Spherical law of cosines

Percentage Accurate: 73.5% → 94.0%

Time: 15.6s

Alternatives: 17

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 94.0

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

3. Applied rewrites 94.0%

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

4. Taylor expanded in lambda1 around inf

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

   1. lower-fma.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-sin.f64 94.0

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

6. Applied rewrites 94.0%

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

Alternative 2: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -8.6 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 3.4 \cdot 10^{+51}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (* (cos phi2) (cos phi1))
            (cos (- lambda2 lambda1))
            (* (sin phi2) (sin phi1))))
          R)))
   (if (<= phi2 -8.6e-26)
     t_0
     (if (<= phi2 3.4e+51)
       (*
        (acos
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))
          (* phi2 (sin phi1))))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(fma((cos(phi2) * cos(phi1)), cos((lambda2 - lambda1)), (sin(phi2) * sin(phi1)))) * R;
	double tmp;
	if (phi2 <= -8.6e-26) {
		tmp = t_0;
	} else if (phi2 <= 3.4e+51) {
		tmp = acos(fma(cos(phi1), (cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))), (phi2 * sin(phi1)))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(Float64(cos(phi2) * cos(phi1)), cos(Float64(lambda2 - lambda1)), Float64(sin(phi2) * sin(phi1)))) * R)
	tmp = 0.0
	if (phi2 <= -8.6e-26)
		tmp = t_0;
	elseif (phi2 <= 3.4e+51)
		tmp = Float64(acos(fma(cos(phi1), Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))), Float64(phi2 * sin(phi1)))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -8.6e-26], t$95$0, If[LessEqual[phi2, 3.4e+51], 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[(phi2 * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -8.59999999999999976e-26 or 3.39999999999999984e51 < phi2

   1. Initial program 73.5%

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

      1. lift-+.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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 73.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.5

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 73.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

   if -8.59999999999999976e-26 < phi2 < 3.39999999999999984e51

   1. Initial program 73.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   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 57.0

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

   9. Applied rewrites 57.0%

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

Alternative 3: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_3 := t\_2 \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -6.2 \cdot 10^{-13}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_3, t\_0, t\_1\right)\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 8.8 \cdot 10^{-26}:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\pi - \cos^{-1} \left(\left(-t\_3\right) \cdot t\_0 - t\_1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (* (sin (fmax phi1 phi2)) (sin (fmin phi1 phi2))))
        (t_2 (cos (fmax phi1 phi2)))
        (t_3 (* t_2 (cos (fmin phi1 phi2)))))
   (if (<= (fmin phi1 phi2) -6.2e-13)
     (* (acos (fma t_3 t_0 t_1)) R)
     (if (<= (fmin phi1 phi2) 8.8e-26)
       (*
        (acos
         (*
          t_2
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2)))))
        R)
       (* (- PI (acos (- (* (- t_3) t_0) t_1))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2));
	double t_2 = cos(fmax(phi1, phi2));
	double t_3 = t_2 * cos(fmin(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -6.2e-13) {
		tmp = acos(fma(t_3, t_0, t_1)) * R;
	} else if (fmin(phi1, phi2) <= 8.8e-26) {
		tmp = acos((t_2 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))) * R;
	} else {
		tmp = (((double) M_PI) - acos(((-t_3 * t_0) - t_1))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2)))
	t_2 = cos(fmax(phi1, phi2))
	t_3 = Float64(t_2 * cos(fmin(phi1, phi2)))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -6.2e-13)
		tmp = Float64(acos(fma(t_3, t_0, t_1)) * R);
	elseif (fmin(phi1, phi2) <= 8.8e-26)
		tmp = Float64(acos(Float64(t_2 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))) * R);
	else
		tmp = Float64(Float64(pi - acos(Float64(Float64(Float64(-t_3) * t_0) - t_1))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -6.2e-13], N[(N[ArcCos[N[(t$95$3 * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 8.8e-26], N[(N[ArcCos[N[(t$95$2 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(Pi - N[ArcCos[N[(N[((-t$95$3) * t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -6.1999999999999998e-13

   1. Initial program 73.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 73.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.5

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 73.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

   if -6.1999999999999998e-13 < phi1 < 8.8000000000000003e-26

   1. Initial program 73.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.1

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

   6. Applied rewrites 53.1%

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

   if 8.8000000000000003e-26 < phi1

   1. Initial program 73.5%

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

      1. lift-acos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. sub-negate-rev N/A

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

      6. acos-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-acos.f64 N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 73.2%

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

Alternative 4: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(t\_0 \cdot \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right), \cos \left(\lambda_2 - \lambda_1\right), \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -6.2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 5.1 \cdot 10^{-27}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmax phi1 phi2)))
        (t_1
         (*
          (acos
           (fma
            (* t_0 (cos (fmin phi1 phi2)))
            (cos (- lambda2 lambda1))
            (* (sin (fmax phi1 phi2)) (sin (fmin phi1 phi2)))))
          R)))
   (if (<= (fmin phi1 phi2) -6.2e-13)
     t_1
     (if (<= (fmin phi1 phi2) 5.1e-27)
       (*
        (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((t_0 * cos(fmin(phi1, phi2))), cos((lambda2 - lambda1)), (sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))))) * R;
	double tmp;
	if (fmin(phi1, phi2) <= -6.2e-13) {
		tmp = t_1;
	} else if (fmin(phi1, phi2) <= 5.1e-27) {
		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(Float64(t_0 * cos(fmin(phi1, phi2))), cos(Float64(lambda2 - lambda1)), Float64(sin(fmax(phi1, phi2)) * sin(fmin(phi1, phi2))))) * R)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -6.2e-13)
		tmp = t_1;
	elseif (fmin(phi1, phi2) <= 5.1e-27)
		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[(t$95$0 * N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -6.2e-13], t$95$1, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 5.1e-27], 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 < -6.1999999999999998e-13 or 5.0999999999999999e-27 < phi1

   1. Initial program 73.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 73.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.5

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 73.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

   if -6.1999999999999998e-13 < phi1 < 5.0999999999999999e-27

   1. Initial program 73.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.1

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

   6. Applied rewrites 53.1%

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

Alternative 5: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \mathsf{fma}\left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right), t\_1, \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -44000:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot t\_3\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 5.1 \cdot 10^{-27}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot t\_3\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_2 \cdot t\_0, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \end{array} \]

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) -44000.0)
     (* (acos (* t_2 t_3)) R)
     (if (<= (fmin phi1 phi2) 5.1e-27)
       (* (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) <= -44000.0) {
		tmp = acos((t_2 * t_3)) * R;
	} else if (fmin(phi1, phi2) <= 5.1e-27) {
		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) <= -44000.0)
		tmp = Float64(acos(Float64(t_2 * t_3)) * R);
	elseif (fmin(phi1, phi2) <= 5.1e-27)
		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], -44000.0], N[(N[ArcCos[N[(t$95$2 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 5.1e-27], 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 < -44000

   1. Initial program 73.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.4

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

   6. Applied rewrites 53.4%

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

   if -44000 < phi1 < 5.0999999999999999e-27

   1. Initial program 73.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.1

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

   6. Applied rewrites 53.1%

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

   if 5.0999999999999999e-27 < phi1

   1. Initial program 73.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in lambda1 around 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 6: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_2 := \mathsf{fma}\left(t\_1, \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\\ t_3 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -44000:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot t\_2\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 7700000:\\ \;\;\;\;\cos^{-1} \left(t\_3 \cdot t\_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_0 \cdot t\_3, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2)))
        (t_1 (cos (fmin lambda1 lambda2)))
        (t_2
         (fma
          t_1
          (cos (fmax lambda1 lambda2))
          (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2)))))
        (t_3 (cos (fmax phi1 phi2))))
   (if (<= (fmin phi1 phi2) -44000.0)
     (* (acos (* t_0 t_2)) R)
     (if (<= (fmin phi1 phi2) 7700000.0)
       (* (acos (* t_3 t_2)) R)
       (*
        (acos
         (fma
          t_1
          (* t_0 t_3)
          (* (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(fmin(phi1, phi2));
	double t_1 = cos(fmin(lambda1, lambda2));
	double t_2 = fma(t_1, cos(fmax(lambda1, lambda2)), (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))));
	double t_3 = cos(fmax(phi1, phi2));
	double tmp;
	if (fmin(phi1, phi2) <= -44000.0) {
		tmp = acos((t_0 * t_2)) * R;
	} else if (fmin(phi1, phi2) <= 7700000.0) {
		tmp = acos((t_3 * t_2)) * R;
	} else {
		tmp = acos(fma(t_1, (t_0 * t_3), (sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	t_1 = cos(fmin(lambda1, lambda2))
	t_2 = fma(t_1, cos(fmax(lambda1, lambda2)), Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2))))
	t_3 = cos(fmax(phi1, phi2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -44000.0)
		tmp = Float64(acos(Float64(t_0 * t_2)) * R);
	elseif (fmin(phi1, phi2) <= 7700000.0)
		tmp = Float64(acos(Float64(t_3 * t_2)) * R);
	else
		tmp = Float64(acos(fma(t_1, Float64(t_0 * t_3), 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[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -44000.0], N[(N[ArcCos[N[(t$95$0 * t$95$2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 7700000.0], N[(N[ArcCos[N[(t$95$3 * t$95$2), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$1 * N[(t$95$0 * t$95$3), $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 < -44000

   1. Initial program 73.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.4

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

   6. Applied rewrites 53.4%

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

   if -44000 < phi1 < 7.7e6

   1. Initial program 73.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.1

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

   6. Applied rewrites 53.1%

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

   if 7.7e6 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.7

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

   4. Applied rewrites 53.7%

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

Alternative 7: 74.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right)\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(t\_0, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(t\_0, \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right), \sin \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin lambda1 lambda2)))
        (t_1
         (*
          (acos (fma t_0 (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
          R)))
   (if (<= phi2 -5.5e-9)
     t_1
     (if (<= phi2 2.1e-6)
       (*
        (acos
         (*
          (cos phi1)
          (fma
           t_0
           (cos (fmax lambda1 lambda2))
           (* (sin (fmin lambda1 lambda2)) (sin (fmax lambda1 lambda2))))))
        R)
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(fmin(lambda1, lambda2));
	double t_1 = acos(fma(t_0, (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))) * R;
	double tmp;
	if (phi2 <= -5.5e-9) {
		tmp = t_1;
	} else if (phi2 <= 2.1e-6) {
		tmp = acos((cos(phi1) * fma(t_0, cos(fmax(lambda1, lambda2)), (sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(lambda1, lambda2))
	t_1 = Float64(acos(fma(t_0, Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) * R)
	tmp = 0.0
	if (phi2 <= -5.5e-9)
		tmp = t_1;
	elseif (phi2 <= 2.1e-6)
		tmp = Float64(acos(Float64(cos(phi1) * fma(t_0, cos(fmax(lambda1, lambda2)), Float64(sin(fmin(lambda1, lambda2)) * sin(fmax(lambda1, lambda2)))))) * R);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -5.5e-9], t$95$1, If[LessEqual[phi2, 2.1e-6], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(t$95$0 * N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -5.4999999999999996e-9 or 2.0999999999999998e-6 < phi2

   1. Initial program 73.5%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.7

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

   4. Applied rewrites 53.7%

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

   if -5.4999999999999996e-9 < phi2 < 2.0999999999999998e-6

   1. Initial program 73.5%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 94.0

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.4

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

   6. Applied rewrites 53.4%

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

Alternative 8: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 850:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right), t\_0 \cdot \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right), \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) \cdot \left(\frac{\mathsf{min}\left(\lambda_1, \lambda_2\right)}{\mathsf{max}\left(\lambda_1, \lambda_2\right)} - 1\right)\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (fmin phi1 phi2))))
   (if (<= (fmax lambda1 lambda2) 850.0)
     (*
      (acos
       (fma
        (cos (fmin lambda1 lambda2))
        (* t_0 (cos (fmax phi1 phi2)))
        (* (sin (fmin phi1 phi2)) (sin (fmax phi1 phi2)))))
      R)
     (*
      (acos
       (*
        t_0
        (cos
         (*
          (fmax lambda1 lambda2)
          (- (/ (fmin lambda1 lambda2) (fmax lambda1 lambda2)) 1.0)))))
      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(lambda1, lambda2) <= 850.0) {
		tmp = acos(fma(cos(fmin(lambda1, lambda2)), (t_0 * cos(fmax(phi1, phi2))), (sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R;
	} else {
		tmp = acos((t_0 * cos((fmax(lambda1, lambda2) * ((fmin(lambda1, lambda2) / fmax(lambda1, lambda2)) - 1.0))))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(fmin(phi1, phi2))
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 850.0)
		tmp = Float64(acos(fma(cos(fmin(lambda1, lambda2)), Float64(t_0 * cos(fmax(phi1, phi2))), Float64(sin(fmin(phi1, phi2)) * sin(fmax(phi1, phi2))))) * R);
	else
		tmp = Float64(acos(Float64(t_0 * cos(Float64(fmax(lambda1, lambda2) * Float64(Float64(fmin(lambda1, lambda2) / fmax(lambda1, lambda2)) - 1.0))))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 850.0], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$0 * N[Cos[N[(N[Max[lambda1, lambda2], $MachinePrecision] * N[(N[(N[Min[lambda1, lambda2], $MachinePrecision] / N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 850

   1. Initial program 73.5%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 53.7

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

   4. Applied rewrites 53.7%

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

   if 850 < lambda2

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 35.2

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

   7. Applied rewrites 35.2%

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

Alternative 9: 59.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.47:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_1\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -2.25 \cdot 10^{-101}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2 \cdot \sin \lambda_1, t\_0, \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot t\_0\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot t\_1\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (* (fmin phi1 phi2) (fmin phi1 phi2)) -0.5 1.0))
        (t_1 (cos (- lambda1 lambda2))))
   (if (<= (fmin phi1 phi2) -0.47)
     (* (acos (* (cos (fmin phi1 phi2)) t_1)) R)
     (if (<= (fmin phi1 phi2) -2.25e-101)
       (*
        (acos
         (fma
          (* (sin lambda2) (sin lambda1))
          t_0
          (* (* (cos lambda2) (cos lambda1)) t_0)))
        R)
       (* (acos (* (cos (fmax phi1 phi2)) t_1)) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0);
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -0.47) {
		tmp = acos((cos(fmin(phi1, phi2)) * t_1)) * R;
	} else if (fmin(phi1, phi2) <= -2.25e-101) {
		tmp = acos(fma((sin(lambda2) * sin(lambda1)), t_0, ((cos(lambda2) * cos(lambda1)) * t_0))) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * t_1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0)
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.47)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * t_1)) * R);
	elseif (fmin(phi1, phi2) <= -2.25e-101)
		tmp = Float64(acos(fma(Float64(sin(lambda2) * sin(lambda1)), t_0, Float64(Float64(cos(lambda2) * cos(lambda1)) * t_0))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * t_1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.47], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -2.25e-101], N[(N[ArcCos[N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.46999999999999997

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   if -0.46999999999999997 < phi1 < -2.2499999999999999e-101

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   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 18.5

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

   7. Applied rewrites 18.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff-rev N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. lower-fma.f64 N/A

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

   9. Applied rewrites 23.8%

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

   if -2.2499999999999999e-101 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.7

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

   4. Applied rewrites 42.7%

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

Alternative 10: 58.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_1 := \cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right)\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -0.95:\\ \;\;\;\;\cos^{-1} \left(t\_2 \cdot t\_3\right) \cdot R\\ \mathbf{elif}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq 0.00085:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right), t\_0, t\_1 \cdot t\_3\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_2, t\_1, \sin \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (fmax phi1 phi2)))
        (t_1 (cos (fmax phi1 phi2)))
        (t_2 (cos (fmin phi1 phi2)))
        (t_3 (cos (- lambda1 lambda2))))
   (if (<= (fmin phi1 phi2) -0.95)
     (* (acos (* t_2 t_3)) R)
     (if (<= (fmin phi1 phi2) 0.00085)
       (* (acos (fma (fmin phi1 phi2) t_0 (* t_1 t_3))) R)
       (* (acos (fma t_2 t_1 (* (sin (fmin phi1 phi2)) t_0))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fmax(phi1, phi2));
	double t_1 = cos(fmax(phi1, phi2));
	double t_2 = cos(fmin(phi1, phi2));
	double t_3 = cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -0.95) {
		tmp = acos((t_2 * t_3)) * R;
	} else if (fmin(phi1, phi2) <= 0.00085) {
		tmp = acos(fma(fmin(phi1, phi2), t_0, (t_1 * t_3))) * R;
	} else {
		tmp = acos(fma(t_2, t_1, (sin(fmin(phi1, phi2)) * t_0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fmax(phi1, phi2))
	t_1 = cos(fmax(phi1, phi2))
	t_2 = cos(fmin(phi1, phi2))
	t_3 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -0.95)
		tmp = Float64(acos(Float64(t_2 * t_3)) * R);
	elseif (fmin(phi1, phi2) <= 0.00085)
		tmp = Float64(acos(fma(fmin(phi1, phi2), t_0, Float64(t_1 * t_3))) * R);
	else
		tmp = Float64(acos(fma(t_2, t_1, Float64(sin(fmin(phi1, phi2)) * t_0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -0.95], N[(N[ArcCos[N[(t$95$2 * t$95$3), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], 0.00085], N[(N[ArcCos[N[(N[Min[phi1, phi2], $MachinePrecision] * t$95$0 + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(t$95$2 * t$95$1 + N[(N[Sin[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.94999999999999996

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   if -0.94999999999999996 < phi1 < 8.49999999999999953e-4

   1. Initial program 73.5%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 35.9

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

   4. Applied rewrites 35.9%

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

   if 8.49999999999999953e-4 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in lambda1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 52.9

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in lambda2 around 0

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

      1. lower-cos.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%

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

Alternative 11: 57.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -44000:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\phi_1, \phi_2\right)\right) \cdot t\_0\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (fmin phi1 phi2) -44000.0)
     (* (acos (* (cos (fmin phi1 phi2)) t_0)) R)
     (* (acos (* (cos (fmax phi1 phi2)) t_0)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -44000.0) {
		tmp = acos((cos(fmin(phi1, phi2)) * t_0)) * R;
	} else {
		tmp = acos((cos(fmax(phi1, phi2)) * t_0)) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    if (fmin(phi1, phi2) <= (-44000.0d0)) then
        tmp = acos((cos(fmin(phi1, phi2)) * t_0)) * r
    else
        tmp = acos((cos(fmax(phi1, phi2)) * t_0)) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (fmin(phi1, phi2) <= -44000.0) {
		tmp = Math.acos((Math.cos(fmin(phi1, phi2)) * t_0)) * R;
	} else {
		tmp = Math.acos((Math.cos(fmax(phi1, phi2)) * t_0)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if fmin(phi1, phi2) <= -44000.0:
		tmp = math.acos((math.cos(fmin(phi1, phi2)) * t_0)) * R
	else:
		tmp = math.acos((math.cos(fmax(phi1, phi2)) * t_0)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (fmin(phi1, phi2) <= -44000.0)
		tmp = Float64(acos(Float64(cos(fmin(phi1, phi2)) * t_0)) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(phi1, phi2)) * t_0)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (min(phi1, phi2) <= -44000.0)
		tmp = acos((cos(min(phi1, phi2)) * t_0)) * R;
	else
		tmp = acos((cos(max(phi1, phi2)) * t_0)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -44000.0], N[(N[ArcCos[N[(N[Cos[N[Min[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[phi1, phi2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -44000

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   if -44000 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 42.7

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

   4. Applied rewrites 42.7%

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

Alternative 12: 43.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 43.1

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

4. Applied rewrites 43.1%

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

Alternative 13: 42.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\lambda_1, \lambda_2\right) \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmax lambda1 lambda2) 2.3e-10)
   (* (acos (* (cos (fmin lambda1 lambda2)) (cos phi1))) R)
   (* (acos (* (cos (fmax lambda1 lambda2)) (cos phi1))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(lambda1, lambda2) <= 2.3e-10) {
		tmp = acos((cos(fmin(lambda1, lambda2)) * cos(phi1))) * R;
	} else {
		tmp = acos((cos(fmax(lambda1, lambda2)) * cos(phi1))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (fmax(lambda1, lambda2) <= 2.3d-10) then
        tmp = acos((cos(fmin(lambda1, lambda2)) * cos(phi1))) * r
    else
        tmp = acos((cos(fmax(lambda1, lambda2)) * cos(phi1))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmax(lambda1, lambda2) <= 2.3e-10) {
		tmp = Math.acos((Math.cos(fmin(lambda1, lambda2)) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((Math.cos(fmax(lambda1, lambda2)) * Math.cos(phi1))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if fmax(lambda1, lambda2) <= 2.3e-10:
		tmp = math.acos((math.cos(fmin(lambda1, lambda2)) * math.cos(phi1))) * R
	else:
		tmp = math.acos((math.cos(fmax(lambda1, lambda2)) * math.cos(phi1))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmax(lambda1, lambda2) <= 2.3e-10)
		tmp = Float64(acos(Float64(cos(fmin(lambda1, lambda2)) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(lambda1, lambda2)) * cos(phi1))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (max(lambda1, lambda2) <= 2.3e-10)
		tmp = acos((cos(min(lambda1, lambda2)) * cos(phi1))) * R;
	else
		tmp = acos((cos(max(lambda1, lambda2)) * cos(phi1))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Max[lambda1, lambda2], $MachinePrecision], 2.3e-10], N[(N[ArcCos[N[(N[Cos[N[Min[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.30000000000000007e-10

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   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 31.6

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

   7. Applied rewrites 31.6%

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

   if 2.30000000000000007e-10 < lambda2

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   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 18.5

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

   7. Applied rewrites 18.5%

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

   9. Applied rewrites 18.5%

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

   10. Taylor expanded in lambda1 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-cos.f64 31.0

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

   12. Applied rewrites 31.0%

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

Alternative 14: 38.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 0.00085:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (acos (* (cos lambda1) (cos phi1))) R)))
   (if (<= phi1 -1.5)
     t_0
     (if (<= phi1 0.00085)
       (* (acos (* (cos (- lambda2 lambda1)) (fma (* phi1 phi1) -0.5 1.0))) R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos((cos(lambda1) * cos(phi1))) * R;
	double tmp;
	if (phi1 <= -1.5) {
		tmp = t_0;
	} else if (phi1 <= 0.00085) {
		tmp = acos((cos((lambda2 - lambda1)) * fma((phi1 * phi1), -0.5, 1.0))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R)
	tmp = 0.0
	if (phi1 <= -1.5)
		tmp = t_0;
	elseif (phi1 <= 0.00085)
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * fma(Float64(phi1 * phi1), -0.5, 1.0))) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -1.5], t$95$0, If[LessEqual[phi1, 0.00085], N[(N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.5 or 8.49999999999999953e-4 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   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 31.6

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

   7. Applied rewrites 31.6%

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

   if -1.5 < phi1 < 8.49999999999999953e-4

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   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 18.5

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

   7. Applied rewrites 18.5%

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

   9. Applied rewrites 18.5%

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

Alternative 15: 25.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -1.8:\\ \;\;\;\;\cos^{-1} \sin \left(\left(\mathsf{min}\left(\lambda_1, \lambda_2\right) + 0.5 \cdot \pi\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right) - \mathsf{min}\left(\lambda_1, \lambda_2\right)\right) \cdot \mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -1.8)
   (*
    (acos
     (sin (- (+ (fmin lambda1 lambda2) (* 0.5 PI)) (fmax lambda1 lambda2))))
    R)
   (*
    (acos
     (*
      (cos (- (fmax lambda1 lambda2) (fmin lambda1 lambda2)))
      (fma (* (fmin phi1 phi2) (fmin phi1 phi2)) -0.5 1.0)))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -1.8) {
		tmp = acos(sin(((fmin(lambda1, lambda2) + (0.5 * ((double) M_PI))) - fmax(lambda1, lambda2)))) * R;
	} else {
		tmp = acos((cos((fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * fma((fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -1.8)
		tmp = Float64(acos(sin(Float64(Float64(fmin(lambda1, lambda2) + Float64(0.5 * pi)) - fmax(lambda1, lambda2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(fmax(lambda1, lambda2) - fmin(lambda1, lambda2))) * fma(Float64(fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -1.8], N[(N[ArcCos[N[Sin[N[(N[(N[Min[lambda1, lambda2], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[(N[Max[lambda1, lambda2], $MachinePrecision] - N[Min[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.80000000000000004

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-+PI/2-rev N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 19.1%

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

   7. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \sin \left(\left(\lambda_1 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) - \lambda_2\right) \cdot R \]
   8. Step-by-step derivation

      1. lower-sin.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 16.5

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

   9. Applied rewrites 16.5%

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

   if -1.80000000000000004 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   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 18.5

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

   7. Applied rewrites 18.5%

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

   9. Applied rewrites 18.5%

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

Alternative 16: 18.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(\phi_1, \phi_2\right) \leq -6.8 \cdot 10^{-23}:\\ \;\;\;\;\cos^{-1} \sin \left(\left(\mathsf{min}\left(\lambda_1, \lambda_2\right) + 0.5 \cdot \pi\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\mathsf{max}\left(\lambda_1, \lambda_2\right)\right) \cdot \mathsf{fma}\left(\mathsf{min}\left(\phi_1, \phi_2\right) \cdot \mathsf{min}\left(\phi_1, \phi_2\right), -0.5, 1\right)\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (fmin phi1 phi2) -6.8e-23)
   (*
    (acos
     (sin (- (+ (fmin lambda1 lambda2) (* 0.5 PI)) (fmax lambda1 lambda2))))
    R)
   (*
    (acos
     (*
      (cos (fmax lambda1 lambda2))
      (fma (* (fmin phi1 phi2) (fmin phi1 phi2)) -0.5 1.0)))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (fmin(phi1, phi2) <= -6.8e-23) {
		tmp = acos(sin(((fmin(lambda1, lambda2) + (0.5 * ((double) M_PI))) - fmax(lambda1, lambda2)))) * R;
	} else {
		tmp = acos((cos(fmax(lambda1, lambda2)) * fma((fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (fmin(phi1, phi2) <= -6.8e-23)
		tmp = Float64(acos(sin(Float64(Float64(fmin(lambda1, lambda2) + Float64(0.5 * pi)) - fmax(lambda1, lambda2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(fmax(lambda1, lambda2)) * fma(Float64(fmin(phi1, phi2) * fmin(phi1, phi2)), -0.5, 1.0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Min[phi1, phi2], $MachinePrecision], -6.8e-23], N[(N[ArcCos[N[Sin[N[(N[(N[Min[lambda1, lambda2], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[N[Max[lambda1, lambda2], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Min[phi1, phi2], $MachinePrecision] * N[Min[phi1, phi2], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.8000000000000001e-23

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-+PI/2-rev N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 19.1%

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

   7. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \sin \left(\left(\lambda_1 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) - \lambda_2\right) \cdot R \]
   8. Step-by-step derivation

      1. lower-sin.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 16.5

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

   9. Applied rewrites 16.5%

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

   if -6.8000000000000001e-23 < phi1

   1. Initial program 73.5%

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

   2. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   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 18.5

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

   7. Applied rewrites 18.5%

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

   9. Applied rewrites 18.5%

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

   10. Taylor expanded in lambda1 around 0

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

   12. Applied rewrites 11.3%

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

Alternative 17: 16.5% accurate, 3.3× speedup

Math

\[\cos^{-1} \sin \left(\left(\mathsf{min}\left(\lambda_1, \lambda_2\right) + 0.5 \cdot \pi\right) - \mathsf{max}\left(\lambda_1, \lambda_2\right)\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos (sin (- (+ (fmin lambda1 lambda2) (* 0.5 PI)) (fmax lambda1 lambda2))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(sin(((fmin(lambda1, lambda2) + (0.5 * ((double) M_PI))) - fmax(lambda1, lambda2)))) * R;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(Math.sin(((fmin(lambda1, lambda2) + (0.5 * Math.PI)) - fmax(lambda1, lambda2)))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(math.sin(((fmin(lambda1, lambda2) + (0.5 * math.pi)) - fmax(lambda1, lambda2)))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(sin(Float64(Float64(fmin(lambda1, lambda2) + Float64(0.5 * pi)) - fmax(lambda1, lambda2)))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(sin(((min(lambda1, lambda2) + (0.5 * pi)) - max(lambda1, lambda2)))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[Sin[N[(N[(N[Min[lambda1, lambda2], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] - N[Max[lambda1, lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower--.f64 43.1

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

4. Applied rewrites 43.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. sin-+PI/2-rev N/A

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

   5. lift-cos.f64 N/A

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

   6. sin-cos-mult N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 19.1%

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

7. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \sin \left(\left(\lambda_1 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) - \lambda_2\right) \cdot R \]
8. Step-by-step derivation

   1. lower-sin.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 16.5

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

9. Applied rewrites 16.5%

   \[\leadsto \cos^{-1} \sin \left(\left(\lambda_1 + 0.5 \cdot \pi\right) - \lambda_2\right) \cdot R \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025178 
(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))