Spherical law of cosines

Percentage Accurate: 74.3% → 94.2%

Time: 16.1s

Alternatives: 29

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 94.2%

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

3. Applied rewrites 94.2%

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

5. Applied rewrites 94.2%

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

Alternative 2: 94.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 94.2%

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

3. Applied rewrites 94.2%

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

4. Taylor expanded in lambda1 around inf

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

   1. lower-fma.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-sin.f64 94.2%

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

6. Applied rewrites 94.2%

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

Alternative 3: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \mathsf{fma}\left(0.5 \cdot \pi, R, \sin^{-1} \left(-\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_0, t\_2, \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot t\_1\right)\right) \cdot R\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq -0.062:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 3.3 \cdot 10^{+30}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_0, t\_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \mathsf{max}\left(\phi_1, \phi_2\right) \cdot t\_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (sin (fmin phi1 phi2)))
        (t_2 (cos (fmax phi1 phi2)))
        (t_3
         (fma
          (* 0.5 PI)
          R
          (*
           (asin
            (-
             (fma
              (* (cos (- lambda2 lambda1)) t_0)
              t_2
              (* (sin (fmax phi1 phi2)) t_1))))
           R))))
   (if (<= (fmax phi1 phi2) -0.062)
     t_3
     (if (<= (fmax phi1 phi2) 3.3e+30)
       (*
        (acos
         (fma
          t_0
          (*
           t_2
           (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))
          (* (fmax phi1 phi2) t_1)))
        R)
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2));
	double t_1 = sin(fmin(phi1, phi2));
	double t_2 = cos(fmax(phi1, phi2));
	double t_3 = fma((0.5 * ((double) M_PI)), R, (asin(-fma((cos((lambda2 - lambda1)) * t_0), t_2, (sin(fmax(phi1, phi2)) * t_1))) * R));
	double tmp;
	if (fmax(phi1, phi2) <= -0.062) {
		tmp = t_3;
	} else if (fmax(phi1, phi2) <= 3.3e+30) {
		tmp = acos(fma(t_0, (t_2 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))), (fmax(phi1, phi2) * t_1))) * R;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	t_1 = sin(fmin(phi1, phi2))
	t_2 = cos(fmax(phi1, phi2))
	t_3 = fma(Float64(0.5 * pi), R, Float64(asin(Float64(-fma(Float64(cos(Float64(lambda2 - lambda1)) * t_0), t_2, Float64(sin(fmax(phi1, phi2)) * t_1)))) * R))
	tmp = 0.0
	if (fmax(phi1, phi2) <= -0.062)
		tmp = t_3;
	elseif (fmax(phi1, phi2) <= 3.3e+30)
		tmp = Float64(acos(fma(t_0, Float64(t_2 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))), Float64(fmax(phi1, phi2) * t_1))) * R);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * Pi), $MachinePrecision] * R + N[(N[ArcSin[(-N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2 + N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], -0.062], t$95$3, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 3.3e+30], N[(N[ArcCos[N[(t$95$0 * N[(t$95$2 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Max[phi1, phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.062 or 3.3000000000000003e30 < phi2

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   5. Applied rewrites 73.8%

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

   if -0.062 < phi2 < 3.3000000000000003e30

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.2%

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

   6. Applied rewrites 94.2%

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

   7. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 56.6%

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

   9. Applied rewrites 56.6%

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

Alternative 4: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_0, t\_1, t\_2 \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.0037:\\ \;\;\;\;\left(1.5707963267948966 - \sin^{-1} t\_3\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 3.45 \cdot 10^{-12}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_0, t\_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \mathsf{min}\left(\phi_1, \phi_2\right) \cdot t\_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \pi, R, \sin^{-1} \left(-t\_3\right) \cdot R\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (cos (fmax phi1 phi2)))
        (t_2 (sin (fmax phi1 phi2)))
        (t_3
         (fma
          (* (cos (- lambda2 lambda1)) t_0)
          t_1
          (* t_2 (sin (fmin phi1 phi2))))))
   (if (<= (fmin phi1 phi2) -0.0037)
     (* (- 1.5707963267948966 (asin t_3)) R)
     (if (<= (fmin phi1 phi2) 3.45e-12)
       (*
        (acos
         (fma
          t_0
          (*
           t_1
           (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))
          (* (fmin phi1 phi2) t_2)))
        R)
       (fma (* 0.5 PI) R (* (asin (- t_3)) R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2));
	double t_1 = cos(fmax(phi1, phi2));
	double t_2 = sin(fmax(phi1, phi2));
	double t_3 = fma((cos((lambda2 - lambda1)) * t_0), t_1, (t_2 * sin(fmin(phi1, phi2))));
	double tmp;
	if (fmin(phi1, phi2) <= -0.0037) {
		tmp = (1.5707963267948966 - asin(t_3)) * R;
	} else if (fmin(phi1, phi2) <= 3.45e-12) {
		tmp = acos(fma(t_0, (t_1 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))), (fmin(phi1, phi2) * t_2))) * R;
	} else {
		tmp = fma((0.5 * ((double) M_PI)), R, (asin(-t_3) * R));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	t_1 = cos(fmax(phi1, phi2))
	t_2 = sin(fmax(phi1, phi2))
	t_3 = fma(Float64(cos(Float64(lambda2 - lambda1)) * t_0), t_1, Float64(t_2 * sin(fmin(phi1, phi2))))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.0037)
		tmp = Float64(Float64(1.5707963267948966 - asin(t_3)) * R);
	elseif (fmin(phi1, phi2) <= 3.45e-12)
		tmp = Float64(acos(fma(t_0, Float64(t_1 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))), Float64(fmin(phi1, phi2) * t_2))) * R);
	else
		tmp = fma(Float64(0.5 * pi), R, Float64(asin(Float64(-t_3)) * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1 + N[(t$95$2 * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.0037], N[(N[(1.5707963267948966 - N[ArcSin[t$95$3], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 3.45e-12], N[(N[ArcCos[N[(t$95$0 * N[(t$95$1 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Min[phi1, phi2], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] * R + N[(N[ArcSin[(-t$95$3)], $MachinePrecision] * R), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.0037000000000000002

   1. Initial program 74.3%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-asin.f64 74.2%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Evaluated real constant 74.2%

      \[\leadsto \left(\color{blue}{1.5707963267948966} - \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1, \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right) \cdot R \]

   if -0.0037000000000000002 < phi1 < 3.45e-12

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.2%

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

   6. Applied rewrites 94.2%

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

   7. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 56.2%

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

   9. Applied rewrites 56.2%

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

   if 3.45e-12 < phi1

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   5. Applied rewrites 73.8%

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

Alternative 5: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right), t\_0, t\_1 \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.00375:\\ \;\;\;\;\left(1.5707963267948966 - \sin^{-1} t\_2\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 3.45 \cdot 10^{-12}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right), t\_1, t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \pi, R, \sin^{-1} \left(-t\_2\right) \cdot R\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (sin (fmax phi1 phi2)))
        (t_2
         (fma
          (* (cos (- lambda2 lambda1)) (cos (fmin phi1 phi2)))
          t_0
          (* t_1 (sin (fmin phi1 phi2))))))
   (if (<= (fmin phi1 phi2) -0.00375)
     (* (- 1.5707963267948966 (asin t_2)) R)
     (if (<= (fmin phi1 phi2) 3.45e-12)
       (*
        (acos
         (fma
          (fmin phi1 phi2)
          t_1
          (*
           t_0
           (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
        R)
       (fma (* 0.5 PI) R (* (asin (- t_2)) R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = sin(fmax(phi1, phi2));
	double t_2 = fma((cos((lambda2 - lambda1)) * cos(fmin(phi1, phi2))), t_0, (t_1 * sin(fmin(phi1, phi2))));
	double tmp;
	if (fmin(phi1, phi2) <= -0.00375) {
		tmp = (1.5707963267948966 - asin(t_2)) * R;
	} else if (fmin(phi1, phi2) <= 3.45e-12) {
		tmp = acos(fma(fmin(phi1, phi2), t_1, (t_0 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))))) * R;
	} else {
		tmp = fma((0.5 * ((double) M_PI)), R, (asin(-t_2) * R));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = sin(fmax(phi1, phi2))
	t_2 = fma(Float64(cos(Float64(lambda2 - lambda1)) * cos(fmin(phi1, phi2))), t_0, Float64(t_1 * sin(fmin(phi1, phi2))))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.00375)
		tmp = Float64(Float64(1.5707963267948966 - asin(t_2)) * R);
	elseif (fmin(phi1, phi2) <= 3.45e-12)
		tmp = Float64(acos(fma(fmin(phi1, phi2), t_1, Float64(t_0 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))))) * R);
	else
		tmp = fma(Float64(0.5 * pi), R, Float64(asin(Float64(-t_2)) * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(t$95$1 * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.00375], N[(N[(1.5707963267948966 - N[ArcSin[t$95$2], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 3.45e-12], N[(N[ArcCos[N[(N[Min[phi1, phi2], $MachinePrecision] * t$95$1 + N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] * R + N[(N[ArcSin[(-t$95$2)], $MachinePrecision] * R), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.0037499999999999999

   1. Initial program 74.3%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-asin.f64 74.2%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Evaluated real constant 74.2%

      \[\leadsto \left(\color{blue}{1.5707963267948966} - \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1, \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right) \cdot R \]

   if -0.0037499999999999999 < phi1 < 3.45e-12

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 46.3%

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

   6. Applied rewrites 46.3%

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

   if 3.45e-12 < phi1

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   5. Applied rewrites 73.8%

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

Alternative 6: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right), t\_0, \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.00375:\\ \;\;\;\;\left(1.5707963267948966 - \sin^{-1} t\_1\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 3.45 \cdot 10^{-12}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \pi, R, \sin^{-1} \left(-t\_1\right) \cdot R\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1
         (fma
          (* (cos (- lambda2 lambda1)) (cos (fmin phi1 phi2)))
          t_0
          (* (sin (fmax phi1 phi2)) (sin (fmin phi1 phi2))))))
   (if (<= (fmin phi1 phi2) -0.00375)
     (* (- 1.5707963267948966 (asin t_1)) R)
     (if (<= (fmin phi1 phi2) 3.45e-12)
       (*
        (acos
         (*
          t_0
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (fma (* 0.5 PI) R (* (asin (- t_1)) R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = fma((cos((lambda2 - lambda1)) * cos(fmin(phi1, phi2))), t_0, (sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))));
	double tmp;
	if (fmin(phi1, phi2) <= -0.00375) {
		tmp = (1.5707963267948966 - asin(t_1)) * R;
	} else if (fmin(phi1, phi2) <= 3.45e-12) {
		tmp = acos((t_0 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = fma((0.5 * ((double) M_PI)), R, (asin(-t_1) * R));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = fma(Float64(cos(Float64(lambda2 - lambda1)) * cos(fmin(phi1, phi2))), t_0, Float64(sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.00375)
		tmp = Float64(Float64(1.5707963267948966 - asin(t_1)) * R);
	elseif (fmin(phi1, phi2) <= 3.45e-12)
		tmp = Float64(acos(Float64(t_0 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = fma(Float64(0.5 * pi), R, Float64(asin(Float64(-t_1)) * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.00375], N[(N[(1.5707963267948966 - N[ArcSin[t$95$1], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 3.45e-12], N[(N[ArcCos[N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] * R + N[(N[ArcSin[(-t$95$1)], $MachinePrecision] * R), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.0037499999999999999

   1. Initial program 74.3%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-asin.f64 74.2%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Evaluated real constant 74.2%

      \[\leadsto \left(\color{blue}{1.5707963267948966} - \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1, \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right) \cdot R \]

   if -0.0037499999999999999 < phi1 < 3.45e-12

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.3%

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

   6. Applied rewrites 53.3%

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

   if 3.45e-12 < phi1

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   5. Applied rewrites 73.8%

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

Alternative 7: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \left(1.5707963267948966 - \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right), t\_0, \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\right)\right) \cdot R\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.00375:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 4 \cdot 10^{-66}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1
         (*
          (-
           1.5707963267948966
           (asin
            (fma
             (* (cos (- lambda2 lambda1)) (cos (fmin phi1 phi2)))
             t_0
             (* (sin (fmax phi1 phi2)) (sin (fmin phi1 phi2))))))
          R)))
   (if (<= (fmin phi1 phi2) -0.00375)
     t_1
     (if (<= (fmin phi1 phi2) 4e-66)
       (*
        (acos
         (*
          t_0
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = (1.5707963267948966 - asin(fma((cos((lambda2 - lambda1)) * cos(fmin(phi1, phi2))), t_0, (sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2)))))) * R;
	double tmp;
	if (fmin(phi1, phi2) <= -0.00375) {
		tmp = t_1;
	} else if (fmin(phi1, phi2) <= 4e-66) {
		tmp = acos((t_0 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = Float64(Float64(1.5707963267948966 - asin(fma(Float64(cos(Float64(lambda2 - lambda1)) * cos(fmin(phi1, phi2))), t_0, Float64(sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2)))))) * R)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.00375)
		tmp = t_1;
	elseif (fmin(phi1, phi2) <= 4e-66)
		tmp = Float64(acos(Float64(t_0 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.5707963267948966 - N[ArcSin[N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.00375], t$95$1, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 4e-66], N[(N[ArcCos[N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0037499999999999999 or 3.9999999999999999e-66 < phi1

   1. Initial program 74.3%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-asin.f64 74.2%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Evaluated real constant 74.2%

      \[\leadsto \left(\color{blue}{1.5707963267948966} - \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1, \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right) \cdot R \]

   if -0.0037499999999999999 < phi1 < 3.9999999999999999e-66

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.3%

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

   6. Applied rewrites 53.3%

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

Alternative 8: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(\sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right), \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right), \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right) \cdot t\_0\right)\right) \cdot R\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.00375:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 4 \cdot 10^{-66}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1
         (*
          (acos
           (fma
            (sin (fmax phi1 phi2))
            (sin (fmin phi1 phi2))
            (* (* (cos (- lambda2 lambda1)) (cos (fmin phi1 phi2))) t_0)))
          R)))
   (if (<= (fmin phi1 phi2) -0.00375)
     t_1
     (if (<= (fmin phi1 phi2) 4e-66)
       (*
        (acos
         (*
          t_0
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = acos(fma(sin(fmax(phi1, phi2)), sin(fmin(phi1, phi2)), ((cos((lambda2 - lambda1)) * cos(fmin(phi1, phi2))) * t_0))) * R;
	double tmp;
	if (fmin(phi1, phi2) <= -0.00375) {
		tmp = t_1;
	} else if (fmin(phi1, phi2) <= 4e-66) {
		tmp = acos((t_0 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = Float64(acos(fma(sin(fmax(phi1, phi2)), sin(fmin(phi1, phi2)), Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(fmin(phi1, phi2))) * t_0))) * R)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.00375)
		tmp = t_1;
	elseif (fmin(phi1, phi2) <= 4e-66)
		tmp = Float64(acos(Float64(t_0 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.00375], t$95$1, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 4e-66], N[(N[ArcCos[N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0037499999999999999 or 3.9999999999999999e-66 < phi1

   1. Initial program 74.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 74.3%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 74.3%

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lower--.f64 74.3%

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

   3. Applied rewrites 74.3%

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

   if -0.0037499999999999999 < phi1 < 3.9999999999999999e-66

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.3%

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

   6. Applied rewrites 53.3%

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

Alternative 9: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \mathsf{fma}\left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right), t\_1, \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1.22:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot t\_3\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 8 \cdot 10^{-24}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot t\_3\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_0 \cdot t\_1, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (fmax lambda1 lambda2)))
        (t_2 (cos (fmin phi1 phi2)))
        (t_3
         (fma
          (cos (fmin lambda1 lambda2))
          t_1
          (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2))))))
   (if (<= (fmin phi1 phi2) -1.22)
     (* (acos (* t_2 t_3)) R)
     (if (<= (fmin phi1 phi2) 8e-24)
       (* (acos (* t_0 t_3)) R)
       (*
        (acos
         (fma
          t_2
          (* t_0 t_1)
          (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos(fmax(lambda1, lambda2));
	double t_2 = cos(fmin(phi1, phi2));
	double t_3 = fma(cos(fmin(lambda1, lambda2)), t_1, (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))));
	double tmp;
	if (fmin(phi1, phi2) <= -1.22) {
		tmp = acos((t_2 * t_3)) * R;
	} else if (fmin(phi1, phi2) <= 8e-24) {
		tmp = acos((t_0 * t_3)) * R;
	} else {
		tmp = acos(fma(t_2, (t_0 * t_1), (sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(fmax(lambda1, lambda2))
	t_2 = cos(fmin(phi1, phi2))
	t_3 = fma(cos(fmin(lambda1, lambda2)), t_1, Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1.22)
		tmp = Float64(acos(Float64(t_2 * t_3)) * R);
	elseif (fmin(phi1, phi2) <= 8e-24)
		tmp = Float64(acos(Float64(t_0 * t_3)) * R);
	else
		tmp = Float64(acos(fma(t_2, Float64(t_0 * t_1), Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * t$95$1 + N[(N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1.22], N[(N[ArcCos[N[(t$95$2 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 8e-24], N[(N[ArcCos[N[(t$95$0 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 * N[(t$95$0 * t$95$1), $MachinePrecision] + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.22

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.5%

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

   6. Applied rewrites 52.5%

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

   if -1.22 < phi1 < 7.9999999999999994e-24

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.3%

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

   6. Applied rewrites 53.3%

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

   if 7.9999999999999994e-24 < phi1

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.2%

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

   6. Applied rewrites 94.2%

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

   7. Taylor expanded in lambda1 around 0

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

      1. lower-cos.f64 52.9%

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

   9. Applied rewrites 52.9%

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

Alternative 10: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \mathsf{fma}\left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right), t\_1, \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1.22:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot t\_3\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 8 \cdot 10^{-24}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot t\_3\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_2 \cdot t\_0, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (fmax lambda1 lambda2)))
        (t_2 (cos (fmin phi1 phi2)))
        (t_3
         (fma
          (cos (fmin lambda1 lambda2))
          t_1
          (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2))))))
   (if (<= (fmin phi1 phi2) -1.22)
     (* (acos (* t_2 t_3)) R)
     (if (<= (fmin phi1 phi2) 8e-24)
       (* (acos (* t_0 t_3)) R)
       (*
        (acos
         (fma
          t_1
          (* t_2 t_0)
          (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos(fmax(lambda1, lambda2));
	double t_2 = cos(fmin(phi1, phi2));
	double t_3 = fma(cos(fmin(lambda1, lambda2)), t_1, (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))));
	double tmp;
	if (fmin(phi1, phi2) <= -1.22) {
		tmp = acos((t_2 * t_3)) * R;
	} else if (fmin(phi1, phi2) <= 8e-24) {
		tmp = acos((t_0 * t_3)) * R;
	} else {
		tmp = acos(fma(t_1, (t_2 * t_0), (sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(fmax(lambda1, lambda2))
	t_2 = cos(fmin(phi1, phi2))
	t_3 = fma(cos(fmin(lambda1, lambda2)), t_1, Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1.22)
		tmp = Float64(acos(Float64(t_2 * t_3)) * R);
	elseif (fmin(phi1, phi2) <= 8e-24)
		tmp = Float64(acos(Float64(t_0 * t_3)) * R);
	else
		tmp = Float64(acos(fma(t_1, Float64(t_2 * t_0), Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * t$95$1 + N[(N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1.22], N[(N[ArcCos[N[(t$95$2 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 8e-24], N[(N[ArcCos[N[(t$95$0 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 * N[(t$95$2 * t$95$0), $MachinePrecision] + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.22

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.5%

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

   6. Applied rewrites 52.5%

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

   if -1.22 < phi1 < 7.9999999999999994e-24

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.3%

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

   6. Applied rewrites 53.3%

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

   if 7.9999999999999994e-24 < phi1

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in lambda1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.9%

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

   6. Applied rewrites 52.9%

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

Alternative 11: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1.22:\\ \;\;\;\;\cos^{-1} \left(t\_1 \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 3.7 \cdot 10^{+42}:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot t\_2, t\_1, \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (cos (fmax phi1 phi2))))
   (if (<= (fmin phi1 phi2) -1.22)
     (* (acos (* t_1 t_0)) R)
     (if (<= (fmin phi1 phi2) 3.7e+42)
       (* (acos (* t_2 t_0)) R)
       (*
        (acos
         (fma
          (* (cos lambda1) t_2)
          t_1
          (* (sin (fmax phi1 phi2)) (sin (fmin phi1 phi2)))))
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = cos(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -1.22) {
		tmp = acos((t_1 * t_0)) * R;
	} else if (fmin(phi1, phi2) <= 3.7e+42) {
		tmp = acos((t_2 * t_0)) * R;
	} else {
		tmp = acos(fma((cos(lambda1) * t_2), t_1, (sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = cos(fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1.22)
		tmp = Float64(acos(Float64(t_1 * t_0)) * R);
	elseif (fmin(phi1, phi2) <= 3.7e+42)
		tmp = Float64(acos(Float64(t_2 * t_0)) * R);
	else
		tmp = Float64(acos(fma(Float64(cos(lambda1) * t_2), t_1, Float64(sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1.22], N[(N[ArcCos[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 3.7e+42], N[(N[ArcCos[N[(t$95$2 * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[lambda1], $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1 + N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.22

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.5%

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

   6. Applied rewrites 52.5%

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

   if -1.22 < phi1 < 3.7e42

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.3%

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

   6. Applied rewrites 53.3%

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

   if 3.7e42 < phi1

   1. Initial program 74.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.4%

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

   4. Applied rewrites 54.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 54.4%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 54.4%

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

   6. Applied rewrites 54.4%

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

Alternative 12: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 0.0042:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right), t\_0, \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2))))
   (if (<= (fmax phi1 phi2) 0.0042)
     (*
      (acos
       (*
        t_0
        (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
      R)
     (*
      (acos
       (fma
        (* (cos lambda1) (cos (fmax phi1 phi2)))
        t_0
        (* (sin (fmax phi1 phi2)) (sin (fmin phi1 phi2)))))
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2));
	double tmp;
	if (fmax(phi1, phi2) <= 0.0042) {
		tmp = acos((t_0 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = acos(fma((cos(lambda1) * cos(fmax(phi1, phi2))), t_0, (sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(phi1, phi2) <= 0.0042)
		tmp = Float64(acos(Float64(t_0 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(acos(fma(Float64(cos(lambda1) * cos(fmax(phi1, phi2))), t_0, Float64(sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 0.0042], N[(N[ArcCos[N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0041999999999999997

   1. Initial program 74.3%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.2%

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

   3. Applied rewrites 94.2%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 52.5%

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

   6. Applied rewrites 52.5%

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

   if 0.0041999999999999997 < phi2

   1. Initial program 74.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.4%

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

   4. Applied rewrites 54.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 54.4%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 54.4%

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

   6. Applied rewrites 54.4%

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

Alternative 13: 62.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 0.23:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot t\_0, t\_1, \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot t\_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot \mathsf{max}\left(\phi_1, \phi_2\right) + \left(t\_1 \cdot t\_0\right) \cdot \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (sin (fmin phi1 phi2))))
   (if (<= (fmax lambda1 lambda2) 0.23)
     (*
      (acos
       (fma
        (* (cos (fmin lambda1 lambda2)) t_0)
        t_1
        (* (sin (fmax phi1 phi2)) t_2)))
      R)
     (*
      (acos
       (+
        (* t_2 (fmax phi1 phi2))
        (*
         (* t_1 t_0)
         (cos (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))))))
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = sin(fmin(phi1, phi2));
	double tmp;
	if (fmax(lambda1, lambda2) <= 0.23) {
		tmp = acos(fma((cos(fmin(lambda1, lambda2)) * t_0), t_1, (sin(fmax(phi1, phi2)) * t_2))) * R;
	} else {
		tmp = acos(((t_2 * fmax(phi1, phi2)) + ((t_1 * t_0) * cos((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = sin(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 0.23)
		tmp = Float64(acos(fma(Float64(cos(fmin(lambda1, lambda2)) * t_0), t_1, Float64(sin(fmax(phi1, phi2)) * t_2))) * R);
	else
		tmp = Float64(acos(Float64(Float64(t_2 * fmax(phi1, phi2)) + Float64(Float64(t_1 * t_0) * cos(Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 0.23], N[(N[ArcCos[N[(N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1 + N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(t$95$2 * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[Cos[N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.23000000000000001

   1. Initial program 74.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.4%

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

   4. Applied rewrites 54.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 54.4%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 54.4%

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

   6. Applied rewrites 54.4%

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

   if 0.23000000000000001 < lambda2

   1. Initial program 74.3%

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

   2. Taylor expanded in phi2 around 0

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

   4. Applied rewrites 44.2%

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

Alternative 14: 62.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 0.23:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right), t\_0, t\_1 \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_1 \cdot \mathsf{max}\left(\phi_1, \phi_2\right) + t\_0 \cdot \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos (fmin phi1 phi2)) (cos (fmax phi1 phi2))))
        (t_1 (sin (fmin phi1 phi2))))
   (if (<= (fmax lambda1 lambda2) 0.23)
     (*
      (acos
       (fma (cos (fmin lambda1 lambda2)) t_0 (* t_1 (sin (fmax phi1 phi2)))))
      R)
     (*
      (acos
       (+
        (* t_1 (fmax phi1 phi2))
        (* t_0 (cos (- (fmin lambda1 lambda2) (fmax lambda1 lambda2))))))
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(phi1, phi2)) * cos(fmax(phi1, phi2));
	double t_1 = sin(fmin(phi1, phi2));
	double tmp;
	if (fmax(lambda1, lambda2) <= 0.23) {
		tmp = acos(fma(cos(fmin(lambda1, lambda2)), t_0, (t_1 * sin(fmax(phi1, phi2))))) * R;
	} else {
		tmp = acos(((t_1 * fmax(phi1, phi2)) + (t_0 * cos((fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(fmin(phi1, phi2)) * cos(fmax(phi1, phi2)))
	t_1 = sin(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 0.23)
		tmp = Float64(acos(fma(cos(fmin(lambda1, lambda2)), t_0, Float64(t_1 * sin(fmax(phi1, phi2))))) * R);
	else
		tmp = Float64(acos(Float64(Float64(t_1 * fmax(phi1, phi2)) + Float64(t_0 * cos(Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2)))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 0.23], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(t$95$1 * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(t$95$1 * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.23000000000000001

   1. Initial program 74.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.4%

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

   4. Applied rewrites 54.4%

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

   if 0.23000000000000001 < lambda2

   1. Initial program 74.3%

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

   2. Taylor expanded in phi2 around 0

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

   4. Applied rewrites 44.2%

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

Alternative 15: 60.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1.25:\\ \;\;\;\;\left(0.5 \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_1\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 1.85:\\ \;\;\;\;\cos^{-1} \left(\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \left(1 + -0.16666666666666666 \cdot {\left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)}^{2}\right)\right) \cdot t\_2 + \left(t\_1 \cdot t\_0\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_0, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (sin (fmax phi1 phi2))))
   (if (<= (fmin phi1 phi2) -1.25)
     (* (- (* 0.5 PI) (asin (* (cos (- lambda2 lambda1)) t_1))) R)
     (if (<= (fmin phi1 phi2) 1.85)
       (*
        (acos
         (+
          (*
           (*
            (fmin phi1 phi2)
            (+ 1.0 (* -0.16666666666666666 (pow (fmin phi1 phi2) 2.0))))
           t_2)
          (* (* t_1 t_0) (cos (- lambda1 lambda2)))))
        R)
       (* (acos (fma t_1 t_0 (* (sin (fmin phi1 phi2)) t_2))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = sin(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -1.25) {
		tmp = ((0.5 * ((double) M_PI)) - asin((cos((lambda2 - lambda1)) * t_1))) * R;
	} else if (fmin(phi1, phi2) <= 1.85) {
		tmp = acos((((fmin(phi1, phi2) * (1.0 + (-0.16666666666666666 * pow(fmin(phi1, phi2), 2.0)))) * t_2) + ((t_1 * t_0) * cos((lambda1 - lambda2))))) * R;
	} else {
		tmp = acos(fma(t_1, t_0, (sin(fmin(phi1, phi2)) * t_2))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = sin(fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1.25)
		tmp = Float64(Float64(Float64(0.5 * pi) - asin(Float64(cos(Float64(lambda2 - lambda1)) * t_1))) * R);
	elseif (fmin(phi1, phi2) <= 1.85)
		tmp = Float64(acos(Float64(Float64(Float64(fmin(phi1, phi2) * Float64(1.0 + Float64(-0.16666666666666666 * (fmin(phi1, phi2) ^ 2.0)))) * t_2) + Float64(Float64(t_1 * t_0) * cos(Float64(lambda1 - lambda2))))) * R);
	else
		tmp = Float64(acos(fma(t_1, t_0, Float64(sin(fmin(phi1, phi2)) * t_2))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1.25], N[(N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 1.85], N[(N[ArcCos[N[(N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Min[phi1, phi2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 * t$95$0 + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.25

   1. Initial program 74.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.7%

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

   4. Applied rewrites 42.7%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-asin.f64 42.7%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-cos.f64 N/A

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

      13. cos-fabs-rev N/A

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

      14. lift-cos.f64 N/A

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

   6. Applied rewrites 42.7%

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

   if -1.25 < phi1 < 1.8500000000000001

   1. Initial program 74.3%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 37.9%

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

   4. Applied rewrites 37.9%

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

   if 1.8500000000000001 < phi1

   1. Initial program 74.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.4%

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 32.2%

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

   7. Applied rewrites 32.2%

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

Alternative 16: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1.25:\\ \;\;\;\;\left(0.5 \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_1\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 0.65:\\ \;\;\;\;\cos^{-1} \left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot t\_2 + \left(t\_1 \cdot t\_0\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_0, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (sin (fmax phi1 phi2))))
   (if (<= (fmin phi1 phi2) -1.25)
     (* (- (* 0.5 PI) (asin (* (cos (- lambda2 lambda1)) t_1))) R)
     (if (<= (fmin phi1 phi2) 0.65)
       (*
        (acos
         (+
          (* (fmin phi1 phi2) t_2)
          (* (* t_1 t_0) (cos (- lambda1 lambda2)))))
        R)
       (* (acos (fma t_1 t_0 (* (sin (fmin phi1 phi2)) t_2))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = sin(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -1.25) {
		tmp = ((0.5 * ((double) M_PI)) - asin((cos((lambda2 - lambda1)) * t_1))) * R;
	} else if (fmin(phi1, phi2) <= 0.65) {
		tmp = acos(((fmin(phi1, phi2) * t_2) + ((t_1 * t_0) * cos((lambda1 - lambda2))))) * R;
	} else {
		tmp = acos(fma(t_1, t_0, (sin(fmin(phi1, phi2)) * t_2))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = sin(fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1.25)
		tmp = Float64(Float64(Float64(0.5 * pi) - asin(Float64(cos(Float64(lambda2 - lambda1)) * t_1))) * R);
	elseif (fmin(phi1, phi2) <= 0.65)
		tmp = Float64(acos(Float64(Float64(fmin(phi1, phi2) * t_2) + Float64(Float64(t_1 * t_0) * cos(Float64(lambda1 - lambda2))))) * R);
	else
		tmp = Float64(acos(fma(t_1, t_0, Float64(sin(fmin(phi1, phi2)) * t_2))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1.25], N[(N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 0.65], N[(N[ArcCos[N[(N[(N[Min[phi1, phi2], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 * t$95$0 + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.25

   1. Initial program 74.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.7%

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

   4. Applied rewrites 42.7%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-asin.f64 42.7%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-cos.f64 N/A

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

      13. cos-fabs-rev N/A

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

      14. lift-cos.f64 N/A

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

   6. Applied rewrites 42.7%

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

   if -1.25 < phi1 < 0.65000000000000002

   1. Initial program 74.3%

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

   2. Taylor expanded in phi1 around 0

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

   4. Applied rewrites 44.2%

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

   if 0.65000000000000002 < phi1

   1. Initial program 74.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.4%

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 32.2%

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

   7. Applied rewrites 32.2%

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

Alternative 17: 60.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.6:\\ \;\;\;\;\left(0.5 \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_1\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 2.55 \cdot 10^{+17}:\\ \;\;\;\;\cos^{-1} \left(t\_2 + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_0, t\_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))))
   (if (<= (fmin phi1 phi2) -0.6)
     (* (- (* 0.5 PI) (asin (* (cos (- lambda2 lambda1)) t_1))) R)
     (if (<= (fmin phi1 phi2) 2.55e+17)
       (* (acos (+ t_2 (* t_0 (cos (- lambda1 lambda2))))) R)
       (* (acos (fma t_1 t_0 t_2)) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -0.6) {
		tmp = ((0.5 * ((double) M_PI)) - asin((cos((lambda2 - lambda1)) * t_1))) * R;
	} else if (fmin(phi1, phi2) <= 2.55e+17) {
		tmp = acos((t_2 + (t_0 * cos((lambda1 - lambda2))))) * R;
	} else {
		tmp = acos(fma(t_1, t_0, t_2)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2)))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.6)
		tmp = Float64(Float64(Float64(0.5 * pi) - asin(Float64(cos(Float64(lambda2 - lambda1)) * t_1))) * R);
	elseif (fmin(phi1, phi2) <= 2.55e+17)
		tmp = Float64(acos(Float64(t_2 + Float64(t_0 * cos(Float64(lambda1 - lambda2))))) * R);
	else
		tmp = Float64(acos(fma(t_1, t_0, t_2)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.6], N[(N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 2.55e+17], N[(N[ArcCos[N[(t$95$2 + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 * t$95$0 + t$95$2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.59999999999999998

   1. Initial program 74.3%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.7%

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

   4. Applied rewrites 42.7%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-asin.f64 42.7%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-cos.f64 N/A

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

      13. cos-fabs-rev N/A

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

      14. lift-cos.f64 N/A

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

   6. Applied rewrites 42.7%

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

   if -0.59999999999999998 < phi1 < 2.55e17

   1. Initial program 74.3%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 43.4%

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

   4. Applied rewrites 43.4%

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

   if 2.55e17 < phi1

   1. Initial program 74.3%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.4%

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in lambda1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-sin.f64 32.2%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   7. Applied rewrites 32.2%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 60.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.6:\\ \;\;\;\;\left(0.5 \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot t\_1\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 1.42 \cdot 10^{+17}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right), t\_2, t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_0, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1 (cos (fmin phi1 phi2)))
        (t_2 (sin (fmax phi1 phi2))))
   (if (<= (fmin phi1 phi2) -0.6)
     (* (- (* 0.5 PI) (asin (* (cos (- lambda2 lambda1)) t_1))) R)
     (if (<= (fmin phi1 phi2) 1.42e+17)
       (*
        (acos (fma (fmin phi1 phi2) t_2 (* t_0 (cos (- lambda1 lambda2)))))
        R)
       (* (acos (fma t_1 t_0 (* (sin (fmin phi1 phi2)) t_2))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmax(phi1, phi2));
	double t_1 = cos(fmin(phi1, phi2));
	double t_2 = sin(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -0.6) {
		tmp = ((0.5 * ((double) M_PI)) - asin((cos((lambda2 - lambda1)) * t_1))) * R;
	} else if (fmin(phi1, phi2) <= 1.42e+17) {
		tmp = acos(fma(fmin(phi1, phi2), t_2, (t_0 * cos((lambda1 - lambda2))))) * R;
	} else {
		tmp = acos(fma(t_1, t_0, (sin(fmin(phi1, phi2)) * t_2))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmax(phi1, phi2))
	t_1 = cos(fmin(phi1, phi2))
	t_2 = sin(fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.6)
		tmp = Float64(Float64(Float64(0.5 * pi) - asin(Float64(cos(Float64(lambda2 - lambda1)) * t_1))) * R);
	elseif (fmin(phi1, phi2) <= 1.42e+17)
		tmp = Float64(acos(fma(fmin(phi1, phi2), t_2, Float64(t_0 * cos(Float64(lambda1 - lambda2))))) * R);
	else
		tmp = Float64(acos(fma(t_1, t_0, Float64(sin(fmin(phi1, phi2)) * t_2))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.6], N[(N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 1.42e+17], N[(N[ArcCos[N[(N[Min[phi1, phi2], $MachinePrecision] * t$95$2 + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 * t$95$0 + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.59999999999999998

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   5. Step-by-step derivation

      1. lift-acos.f64 N/A

         \[\leadsto \color{blue}{\cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. acos-asin N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\pi}}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. mult-flip-rev N/A

         \[\leadsto \left(\color{blue}{\pi \cdot \frac{1}{2}} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. metadata-eval N/A

         \[\leadsto \left(\pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \pi} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \pi} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      9. lower-asin.f64 42.7%

         \[\leadsto \left(0.5 \cdot \pi - \color{blue}{\sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot R \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right) \cdot R \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right)\right) \cdot R \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\phi_1}\right)\right) \cdot R \]

      13. cos-fabs-rev N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \left(\left|\lambda_1 - \lambda_2\right|\right) \cdot \cos \color{blue}{\phi_1}\right)\right) \cdot R \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \left(\left|\lambda_1 - \lambda_2\right|\right) \cdot \cos \phi_1\right)\right) \cdot R \]

   6. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)\right)} \cdot R \]

   if -0.59999999999999998 < phi1 < 1.42e17

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. lower--.f64 36.7%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

   4. Applied rewrites 36.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

   if 1.42e17 < phi1

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-sin.f64 54.4%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. Applied rewrites 54.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-sin.f64 32.2%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   7. Applied rewrites 32.2%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 58.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 0.0042:\\ \;\;\;\;\left(0.5 \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmax phi1 phi2) 0.0042)
   (*
    (- (* 0.5 PI) (asin (* (cos (- lambda2 lambda1)) (cos (fmin phi1 phi2)))))
    R)
   (* (acos (* (cos (fmax phi1 phi2)) (cos (- lambda1 lambda2)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(phi1, phi2) <= 0.0042) {
		tmp = ((0.5 * ((double) M_PI)) - asin((cos((lambda2 - lambda1)) * cos(fmin(phi1, phi2))))) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * cos((lambda1 - lambda2)))) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(phi1, phi2) <= 0.0042) {
		tmp = ((0.5 * Math.PI) - Math.asin((Math.cos((lambda2 - lambda1)) * Math.cos(fmin(phi1, phi2))))) * R;
	} else {
		tmp = Math.acos((Math.cos(fmax(phi1, phi2)) * Math.cos((lambda1 - lambda2)))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmax(phi1, phi2) <= 0.0042:
		tmp = ((0.5 * math.pi) - math.asin((math.cos((lambda2 - lambda1)) * math.cos(fmin(phi1, phi2))))) * R
	else:
		tmp = math.acos((math.cos(fmax(phi1, phi2)) * math.cos((lambda1 - lambda2)))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(phi1, phi2) <= 0.0042)
		tmp = Float64(Float64(Float64(0.5 * pi) - asin(Float64(cos(Float64(lambda2 - lambda1)) * cos(fmin(phi1, phi2))))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * cos(Float64(lambda1 - lambda2)))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (max(phi1, phi2) <= 0.0042)
		tmp = ((0.5 * pi) - asin((cos((lambda2 - lambda1)) * cos(min(phi1, phi2))))) * R;
	else
		tmp = acos((cos(max(phi1, phi2)) * cos((lambda1 - lambda2)))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 0.0042], N[(N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0041999999999999997

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   5. Step-by-step derivation

      1. lift-acos.f64 N/A

         \[\leadsto \color{blue}{\cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. acos-asin N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\pi}}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. mult-flip-rev N/A

         \[\leadsto \left(\color{blue}{\pi \cdot \frac{1}{2}} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. metadata-eval N/A

         \[\leadsto \left(\pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \pi} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \pi} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      9. lower-asin.f64 42.7%

         \[\leadsto \left(0.5 \cdot \pi - \color{blue}{\sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot R \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right) \cdot R \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \phi_1}\right)\right) \cdot R \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\phi_1}\right)\right) \cdot R \]

      13. cos-fabs-rev N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \left(\left|\lambda_1 - \lambda_2\right|\right) \cdot \cos \color{blue}{\phi_1}\right)\right) \cdot R \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \pi - \sin^{-1} \left(\cos \left(\left|\lambda_1 - \lambda_2\right|\right) \cdot \cos \phi_1\right)\right) \cdot R \]

   6. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot \pi - \sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)\right)} \cdot R \]

   if 0.0041999999999999997 < phi2

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 43.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 43.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 58.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 0.0042:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (fmax phi1 phi2) 0.0042)
     (* (acos (* (cos (fmin phi1 phi2)) t_0)) R)
     (* (acos (* (cos (fmax phi1 phi2)) t_0)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (fmax(phi1, phi2) <= 0.0042) {
		tmp = acos((cos(fmin(phi1, phi2)) * t_0)) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * t_0)) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    if (fmax(phi1, phi2) <= 0.0042d0) then
        tmp = acos((cos(fmin(phi1, phi2)) * t_0)) * r
    else
        tmp = acos((cos(fmax(phi1, phi2)) * t_0)) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (fmax(phi1, phi2) <= 0.0042) {
		tmp = Math.acos((Math.cos(fmin(phi1, phi2)) * t_0)) * R;
	} else {
		tmp = Math.acos((Math.cos(fmax(phi1, phi2)) * t_0)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if fmax(phi1, phi2) <= 0.0042:
		tmp = math.acos((math.cos(fmin(phi1, phi2)) * t_0)) * R
	else:
		tmp = math.acos((math.cos(fmax(phi1, phi2)) * t_0)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmax(phi1, phi2) <= 0.0042)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * t_0)) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * t_0)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (max(phi1, phi2) <= 0.0042)
		tmp = acos((cos(min(phi1, phi2)) * t_0)) * R;
	else
		tmp = acos((cos(max(phi1, phi2)) * t_0)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 0.0042], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0041999999999999997

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   if 0.0041999999999999997 < phi2

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 43.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 43.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 54.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 0.0042:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmax phi1 phi2) 0.0042)
   (*
    (acos
     (*
      (cos (fmin phi1 phi2))
      (cos (- (fmin lambda1 lambda2) (fmax lambda1 lambda2)))))
    R)
   (* (acos (* (cos (fmax phi1 phi2)) (cos (fmax lambda1 lambda2)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(phi1, phi2) <= 0.0042) {
		tmp = acos((cos(fmin(phi1, phi2)) * cos((fmin(lambda1, lambda2) - fmax(lambda1, lambda2))))) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * cos(fmax(lambda1, lambda2)))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (fmax(phi1, phi2) <= 0.0042d0) then
        tmp = acos((cos(fmin(phi1, phi2)) * cos((fmin(lambda1, lambda2) - fmax(lambda1, lambda2))))) * r
    else
        tmp = acos((cos(fmax(phi1, phi2)) * cos(fmax(lambda1, lambda2)))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(phi1, phi2) <= 0.0042) {
		tmp = Math.acos((Math.cos(fmin(phi1, phi2)) * Math.cos((fmin(lambda1, lambda2) - fmax(lambda1, lambda2))))) * R;
	} else {
		tmp = Math.acos((Math.cos(fmax(phi1, phi2)) * Math.cos(fmax(lambda1, lambda2)))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmax(phi1, phi2) <= 0.0042:
		tmp = math.acos((math.cos(fmin(phi1, phi2)) * math.cos((fmin(lambda1, lambda2) - fmax(lambda1, lambda2))))) * R
	else:
		tmp = math.acos((math.cos(fmax(phi1, phi2)) * math.cos(fmax(lambda1, lambda2)))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(phi1, phi2) <= 0.0042)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * cos(Float64(fmin(lambda1, lambda2) - fmax(lambda1, lambda2))))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * cos(fmax(lambda1, lambda2)))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (max(phi1, phi2) <= 0.0042)
		tmp = acos((cos(min(phi1, phi2)) * cos((min(lambda1, lambda2) - max(lambda1, lambda2))))) * R;
	else
		tmp = acos((cos(max(phi1, phi2)) * cos(max(lambda1, lambda2)))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 0.0042], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Min[lambda1, lambda2], $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0041999999999999997

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   if 0.0041999999999999997 < phi2

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 94.2%

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

   3. Applied rewrites 94.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]

   4. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \color{blue}{\lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. lower-sin.f64 53.3%

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   6. Applied rewrites 53.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

   7. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]
   8. Step-by-step derivation

      1. lower-cos.f64 31.4%

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]

   9. Applied rewrites 31.4%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 43.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -265000000000:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin lambda1 lambda2) -265000000000.0)
   (* (acos (* (cos (fmin lambda1 lambda2)) (cos (fmin phi1 phi2)))) R)
   (* (acos (* (cos (fmax phi1 phi2)) (cos (fmax lambda1 lambda2)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(lambda1, lambda2) <= -265000000000.0) {
		tmp = acos((cos(fmin(lambda1, lambda2)) * cos(fmin(phi1, phi2)))) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * cos(fmax(lambda1, lambda2)))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (fmin(lambda1, lambda2) <= (-265000000000.0d0)) then
        tmp = acos((cos(fmin(lambda1, lambda2)) * cos(fmin(phi1, phi2)))) * r
    else
        tmp = acos((cos(fmax(phi1, phi2)) * cos(fmax(lambda1, lambda2)))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(lambda1, lambda2) <= -265000000000.0) {
		tmp = Math.acos((Math.cos(fmin(lambda1, lambda2)) * Math.cos(fmin(phi1, phi2)))) * R;
	} else {
		tmp = Math.acos((Math.cos(fmax(phi1, phi2)) * Math.cos(fmax(lambda1, lambda2)))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmin(lambda1, lambda2) <= -265000000000.0:
		tmp = math.acos((math.cos(fmin(lambda1, lambda2)) * math.cos(fmin(phi1, phi2)))) * R
	else:
		tmp = math.acos((math.cos(fmax(phi1, phi2)) * math.cos(fmax(lambda1, lambda2)))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(lambda1, lambda2) <= -265000000000.0)
		tmp = Float64(acos(Float64(cos(fmin(lambda1, lambda2)) * cos(fmin(phi1, phi2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * cos(fmax(lambda1, lambda2)))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (min(lambda1, lambda2) <= -265000000000.0)
		tmp = acos((cos(min(lambda1, lambda2)) * cos(min(phi1, phi2)))) * R;
	else
		tmp = acos((cos(max(phi1, phi2)) * cos(max(lambda1, lambda2)))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -265000000000.0], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.65e11

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 31.5%

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

   7. Applied rewrites 31.5%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   if -2.65e11 < lambda1

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 94.2%

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

   3. Applied rewrites 94.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]

   4. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \color{blue}{\lambda_2}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. lower-sin.f64 53.3%

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   6. Applied rewrites 53.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]

   7. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]
   8. Step-by-step derivation

      1. lower-cos.f64 31.4%

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]

   9. Applied rewrites 31.4%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 42.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\phi_1, \phi_2\right) \leq 0.0042:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmax phi1 phi2) 0.0042)
   (* (acos (* (cos lambda1) (cos (fmin phi1 phi2)))) R)
   (* (acos (* (cos lambda1) (cos (fmax phi1 phi2)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(phi1, phi2) <= 0.0042) {
		tmp = acos((cos(lambda1) * cos(fmin(phi1, phi2)))) * R;
	} else {
		tmp = acos((cos(lambda1) * cos(fmax(phi1, phi2)))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (fmax(phi1, phi2) <= 0.0042d0) then
        tmp = acos((cos(lambda1) * cos(fmin(phi1, phi2)))) * r
    else
        tmp = acos((cos(lambda1) * cos(fmax(phi1, phi2)))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(phi1, phi2) <= 0.0042) {
		tmp = Math.acos((Math.cos(lambda1) * Math.cos(fmin(phi1, phi2)))) * R;
	} else {
		tmp = Math.acos((Math.cos(lambda1) * Math.cos(fmax(phi1, phi2)))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmax(phi1, phi2) <= 0.0042:
		tmp = math.acos((math.cos(lambda1) * math.cos(fmin(phi1, phi2)))) * R
	else:
		tmp = math.acos((math.cos(lambda1) * math.cos(fmax(phi1, phi2)))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(phi1, phi2) <= 0.0042)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(fmin(phi1, phi2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(lambda1) * cos(fmax(phi1, phi2)))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (max(phi1, phi2) <= 0.0042)
		tmp = acos((cos(lambda1) * cos(min(phi1, phi2)))) * R;
	else
		tmp = acos((cos(lambda1) * cos(max(phi1, phi2)))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[phi1, phi2], $MachinePrecision], 0.0042], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0041999999999999997

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 31.5%

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

   7. Applied rewrites 31.5%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   if 0.0041999999999999997 < phi2

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-sin.f64 54.4%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. Applied rewrites 54.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot R \]

      3. lower-cos.f64 32.1%

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot R \]

   7. Applied rewrites 32.1%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 37.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1.02:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right) \cdot \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -1.02)
   (* (acos (* (cos (fmin lambda1 lambda2)) (cos (fmin phi1 phi2)))) R)
   (*
    (-
     (* PI 0.5)
     (asin
      (*
       (fma (* (fmin phi1 phi2) (fmin phi1 phi2)) -0.5 1.0)
       (cos (- (fmax lambda1 lambda2) (fmin lambda1 lambda2))))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -1.02) {
		tmp = acos((cos(fmin(lambda1, lambda2)) * cos(fmin(phi1, phi2)))) * R;
	} else {
		tmp = ((((double) M_PI) * 0.5) - asin((fma((fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0) * cos((fmax(lambda1, lambda2) - fmin(lambda1, lambda2)))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1.02)
		tmp = Float64(acos(Float64(cos(fmin(lambda1, lambda2)) * cos(fmin(phi1, phi2)))) * R);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(Float64(fma(Float64(fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0) * cos(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2)))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1.02], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Cos[N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.02

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 31.5%

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R \]

   7. Applied rewrites 31.5%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   if -1.02 < phi1

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-pow.f64 18.4%

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.4%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   8. Step-by-step derivation

      1. lift-acos.f64 N/A

         \[\leadsto \color{blue}{\cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. acos-asin N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      4. mult-flip N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      8. lower-PI.f64 N/A

         \[\leadsto \left(\color{blue}{\pi} \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      9. metadata-eval N/A

         \[\leadsto \left(\pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      10. lower-asin.f64 18.4%

          \[\leadsto \left(\pi \cdot 0.5 - \color{blue}{\sin^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot R \]

   9. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 21.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -2.2:\\ \;\;\;\;\cos^{-1} \left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -2.2)
   (* (acos (* (fmin phi1 phi2) (fmax phi1 phi2))) R)
   (*
    (-
     (* PI 0.5)
     (asin
      (*
       (fma (* (fmin phi1 phi2) (fmin phi1 phi2)) -0.5 1.0)
       (cos (- lambda2 lambda1)))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -2.2) {
		tmp = acos((fmin(phi1, phi2) * fmax(phi1, phi2))) * R;
	} else {
		tmp = ((((double) M_PI) * 0.5) - asin((fma((fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0) * cos((lambda2 - lambda1))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -2.2)
		tmp = Float64(acos(Float64(fmin(phi1, phi2) * fmax(phi1, phi2))) * R);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) - asin(Float64(fma(Float64(fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0) * cos(Float64(lambda2 - lambda1))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -2.2], N[(N[ArcCos[N[(N[Min[phi1, phi2], $MachinePrecision] * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.2000000000000002

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-sin.f64 54.4%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. Applied rewrites 54.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_1 \cdot \cos \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 25.7%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   7. Applied rewrites 25.7%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   8. Taylor expanded in phi1 around inf

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-sin.f64 12.3%

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

   10. Applied rewrites 12.3%

       \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]
   12. Step-by-step derivation

       1. lower-*.f64 9.7%

          \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]

   13. Applied rewrites 9.7%

       \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]

   if -2.2000000000000002 < phi1

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-pow.f64 18.4%

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.4%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   8. Step-by-step derivation

      1. lift-acos.f64 N/A

         \[\leadsto \color{blue}{\cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. acos-asin N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      4. mult-flip N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      8. lower-PI.f64 N/A

         \[\leadsto \left(\color{blue}{\pi} \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      9. metadata-eval N/A

         \[\leadsto \left(\pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      10. lower-asin.f64 18.4%

          \[\leadsto \left(\pi \cdot 0.5 - \color{blue}{\sin^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot R \]

   9. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 21.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -2.2:\\ \;\;\;\;\cos^{-1} \left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -2.2)
   (* (acos (* (fmin phi1 phi2) (fmax phi1 phi2))) R)
   (*
    (acos
     (*
      (fma (* (fmin phi1 phi2) (fmin phi1 phi2)) -0.5 1.0)
      (cos (- lambda2 lambda1))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -2.2) {
		tmp = acos((fmin(phi1, phi2) * fmax(phi1, phi2))) * R;
	} else {
		tmp = acos((fma((fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0) * cos((lambda2 - lambda1)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -2.2)
		tmp = Float64(acos(Float64(fmin(phi1, phi2) * fmax(phi1, phi2))) * R);
	else
		tmp = Float64(acos(Float64(fma(Float64(fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0) * cos(Float64(lambda2 - lambda1)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -2.2], N[(N[ArcCos[N[(N[Min[phi1, phi2], $MachinePrecision] * N[Max[phi1, phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.2000000000000002

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-sin.f64 54.4%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. Applied rewrites 54.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_1 \cdot \cos \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 25.7%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   7. Applied rewrites 25.7%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   8. Taylor expanded in phi1 around inf

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-sin.f64 12.3%

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

   10. Applied rewrites 12.3%

       \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]
   12. Step-by-step derivation

       1. lower-*.f64 9.7%

          \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]

   13. Applied rewrites 9.7%

       \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]

   if -2.2000000000000002 < phi1

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower--.f64 42.7%

         \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied rewrites 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \color{blue}{\lambda_2}\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-pow.f64 18.4%

         \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 18.4%

      \[\leadsto \cos^{-1} \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
   8. Step-by-step derivation

   9. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 15.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\lambda_1, \lambda_2\right) \leq -2.15 \cdot 10^{-19}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) + \phi_1 \cdot \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin lambda1 lambda2) -2.15e-19)
   (* (acos (+ (cos (fmin lambda1 lambda2)) (* phi1 phi2))) R)
   (* (acos (* phi1 (sin phi2))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(lambda1, lambda2) <= -2.15e-19) {
		tmp = acos((cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * R;
	} else {
		tmp = acos((phi1 * sin(phi2))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (fmin(lambda1, lambda2) <= (-2.15d-19)) then
        tmp = acos((cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * r
    else
        tmp = acos((phi1 * sin(phi2))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(lambda1, lambda2) <= -2.15e-19) {
		tmp = Math.acos((Math.cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * R;
	} else {
		tmp = Math.acos((phi1 * Math.sin(phi2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmin(lambda1, lambda2) <= -2.15e-19:
		tmp = math.acos((math.cos(fmin(lambda1, lambda2)) + (phi1 * phi2))) * R
	else:
		tmp = math.acos((phi1 * math.sin(phi2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(lambda1, lambda2) <= -2.15e-19)
		tmp = Float64(acos(Float64(cos(fmin(lambda1, lambda2)) + Float64(phi1 * phi2))) * R);
	else
		tmp = Float64(acos(Float64(phi1 * sin(phi2))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (min(lambda1, lambda2) <= -2.15e-19)
		tmp = acos((cos(min(lambda1, lambda2)) + (phi1 * phi2))) * R;
	else
		tmp = acos((phi1 * sin(phi2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[lambda1, lambda2], $MachinePrecision], -2.15e-19], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.1500000000000001e-19

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-sin.f64 54.4%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. Applied rewrites 54.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_1 \cdot \cos \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 25.7%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   7. Applied rewrites 25.7%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \phi_2\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \phi_2\right) \cdot R \]

      3. lower-*.f64 11.5%

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \phi_2\right) \cdot R \]

   10. Applied rewrites 11.5%

       \[\leadsto \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \color{blue}{\phi_2}\right) \cdot R \]

   if -2.1500000000000001e-19 < lambda1

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-sin.f64 54.4%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. Applied rewrites 54.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_1 \cdot \cos \phi_2}\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 25.7%

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   7. Applied rewrites 25.7%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   8. Taylor expanded in phi1 around inf

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-sin.f64 12.3%

         \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

   10. Applied rewrites 12.3%

       \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 12.3% accurate, 4.3× speedup

Math

\[\cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (* phi1 (sin phi2))) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos((phi1 * sin(phi2))) * R;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos((phi1 * sin(phi2))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos((phi1 * Math.sin(phi2))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos((phi1 * math.sin(phi2))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(phi1 * sin(phi2))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos((phi1 * sin(phi2))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 74.3%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   3. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   6. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   7. lower-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   8. lower-sin.f64 54.4%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

4. Applied rewrites 54.4%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_1 \cdot \cos \phi_2}\right) \cdot R \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   2. lower-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   3. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   4. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   5. lower-cos.f64 25.7%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

7. Applied rewrites 25.7%

   \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

8. Taylor expanded in phi1 around inf

   \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

   2. lower-sin.f64 12.3%

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

10. Applied rewrites 12.3%

    \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]
11. Add Preprocessing

Alternative 29: 9.7% accurate, 17.3× speedup

Math

\[\cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (* phi1 phi2)) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos((phi1 * phi2)) * R;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos((phi1 * phi2)) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos((phi1 * phi2)) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos((phi1 * phi2)) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(phi1 * phi2)) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos((phi1 * phi2)) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(phi1 * phi2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 74.3%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   3. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \color{blue}{\phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   6. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   7. lower-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   8. lower-sin.f64 54.4%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

4. Applied rewrites 54.4%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_1 \cdot \cos \phi_2}\right) \cdot R \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   2. lower-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   3. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   4. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

   5. lower-cos.f64 25.7%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

7. Applied rewrites 25.7%

   \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \color{blue}{\sin \phi_2}, \cos \lambda_1 \cdot \cos \phi_2\right)\right) \cdot R \]

8. Taylor expanded in phi1 around inf

   \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

   2. lower-sin.f64 12.3%

      \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

10. Applied rewrites 12.3%

    \[\leadsto \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right) \cdot R \]

11. Taylor expanded in phi2 around 0

    \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]
12. Step-by-step derivation

    1. lower-*.f64 9.7%

       \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]

13. Applied rewrites 9.7%

    \[\leadsto \cos^{-1} \left(\phi_1 \cdot \phi_2\right) \cdot R \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025197 
(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))