		tmp = Math.acos((Math.cos(phi1) * Math.cos(lambda1))) * R;
	} else {
		tmp = Math.acos(Math.cos((lambda2 - lambda1))) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -5.1e-8:
		tmp = math.acos((math.cos(phi1) * math.cos(lambda1))) * R
	else:
		tmp = math.acos(math.cos((lambda2 - lambda1))) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -5.1e-8)
		tmp = Float64(acos(Float64(cos(phi1) * cos(lambda1))) * R);
	else
		tmp = Float64(acos(cos(Float64(lambda2 - lambda1))) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -5.1e-8)
		tmp = acos((cos(phi1) * cos(lambda1))) * R;
	else
		tmp = acos(cos((lambda2 - lambda1))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5.1e-8], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.10000000000000001e-8

   1. Initial program 78.9%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 39.8%

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

   if -5.10000000000000001e-8 < phi1

   1. Initial program 71.2%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 31.6%

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

Alternative 19: 42.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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((lambda2 - lambda1)))) * r
end function

Java

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

Python

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

Julia

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

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos((cos(phi1) * cos((lambda2 - lambda1)))) * R;
end

Wolfram

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

Derivation

1. Initial program 73.6%

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

3. Taylor expanded in phi2 around 0

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

5. Applied rewrites 43.0%

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

Alternative 20: 26.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.4 \cdot 10^{-8} \lor \neg \left(\lambda_1 \leq 1.65 \cdot 10^{-10}\right):\\ \;\;\;\;\cos^{-1} \cos \lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \lambda_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda1 -4.4e-8) (not (<= lambda1 1.65e-10)))
   (* (acos (cos lambda1)) R)
   (* (acos (cos lambda2)) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 <= -4.4e-8) || !(lambda1 <= 1.65e-10)) {
		tmp = acos(cos(lambda1)) * R;
	} else {
		tmp = acos(cos(lambda2)) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda1 <= (-4.4d-8)) .or. (.not. (lambda1 <= 1.65d-10))) then
        tmp = acos(cos(lambda1)) * r
    else
        tmp = acos(cos(lambda2)) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 <= -4.4e-8) || !(lambda1 <= 1.65e-10)) {
		tmp = Math.acos(Math.cos(lambda1)) * R;
	} else {
		tmp = Math.acos(Math.cos(lambda2)) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 <= -4.4e-8) or not (lambda1 <= 1.65e-10):
		tmp = math.acos(math.cos(lambda1)) * R
	else:
		tmp = math.acos(math.cos(lambda2)) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda1 <= -4.4e-8) || !(lambda1 <= 1.65e-10))
		tmp = Float64(acos(cos(lambda1)) * R);
	else
		tmp = Float64(acos(cos(lambda2)) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 <= -4.4e-8) || ~((lambda1 <= 1.65e-10)))
		tmp = acos(cos(lambda1)) * R;
	else
		tmp = acos(cos(lambda2)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda1, -4.4e-8], N[Not[LessEqual[lambda1, 1.65e-10]], $MachinePrecision]], N[(N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.3999999999999997e-8 or 1.65e-10 < lambda1

   1. Initial program 63.0%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 30.8%

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

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 30.9%

       \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]

   if -4.3999999999999997e-8 < lambda1 < 1.65e-10

   1. Initial program 86.8%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 22.1%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \cos \lambda_2 \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 22.1%

       \[\leadsto \cos^{-1} \cos \lambda_2 \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.4 \cdot 10^{-8} \lor \neg \left(\lambda_1 \leq 1.65 \cdot 10^{-10}\right):\\ \;\;\;\;\cos^{-1} \cos \lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(cos((lambda2 - lambda1))) * R;
end

Wolfram

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

Derivation

1. Initial program 73.6%

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

3. Taylor expanded in phi2 around 0

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

5. Applied rewrites 43.0%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 27.0%

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

Alternative 22: 17.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \cos^{-1} \cos \lambda_1 \cdot R \end{array} \]

FPCore

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

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(math.cos(lambda1)) * R

Julia

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

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(cos(lambda1)) * R;
end

Wolfram

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

Derivation

1. Initial program 73.6%

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

3. Taylor expanded in phi2 around 0

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

5. Applied rewrites 43.0%

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

6. Taylor expanded in phi1 around 0

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

8. Applied rewrites 27.0%

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

9. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot R \]
10. Step-by-step derivation

11. Applied rewrites 19.0%

    \[\leadsto \cos^{-1} \cos \lambda_1 \cdot R \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025021 
(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))