Spherical law of cosines

Percentage Accurate: 74.1% → 93.5%

Time: 18.1s

Alternatives: 27

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.1%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. 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. sum-to-mult N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-cos.f64 93.5

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

3. Applied rewrites 93.5%

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

5. Applied rewrites 93.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 93.5

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 93.5

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 93.5

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

7. Applied rewrites 93.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 93.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 93.5

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

9. Applied rewrites 93.5%

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

Alternative 2: 93.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 93.5

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

3. Applied rewrites 93.5%

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

Alternative 3: 93.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1)))))) * 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[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 74.1%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. 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 93.5

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

3. Applied rewrites 93.5%

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

Alternative 4: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right), \cos \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \left(1 + \frac{\sin \lambda_1 \cdot \sin \lambda_2}{\cos \lambda_1 \cdot \cos \lambda_2}\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (* (cos lambda1) (* (cos phi1) (cos phi2)))
            (cos lambda2)
            (* (sin phi2) (sin phi1))))
          R)))
   (if (<= phi2 -1.35e-12)
     t_0
     (if (<= phi2 2.95e-5)
       (*
        (acos
         (*
          (cos lambda1)
          (*
           (cos lambda2)
           (*
            (cos phi1)
            (+
             1.0
             (/
              (* (sin lambda1) (sin lambda2))
              (* (cos lambda1) (cos lambda2))))))))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(fma((cos(lambda1) * (cos(phi1) * cos(phi2))), cos(lambda2), (sin(phi2) * sin(phi1)))) * R;
	double tmp;
	if (phi2 <= -1.35e-12) {
		tmp = t_0;
	} else if (phi2 <= 2.95e-5) {
		tmp = acos((cos(lambda1) * (cos(lambda2) * (cos(phi1) * (1.0 + ((sin(lambda1) * sin(lambda2)) / (cos(lambda1) * cos(lambda2)))))))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(Float64(cos(lambda1) * Float64(cos(phi1) * cos(phi2))), cos(lambda2), Float64(sin(phi2) * sin(phi1)))) * R)
	tmp = 0.0
	if (phi2 <= -1.35e-12)
		tmp = t_0;
	elseif (phi2 <= 2.95e-5)
		tmp = Float64(acos(Float64(cos(lambda1) * Float64(cos(lambda2) * Float64(cos(phi1) * Float64(1.0 + Float64(Float64(sin(lambda1) * sin(lambda2)) / Float64(cos(lambda1) * cos(lambda2)))))))) * 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[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -1.35e-12], t$95$0, If[LessEqual[phi2, 2.95e-5], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(1.0 + N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.3499999999999999e-12 or 2.9499999999999999e-5 < phi2

   1. Initial program 74.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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. sum-to-mult N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-cos.f64 93.5

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

   3. Applied rewrites 93.5%

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in lambda2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 74.3

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

   8. Applied rewrites 74.3%

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

   if -1.3499999999999999e-12 < phi2 < 2.9499999999999999e-5

   1. Initial program 74.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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. sum-to-mult N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-cos.f64 93.5

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 52.7%

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

Alternative 5: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right), \cos \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\cos^{-1} \left(\left(\cos \phi_1 \cdot \left(\mathsf{fma}\left(\tan \lambda_1, \tan \lambda_2, 1\right) \cdot \cos \lambda_2\right)\right) \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (* (cos lambda1) (* (cos phi1) (cos phi2)))
            (cos lambda2)
            (* (sin phi2) (sin phi1))))
          R)))
   (if (<= phi2 -1.35e-12)
     t_0
     (if (<= phi2 2.95e-5)
       (*
        (acos
         (*
          (*
           (cos phi1)
           (* (fma (tan lambda1) (tan lambda2) 1.0) (cos lambda2)))
          (cos lambda1)))
        R)
       t_0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(fma((cos(lambda1) * (cos(phi1) * cos(phi2))), cos(lambda2), (sin(phi2) * sin(phi1)))) * R;
	double tmp;
	if (phi2 <= -1.35e-12) {
		tmp = t_0;
	} else if (phi2 <= 2.95e-5) {
		tmp = acos(((cos(phi1) * (fma(tan(lambda1), tan(lambda2), 1.0) * cos(lambda2))) * cos(lambda1))) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(Float64(cos(lambda1) * Float64(cos(phi1) * cos(phi2))), cos(lambda2), Float64(sin(phi2) * sin(phi1)))) * R)
	tmp = 0.0
	if (phi2 <= -1.35e-12)
		tmp = t_0;
	elseif (phi2 <= 2.95e-5)
		tmp = Float64(acos(Float64(Float64(cos(phi1) * Float64(fma(tan(lambda1), tan(lambda2), 1.0) * cos(lambda2))) * cos(lambda1))) * 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[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -1.35e-12], t$95$0, If[LessEqual[phi2, 2.95e-5], N[(N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Tan[lambda1], $MachinePrecision] * N[Tan[lambda2], $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.3499999999999999e-12 or 2.9499999999999999e-5 < phi2

   1. Initial program 74.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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. sum-to-mult N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-cos.f64 93.5

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

   3. Applied rewrites 93.5%

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in lambda2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 74.3

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

   8. Applied rewrites 74.3%

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

   if -1.3499999999999999e-12 < phi2 < 2.9499999999999999e-5

   1. Initial program 74.1%

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

   2. 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. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. sum-to-mult-rev N/A

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

      14. lift-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

   6. Applied rewrites 52.7%

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

Alternative 6: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{-12}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(t\_1, t\_0, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\cos^{-1} \left(\left(\cos \phi_1 \cdot \left(\mathsf{fma}\left(\tan \lambda_1, \tan \lambda_2, 1\right) \cdot \cos \lambda_2\right)\right) \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(-\left(-t\_1\right)\right) \cdot t\_0\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))) (t_1 (* (cos phi2) (cos phi1))))
   (if (<= phi2 -1.35e-12)
     (* (acos (fma t_1 t_0 (* (sin phi2) (sin phi1)))) R)
     (if (<= phi2 2.95e-5)
       (*
        (acos
         (*
          (*
           (cos phi1)
           (* (fma (tan lambda1) (tan lambda2) 1.0) (cos lambda2)))
          (cos lambda1)))
        R)
       (* (acos (fma (sin phi2) (sin phi1) (* (- (- t_1)) t_0))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = cos(phi2) * cos(phi1);
	double tmp;
	if (phi2 <= -1.35e-12) {
		tmp = acos(fma(t_1, t_0, (sin(phi2) * sin(phi1)))) * R;
	} else if (phi2 <= 2.95e-5) {
		tmp = acos(((cos(phi1) * (fma(tan(lambda1), tan(lambda2), 1.0) * cos(lambda2))) * cos(lambda1))) * R;
	} else {
		tmp = acos(fma(sin(phi2), sin(phi1), (-(-t_1) * t_0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(cos(phi2) * cos(phi1))
	tmp = 0.0
	if (phi2 <= -1.35e-12)
		tmp = Float64(acos(fma(t_1, t_0, Float64(sin(phi2) * sin(phi1)))) * R);
	elseif (phi2 <= 2.95e-5)
		tmp = Float64(acos(Float64(Float64(cos(phi1) * Float64(fma(tan(lambda1), tan(lambda2), 1.0) * cos(lambda2))) * cos(lambda1))) * R);
	else
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(-Float64(-t_1)) * t_0))) * 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[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.35e-12], N[(N[ArcCos[N[(t$95$1 * t$95$0 + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 2.95e-5], N[(N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Tan[lambda1], $MachinePrecision] * N[Tan[lambda2], $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[((-(-t$95$1)) * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.3499999999999999e-12

   1. Initial program 74.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 74.1

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

      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 74.1

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

      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 74.1

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

      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 74.1

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

   3. Applied rewrites 74.1%

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

   if -1.3499999999999999e-12 < phi2 < 2.9499999999999999e-5

   1. Initial program 74.1%

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

   2. 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. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. sum-to-mult-rev N/A

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

      14. lift-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

   6. Applied rewrites 52.7%

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

   if 2.9499999999999999e-5 < phi2

   1. Initial program 74.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-neg.f64 74.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 74.1

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

      14. lift-cos.f64 N/A

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

      15. cos-neg-rev N/A

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

      16. lower-cos.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

   3. Applied rewrites 74.1%

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

Alternative 7: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, t\_0\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(-\lambda_2\right), t\_0\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -6.5e-8)
     (* (acos (fma (cos lambda1) (* (cos phi1) (cos phi2)) t_0)) R)
     (* (acos (fma (cos phi1) (* (cos phi2) (cos (- lambda2))) t_0)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -6.5e-8) {
		tmp = acos(fma(cos(lambda1), (cos(phi1) * cos(phi2)), t_0)) * R;
	} else {
		tmp = acos(fma(cos(phi1), (cos(phi2) * cos(-lambda2)), t_0)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -6.5e-8)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(phi1) * cos(phi2)), t_0)) * R);
	else
		tmp = Float64(acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(-lambda2))), t_0)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -6.5e-8], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.49999999999999997e-8

   1. Initial program 74.1%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.0

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

   4. Applied rewrites 54.0%

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

   if -6.49999999999999997e-8 < lambda1

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

Alternative 8: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-neg.f64 74.1

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 74.1

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

   14. lift-cos.f64 N/A

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

   15. cos-neg-rev N/A

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

   16. lower-cos.f64 N/A

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

   17. lift--.f64 N/A

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

   18. sub-negate-rev N/A

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

3. Applied rewrites 74.1%

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

Alternative 9: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := 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]

Derivation

1. Initial program 74.1%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. 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 74.1

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

   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 74.1

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

   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 74.1

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

   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 74.1

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

3. Applied rewrites 74.1%

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

Alternative 10: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \lambda_2, \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -6.5e-8)
   (*
    (acos
     (fma (cos lambda1) (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
    R)
   (*
    (acos
     (fma (* (cos phi1) (cos lambda2)) (cos phi2) (* (sin phi2) (sin phi1))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.5e-8) {
		tmp = acos(fma(cos(lambda1), (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))) * R;
	} else {
		tmp = acos(fma((cos(phi1) * cos(lambda2)), cos(phi2), (sin(phi2) * sin(phi1)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -6.5e-8)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) * R);
	else
		tmp = Float64(acos(fma(Float64(cos(phi1) * cos(lambda2)), cos(phi2), Float64(sin(phi2) * sin(phi1)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -6.5e-8], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.49999999999999997e-8

   1. Initial program 74.1%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.0

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

   4. Applied rewrites 54.0%

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

   if -6.49999999999999997e-8 < lambda1

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lift-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 53.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 53.3

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

   6. Applied rewrites 53.3%

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

Alternative 11: 63.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 74.1

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 74.1

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

   11. lift-cos.f64 N/A

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

   12. cos-neg-rev N/A

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

   13. lower-cos.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-negate-rev N/A

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

   16. lower--.f64 74.1

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

3. Applied rewrites 74.1%

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

Alternative 12: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -6.5e-8)
   (*
    (acos
     (fma (cos lambda1) (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
    R)
   (*
    (acos
     (fma (sin phi2) (sin phi1) (* (* (cos phi2) (cos phi1)) (cos lambda2))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.5e-8) {
		tmp = acos(fma(cos(lambda1), (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))) * R;
	} else {
		tmp = acos(fma(sin(phi2), sin(phi1), ((cos(phi2) * cos(phi1)) * cos(lambda2)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -6.5e-8)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) * R);
	else
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi2) * cos(phi1)) * cos(lambda2)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -6.5e-8], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.49999999999999997e-8

   1. Initial program 74.1%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.0

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

   4. Applied rewrites 54.0%

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

   if -6.49999999999999997e-8 < lambda1

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 53.3

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

   6. Applied rewrites 53.3%

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

Alternative 13: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.0027:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \frac{\sin^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)}{0.5 \cdot \pi}\right) \cdot \left(0.5 \cdot \pi\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 0.0027)
   (*
    (acos
     (fma (cos lambda1) (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
    R)
   (*
    (*
     (- 1.0 (/ (asin (* (cos (- lambda2 lambda1)) (cos phi1))) (* 0.5 PI)))
     (* 0.5 PI))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.0027) {
		tmp = acos(fma(cos(lambda1), (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))) * R;
	} else {
		tmp = ((1.0 - (asin((cos((lambda2 - lambda1)) * cos(phi1))) / (0.5 * ((double) M_PI)))) * (0.5 * ((double) M_PI))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 0.0027)
		tmp = Float64(acos(fma(cos(lambda1), Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) * R);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(asin(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1))) / Float64(0.5 * pi))) * Float64(0.5 * pi)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 0.0027], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(1.0 - N[(N[ArcSin[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.0027000000000000001

   1. Initial program 74.1%

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

   2. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 54.0

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

   4. Applied rewrites 54.0%

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

   if 0.0027000000000000001 < lambda2

   1. Initial program 74.1%

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

   2. 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-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. sub-to-mult N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 43.1%

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

Alternative 14: 56.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.36:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot t\_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.136:\\ \;\;\;\;\cos^{-1} \left(t\_0 + \left(\left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, t\_0\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= phi1 -0.36)
     (* (acos (* (cos phi1) t_1)) R)
     (if (<= phi1 0.136)
       (*
        (acos (+ t_0 (* (* (+ 1.0 (* -0.5 (pow phi1 2.0))) (cos phi2)) t_1)))
        R)
       (* (acos (fma (cos phi1) (cos phi2) t_0)) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.36) {
		tmp = acos((cos(phi1) * t_1)) * R;
	} else if (phi1 <= 0.136) {
		tmp = acos((t_0 + (((1.0 + (-0.5 * pow(phi1, 2.0))) * cos(phi2)) * t_1))) * R;
	} else {
		tmp = acos(fma(cos(phi1), cos(phi2), t_0)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -0.36)
		tmp = Float64(acos(Float64(cos(phi1) * t_1)) * R);
	elseif (phi1 <= 0.136)
		tmp = Float64(acos(Float64(t_0 + Float64(Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) * cos(phi2)) * t_1))) * R);
	else
		tmp = Float64(acos(fma(cos(phi1), cos(phi2), t_0)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.36], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 0.136], N[(N[ArcCos[N[(t$95$0 + N[(N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.35999999999999999

   1. Initial program 74.1%

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

   2. 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 -0.35999999999999999 < phi1 < 0.13600000000000001

   1. Initial program 74.1%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 34.1

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

   4. Applied rewrites 34.1%

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

   if 0.13600000000000001 < phi1

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

   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 32.4

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

   7. Applied rewrites 32.4%

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

Alternative 15: 56.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.62:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 15600:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 -0.62)
     (* (acos (* (cos phi1) t_0)) R)
     (if (<= phi1 15600.0)
       (* (acos (+ (* phi1 (sin phi2)) (* (* (cos phi1) (cos phi2)) t_0))) R)
       (* (acos (fma (cos phi1) (cos phi2) (* (sin phi1) (sin phi2)))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.62) {
		tmp = acos((cos(phi1) * t_0)) * R;
	} else if (phi1 <= 15600.0) {
		tmp = acos(((phi1 * sin(phi2)) + ((cos(phi1) * cos(phi2)) * t_0))) * R;
	} else {
		tmp = acos(fma(cos(phi1), cos(phi2), (sin(phi1) * sin(phi2)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -0.62)
		tmp = Float64(acos(Float64(cos(phi1) * t_0)) * R);
	elseif (phi1 <= 15600.0)
		tmp = Float64(acos(Float64(Float64(phi1 * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))) * R);
	else
		tmp = Float64(acos(fma(cos(phi1), cos(phi2), Float64(sin(phi1) * sin(phi2)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.62], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 15600.0], N[(N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.619999999999999996

   1. Initial program 74.1%

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

   2. 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 -0.619999999999999996 < phi1 < 15600

   1. Initial program 74.1%

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

   2. Taylor expanded in phi1 around 0

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

   4. Applied rewrites 43.8%

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

   if 15600 < phi1

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

   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 32.4

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

   7. Applied rewrites 32.4%

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

Alternative 16: 56.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot t\_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{+21}:\\ \;\;\;\;\cos^{-1} \left(t\_0 + \cos \phi_2 \cdot t\_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, t\_0\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= phi1 -2.9e-8)
     (* (acos (* (cos phi1) t_1)) R)
     (if (<= phi1 1.55e+21)
       (* (acos (+ t_0 (* (cos phi2) t_1))) R)
       (* (acos (fma (cos phi1) (cos phi2) t_0)) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.9e-8) {
		tmp = acos((cos(phi1) * t_1)) * R;
	} else if (phi1 <= 1.55e+21) {
		tmp = acos((t_0 + (cos(phi2) * t_1))) * R;
	} else {
		tmp = acos(fma(cos(phi1), cos(phi2), t_0)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -2.9e-8)
		tmp = Float64(acos(Float64(cos(phi1) * t_1)) * R);
	elseif (phi1 <= 1.55e+21)
		tmp = Float64(acos(Float64(t_0 + Float64(cos(phi2) * t_1))) * R);
	else
		tmp = Float64(acos(fma(cos(phi1), cos(phi2), t_0)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -2.9e-8], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 1.55e+21], N[(N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.9000000000000002e-8

   1. Initial program 74.1%

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

   2. 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 -2.9000000000000002e-8 < phi1 < 1.55e21

   1. Initial program 74.1%

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

   2. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 42.5

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

   4. Applied rewrites 42.5%

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

   if 1.55e21 < phi1

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

   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 32.4

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

   7. Applied rewrites 32.4%

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

Alternative 17: 56.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.055:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot t\_0\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 -2.9e-8)
     (* (acos (* (cos phi1) t_0)) R)
     (if (<= phi1 0.055)
       (* (acos (fma phi1 (sin phi2) (* (cos phi2) t_0))) R)
       (* (acos (fma (cos phi1) (cos phi2) (* (sin phi1) (sin phi2)))) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.9e-8) {
		tmp = acos((cos(phi1) * t_0)) * R;
	} else if (phi1 <= 0.055) {
		tmp = acos(fma(phi1, sin(phi2), (cos(phi2) * t_0))) * R;
	} else {
		tmp = acos(fma(cos(phi1), cos(phi2), (sin(phi1) * sin(phi2)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -2.9e-8)
		tmp = Float64(acos(Float64(cos(phi1) * t_0)) * R);
	elseif (phi1 <= 0.055)
		tmp = Float64(acos(fma(phi1, sin(phi2), Float64(cos(phi2) * t_0))) * R);
	else
		tmp = Float64(acos(fma(cos(phi1), cos(phi2), Float64(sin(phi1) * sin(phi2)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -2.9e-8], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 0.055], N[(N[ArcCos[N[(phi1 * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.9000000000000002e-8

   1. Initial program 74.1%

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

   2. 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 -2.9000000000000002e-8 < phi1 < 0.0550000000000000003

   1. Initial program 74.1%

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

   2. 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 0.0550000000000000003 < phi1

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

   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 32.4

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

   7. Applied rewrites 32.4%

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

Alternative 18: 50.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot t\_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot t\_0\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi2 3.1e-9)
     (* (acos (* (cos phi1) t_0)) R)
     (* (acos (* (cos 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 (phi2 <= 3.1e-9) {
		tmp = acos((cos(phi1) * t_0)) * R;
	} else {
		tmp = acos((cos(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 (phi2 <= 3.1d-9) then
        tmp = acos((cos(phi1) * t_0)) * r
    else
        tmp = acos((cos(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 (phi2 <= 3.1e-9) {
		tmp = Math.acos((Math.cos(phi1) * t_0)) * R;
	} else {
		tmp = Math.acos((Math.cos(phi2) * t_0)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi2 <= 3.1e-9:
		tmp = math.acos((math.cos(phi1) * t_0)) * R
	else:
		tmp = math.acos((math.cos(phi2) * t_0)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= 3.1e-9)
		tmp = Float64(acos(Float64(cos(phi1) * t_0)) * R);
	else
		tmp = Float64(acos(Float64(cos(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 (phi2 <= 3.1e-9)
		tmp = acos((cos(phi1) * t_0)) * R;
	else
		tmp = acos((cos(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[phi2, 3.1e-9], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.10000000000000005e-9

   1. Initial program 74.1%

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

   2. 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 3.10000000000000005e-9 < phi2

   1. Initial program 74.1%

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

   2. 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 42.8

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

   4. Applied rewrites 42.8%

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

Alternative 19: 48.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \left(-\lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.9e-5)
   (* (acos (* (cos phi1) (cos (- lambda1 lambda2)))) R)
   (* (acos (* (cos phi2) (cos (- lambda2)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.9e-5) {
		tmp = acos((cos(phi1) * cos((lambda1 - lambda2)))) * R;
	} else {
		tmp = acos((cos(phi2) * cos(-lambda2))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 2.9d-5) then
        tmp = acos((cos(phi1) * cos((lambda1 - lambda2)))) * r
    else
        tmp = acos((cos(phi2) * cos(-lambda2))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.9e-5) {
		tmp = Math.acos((Math.cos(phi1) * Math.cos((lambda1 - lambda2)))) * R;
	} else {
		tmp = Math.acos((Math.cos(phi2) * Math.cos(-lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.9e-5:
		tmp = math.acos((math.cos(phi1) * math.cos((lambda1 - lambda2)))) * R
	else:
		tmp = math.acos((math.cos(phi2) * math.cos(-lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.9e-5)
		tmp = Float64(acos(Float64(cos(phi1) * cos(Float64(lambda1 - lambda2)))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi2) * cos(Float64(-lambda2)))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.9e-5)
		tmp = acos((cos(phi1) * cos((lambda1 - lambda2)))) * R;
	else
		tmp = acos((cos(phi2) * cos(-lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.9e-5], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.9e-5

   1. Initial program 74.1%

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

   2. 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 2.9e-5 < phi2

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 25.0

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-neg.f64 30.7

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

   10. Applied rewrites 30.7%

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

Alternative 20: 37.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \left(-\lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -6.5e-8)
   (* (acos (* (cos lambda1) (cos phi1))) R)
   (* (acos (* (cos phi1) (cos (- lambda2)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.5e-8) {
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	} else {
		tmp = acos((cos(phi1) * cos(-lambda2))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-6.5d-8)) then
        tmp = acos((cos(lambda1) * cos(phi1))) * r
    else
        tmp = acos((cos(phi1) * cos(-lambda2))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.5e-8) {
		tmp = Math.acos((Math.cos(lambda1) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((Math.cos(phi1) * Math.cos(-lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -6.5e-8:
		tmp = math.acos((math.cos(lambda1) * math.cos(phi1))) * R
	else:
		tmp = math.acos((math.cos(phi1) * math.cos(-lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -6.5e-8)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi1) * cos(Float64(-lambda2)))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -6.5e-8)
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	else
		tmp = acos((cos(phi1) * cos(-lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -6.5e-8], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.49999999999999997e-8

   1. Initial program 74.1%

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

   2. 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 31.5

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

   7. Applied rewrites 31.5%

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

   if -6.49999999999999997e-8 < lambda1

   1. Initial program 74.1%

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

   2. 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 lambda1 around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-neg.f64 31.1

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

   7. Applied rewrites 31.1%

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

Alternative 21: 37.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \left(-\lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2e-7)
   (* (acos (* (cos lambda1) (cos phi1))) R)
   (* (acos (* (cos phi2) (cos (- lambda2)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2e-7) {
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	} else {
		tmp = acos((cos(phi2) * cos(-lambda2))) * R;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-2d-7)) then
        tmp = acos((cos(lambda1) * cos(phi1))) * r
    else
        tmp = acos((cos(phi2) * cos(-lambda2))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2e-7) {
		tmp = Math.acos((Math.cos(lambda1) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos((Math.cos(phi2) * Math.cos(-lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2e-7:
		tmp = math.acos((math.cos(lambda1) * math.cos(phi1))) * R
	else:
		tmp = math.acos((math.cos(phi2) * math.cos(-lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2e-7)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(phi2) * cos(Float64(-lambda2)))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2e-7)
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	else
		tmp = acos((cos(phi2) * cos(-lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2e-7], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.9999999999999999e-7

   1. Initial program 74.1%

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

   2. 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 31.5

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

   7. Applied rewrites 31.5%

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

   if -1.9999999999999999e-7 < lambda1

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 25.0

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-neg.f64 30.7

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

   10. Applied rewrites 30.7%

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

Alternative 22: 27.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.36:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.36)
   (* (acos (* (cos lambda1) (cos phi1))) R)
   (* (acos (* (cos (- lambda2 lambda1)) (fma (* phi1 phi1) -0.5 1.0))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.36) {
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	} else {
		tmp = acos((cos((lambda2 - lambda1)) * fma((phi1 * phi1), -0.5, 1.0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.36)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * fma(Float64(phi1 * phi1), -0.5, 1.0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.36], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.35999999999999999

   1. Initial program 74.1%

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

   2. 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 31.5

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

   7. Applied rewrites 31.5%

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

   if -0.35999999999999999 < phi1

   1. Initial program 74.1%

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

   2. 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 18.1

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

   7. Applied rewrites 18.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 18.1

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift--.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lift-cos.f64 18.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 18.1

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 18.1

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

   9. Applied rewrites 18.1%

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

Alternative 23: 18.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;\cos^{-1} \left(\left(1 + \frac{\cos \lambda_2}{\phi_2 \cdot \phi_1}\right) \cdot \left(\phi_2 \cdot \phi_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.3e+24)
   (* (acos (* (+ 1.0 (/ (cos lambda2) (* phi2 phi1))) (* phi2 phi1))) R)
   (* (acos (* (cos (- lambda2 lambda1)) (fma (* phi1 phi1) -0.5 1.0))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.3e+24) {
		tmp = acos(((1.0 + (cos(lambda2) / (phi2 * phi1))) * (phi2 * phi1))) * R;
	} else {
		tmp = acos((cos((lambda2 - lambda1)) * fma((phi1 * phi1), -0.5, 1.0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.3e+24)
		tmp = Float64(acos(Float64(Float64(1.0 + Float64(cos(lambda2) / Float64(phi2 * phi1))) * Float64(phi2 * phi1))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * fma(Float64(phi1 * phi1), -0.5, 1.0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.3e+24], N[(N[ArcCos[N[(N[(1.0 + N[(N[Cos[lambda2], $MachinePrecision] / N[(phi2 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(phi2 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.2999999999999999e24

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 25.0

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 11.0

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

   10. Applied rewrites 11.0%

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

       1. lift-+.f64 N/A

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

       2. lift-cos.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. cos-neg-rev N/A

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

       5. lift-cos.f64 N/A

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

       6. +-commutative N/A

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

       7. sum-to-mult N/A

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

       8. lower-*.f64 N/A

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

       9. lower-+.f64 N/A

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

       10. lower-/.f64 7.1

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 7.1

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 7.1

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

   12. Applied rewrites 7.1%

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

   if -2.2999999999999999e24 < phi1

   1. Initial program 74.1%

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

   2. 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 18.1

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

   7. Applied rewrites 18.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 18.1

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift--.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lift-cos.f64 18.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 18.1

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 18.1

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

   9. Applied rewrites 18.1%

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

Alternative 24: 18.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \left(\phi_2 + \frac{\cos \left(-\lambda_2\right)}{\phi_1}\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.3e+24)
   (* (acos (* phi1 (+ phi2 (/ (cos (- lambda2)) phi1)))) R)
   (* (acos (* (cos (- lambda2 lambda1)) (fma (* phi1 phi1) -0.5 1.0))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.3e+24) {
		tmp = acos((phi1 * (phi2 + (cos(-lambda2) / phi1)))) * R;
	} else {
		tmp = acos((cos((lambda2 - lambda1)) * fma((phi1 * phi1), -0.5, 1.0))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.3e+24)
		tmp = Float64(acos(Float64(phi1 * Float64(phi2 + Float64(cos(Float64(-lambda2)) / phi1)))) * R);
	else
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * fma(Float64(phi1 * phi1), -0.5, 1.0))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.3e+24], N[(N[ArcCos[N[(phi1 * N[(phi2 + N[(N[Cos[(-lambda2)], $MachinePrecision] / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.2999999999999999e24

   1. Initial program 74.1%

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

   2. 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 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 25.0

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 11.0

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

   10. Applied rewrites 11.0%

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

   11. Taylor expanded in phi1 around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower-neg.f64 11.2

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

   13. Applied rewrites 11.2%

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

   if -2.2999999999999999e24 < phi1

   1. Initial program 74.1%

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

   2. 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 18.1

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

   7. Applied rewrites 18.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 18.1

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

      4. lift-cos.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. lift--.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lift-cos.f64 18.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 18.1

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 18.1

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

   9. Applied rewrites 18.1%

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

Alternative 25: 11.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

2. 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 53.3

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

4. Applied rewrites 53.3%

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

5. Taylor expanded in phi2 around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-neg.f64 25.0

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

7. Applied rewrites 25.0%

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

8. Taylor expanded in phi1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-*.f64 11.0

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

10. Applied rewrites 11.0%

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

11. Taylor expanded in phi1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-neg.f64 11.2

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

13. Applied rewrites 11.2%

    \[\leadsto \cos^{-1} \left(\phi_1 \cdot \left(\phi_2 + \frac{\cos \left(-\lambda_2\right)}{\color{blue}{\phi_1}}\right)\right) \cdot R \]
14. Add Preprocessing

Alternative 26: 11.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\mathsf{fma}\left(\phi_2, \phi_1, \cos \lambda_2\right)\right) \cdot R \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

2. 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 53.3

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

4. Applied rewrites 53.3%

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

5. Taylor expanded in phi2 around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-neg.f64 25.0

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

7. Applied rewrites 25.0%

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

8. Taylor expanded in phi1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-*.f64 11.0

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

10. Applied rewrites 11.0%

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

12. Applied rewrites 11.0%

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

Alternative 27: 2.5% accurate, 13.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

2. 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 53.3

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

4. Applied rewrites 53.3%

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

5. Taylor expanded in phi2 around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-neg.f64 25.0

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

7. Applied rewrites 25.0%

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

8. Taylor expanded in phi1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-*.f64 11.0

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

10. Applied rewrites 11.0%

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

11. Taylor expanded in lambda2 around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 2.5

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

13. Applied rewrites 2.5%

    \[\leadsto \cos^{-1} \left(1 + \phi_1 \cdot \phi_2\right) \cdot R \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(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))