Bearing on a great circle

Percentage Accurate: 78.5% → 99.7%

Time: 18.2s

Alternatives: 25

Speedup: 1.0×

Specification

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}

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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 78.5% accurate, 1.0× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}

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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (fma (sin lambda1) (cos lambda2) (* (- (cos lambda1)) (sin lambda2)))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (sin phi1)
    (fma
     (* (cos phi2) (sin lambda1))
     (sin lambda2)
     (* (* (cos phi2) (cos lambda1)) (cos lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((fma(sin(lambda1), cos(lambda2), (-cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (sin(phi1) * fma((cos(phi2) * sin(lambda1)), sin(lambda2), ((cos(phi2) * cos(lambda1)) * cos(lambda2))))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

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

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

   6. lower-unsound--.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 89.3%

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

3. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-*.f64 99.7%

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

5. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

7. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-to-mult-rev N/A

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

   5. sub-flip N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. lower-*.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 99.7%

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

9. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (- (* (cos lambda2) (sin lambda1)) (* (cos lambda1) (sin lambda2)))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (sin phi1)
    (fma
     (* (cos phi2) (sin lambda1))
     (sin lambda2)
     (* (* (cos phi2) (cos lambda1)) (cos lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((((cos(lambda2) * sin(lambda1)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (sin(phi1) * fma((cos(phi2) * sin(lambda1)), sin(lambda2), ((cos(phi2) * cos(lambda1)) * cos(lambda2))))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. sub-to-mult N/A

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

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

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

   6. lower-unsound--.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 89.3%

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

3. Applied rewrites 89.3%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift--.f64 N/A

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

   4. cos-diff N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-*.f64 99.7%

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

5. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

7. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-to-mult-rev N/A

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

   5. lower--.f64 99.7%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.7%

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7%

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

9. Applied rewrites 99.7%

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

Alternative 3: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{t\_0}\right) \cdot t\_0\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{-78}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_1}\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-66}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{-1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \sin \phi_1, \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \sin \phi_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, t\_0\right) \cdot \cos \phi_2}{t\_1}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda1) (cos lambda2)))
        (t_1
         (-
          (* (cos phi1) (sin phi2))
          (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        (t_2
         (*
          (* (- 1.0 (/ (* (sin lambda2) (cos lambda1)) t_0)) t_0)
          (cos phi2))))
   (if (<= phi2 -4.2e-78)
     (atan2 t_2 t_1)
     (if (<= phi2 6.5e-66)
       (atan2
        t_2
        (*
         -1.0
         (fma
          (cos lambda1)
          (* (cos lambda2) (sin phi1))
          (* (sin lambda1) (* (sin lambda2) (sin phi1))))))
       (atan2 (* (fma (- (sin lambda2)) (cos lambda1) t_0) (cos phi2)) t_1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(lambda2);
	double t_1 = (cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)));
	double t_2 = ((1.0 - ((sin(lambda2) * cos(lambda1)) / t_0)) * t_0) * cos(phi2);
	double tmp;
	if (phi2 <= -4.2e-78) {
		tmp = atan2(t_2, t_1);
	} else if (phi2 <= 6.5e-66) {
		tmp = atan2(t_2, (-1.0 * fma(cos(lambda1), (cos(lambda2) * sin(phi1)), (sin(lambda1) * (sin(lambda2) * sin(phi1))))));
	} else {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), t_0) * cos(phi2)), t_1);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(lambda2))
	t_1 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))
	t_2 = Float64(Float64(Float64(1.0 - Float64(Float64(sin(lambda2) * cos(lambda1)) / t_0)) * t_0) * cos(phi2))
	tmp = 0.0
	if (phi2 <= -4.2e-78)
		tmp = atan(t_2, t_1);
	elseif (phi2 <= 6.5e-66)
		tmp = atan(t_2, Float64(-1.0 * fma(cos(lambda1), Float64(cos(lambda2) * sin(phi1)), Float64(sin(lambda1) * Float64(sin(lambda2) * sin(phi1))))));
	else
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), t_0) * cos(phi2)), t_1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1.0 - N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.2e-78], N[ArcTan[t$95$2 / t$95$1], $MachinePrecision], If[LessEqual[phi2, 6.5e-66], N[ArcTan[t$95$2 / N[(-1.0 * N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.2000000000000001e-78

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

   if -4.2000000000000001e-78 < phi2 < 6.5000000000000002e-66

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 58.1%

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

   8. Applied rewrites 58.1%

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

   if 6.5000000000000002e-66 < phi2

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

Alternative 4: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{t\_0}\right) \cdot t\_0\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{-78}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_1}\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-66}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{-1 \cdot \left(\sin \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, t\_0\right) \cdot \cos \phi_2}{t\_1}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda1) (cos lambda2)))
        (t_1
         (-
          (* (cos phi1) (sin phi2))
          (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        (t_2
         (*
          (* (- 1.0 (/ (* (sin lambda2) (cos lambda1)) t_0)) t_0)
          (cos phi2))))
   (if (<= phi2 -4.2e-78)
     (atan2 t_2 t_1)
     (if (<= phi2 6.5e-66)
       (atan2
        t_2
        (*
         -1.0
         (*
          (sin phi1)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
       (atan2 (* (fma (- (sin lambda2)) (cos lambda1) t_0) (cos phi2)) t_1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(lambda2);
	double t_1 = (cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)));
	double t_2 = ((1.0 - ((sin(lambda2) * cos(lambda1)) / t_0)) * t_0) * cos(phi2);
	double tmp;
	if (phi2 <= -4.2e-78) {
		tmp = atan2(t_2, t_1);
	} else if (phi2 <= 6.5e-66) {
		tmp = atan2(t_2, (-1.0 * (sin(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), t_0) * cos(phi2)), t_1);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(lambda2))
	t_1 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))
	t_2 = Float64(Float64(Float64(1.0 - Float64(Float64(sin(lambda2) * cos(lambda1)) / t_0)) * t_0) * cos(phi2))
	tmp = 0.0
	if (phi2 <= -4.2e-78)
		tmp = atan(t_2, t_1);
	elseif (phi2 <= 6.5e-66)
		tmp = atan(t_2, Float64(-1.0 * Float64(sin(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), t_0) * cos(phi2)), t_1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1.0 - N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.2e-78], N[ArcTan[t$95$2 / t$95$1], $MachinePrecision], If[LessEqual[phi2, 6.5e-66], N[ArcTan[t$95$2 / N[(-1.0 * N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.2000000000000001e-78

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

   if -4.2000000000000001e-78 < phi2 < 6.5000000000000002e-66

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 58.1%

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

   10. Applied rewrites 58.1%

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

   if 6.5000000000000002e-66 < phi2

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

Alternative 5: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{-78}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \left(1 - \frac{\cos \lambda_1 \cdot \sin \lambda_2}{\cos \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)}{t\_0}\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-66}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{t\_1}\right) \cdot t\_1\right) \cdot \cos \phi_2}{-1 \cdot \left(\sin \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, t\_1\right) \cdot \cos \phi_2}{t\_0}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (-
          (* (cos phi1) (sin phi2))
          (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        (t_1 (* (sin lambda1) (cos lambda2))))
   (if (<= phi2 -4.2e-78)
     (atan2
      (*
       (cos lambda2)
       (*
        (cos phi2)
        (*
         (sin lambda1)
         (-
          1.0
          (/
           (* (cos lambda1) (sin lambda2))
           (* (cos lambda2) (sin lambda1)))))))
      t_0)
     (if (<= phi2 6.5e-66)
       (atan2
        (* (* (- 1.0 (/ (* (sin lambda2) (cos lambda1)) t_1)) t_1) (cos phi2))
        (*
         -1.0
         (*
          (sin phi1)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
       (atan2 (* (fma (- (sin lambda2)) (cos lambda1) t_1) (cos phi2)) t_0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)));
	double t_1 = sin(lambda1) * cos(lambda2);
	double tmp;
	if (phi2 <= -4.2e-78) {
		tmp = atan2((cos(lambda2) * (cos(phi2) * (sin(lambda1) * (1.0 - ((cos(lambda1) * sin(lambda2)) / (cos(lambda2) * sin(lambda1))))))), t_0);
	} else if (phi2 <= 6.5e-66) {
		tmp = atan2((((1.0 - ((sin(lambda2) * cos(lambda1)) / t_1)) * t_1) * cos(phi2)), (-1.0 * (sin(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), t_1) * cos(phi2)), t_0);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))
	t_1 = Float64(sin(lambda1) * cos(lambda2))
	tmp = 0.0
	if (phi2 <= -4.2e-78)
		tmp = atan(Float64(cos(lambda2) * Float64(cos(phi2) * Float64(sin(lambda1) * Float64(1.0 - Float64(Float64(cos(lambda1) * sin(lambda2)) / Float64(cos(lambda2) * sin(lambda1))))))), t_0);
	elseif (phi2 <= 6.5e-66)
		tmp = atan(Float64(Float64(Float64(1.0 - Float64(Float64(sin(lambda2) * cos(lambda1)) / t_1)) * t_1) * cos(phi2)), Float64(-1.0 * Float64(sin(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), t_1) * cos(phi2)), t_0);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.2e-78], N[ArcTan[N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[(1.0 - N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision], If[LessEqual[phi2, 6.5e-66], N[ArcTan[N[(N[(N[(1.0 - N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + t$95$1), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.2000000000000001e-78

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

   4. Taylor expanded in lambda1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \left(1 - \frac{\cos \lambda_1 \cdot \sin \lambda_2}{\cos \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      12. lower-*.f64 N/A

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

   6. Applied rewrites 89.3%

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

   if -4.2000000000000001e-78 < phi2 < 6.5000000000000002e-66

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 58.1%

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

   10. Applied rewrites 58.1%

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

   if 6.5000000000000002e-66 < phi2

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

Alternative 6: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \cos \lambda_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{-78}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(t\_2 - t\_0\right) \cdot \cos \phi_2}{t\_1}\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-66}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\left(1 - \frac{t\_0}{t\_2}\right) \cdot t\_2\right) \cdot \cos \phi_2}{-1 \cdot \left(\sin \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, t\_2\right) \cdot \cos \phi_2}{t\_1}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda2) (cos lambda1)))
        (t_1
         (-
          (* (cos phi1) (sin phi2))
          (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        (t_2 (* (sin lambda1) (cos lambda2))))
   (if (<= phi2 -4.2e-78)
     (atan2 (* (- t_2 t_0) (cos phi2)) t_1)
     (if (<= phi2 6.5e-66)
       (atan2
        (* (* (- 1.0 (/ t_0 t_2)) t_2) (cos phi2))
        (*
         -1.0
         (*
          (sin phi1)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
       (atan2 (* (fma (- (sin lambda2)) (cos lambda1) t_2) (cos phi2)) t_1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda2) * cos(lambda1);
	double t_1 = (cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)));
	double t_2 = sin(lambda1) * cos(lambda2);
	double tmp;
	if (phi2 <= -4.2e-78) {
		tmp = atan2(((t_2 - t_0) * cos(phi2)), t_1);
	} else if (phi2 <= 6.5e-66) {
		tmp = atan2((((1.0 - (t_0 / t_2)) * t_2) * cos(phi2)), (-1.0 * (sin(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2((fma(-sin(lambda2), cos(lambda1), t_2) * cos(phi2)), t_1);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda2) * cos(lambda1))
	t_1 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))
	t_2 = Float64(sin(lambda1) * cos(lambda2))
	tmp = 0.0
	if (phi2 <= -4.2e-78)
		tmp = atan(Float64(Float64(t_2 - t_0) * cos(phi2)), t_1);
	elseif (phi2 <= 6.5e-66)
		tmp = atan(Float64(Float64(Float64(1.0 - Float64(t_0 / t_2)) * t_2) * cos(phi2)), Float64(-1.0 * Float64(sin(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(Float64(fma(Float64(-sin(lambda2)), cos(lambda1), t_2) * cos(phi2)), t_1);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.2e-78], N[ArcTan[N[(N[(t$95$2 - t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision], If[LessEqual[phi2, 6.5e-66], N[ArcTan[N[(N[(N[(1.0 - N[(t$95$0 / t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + t$95$2), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.2000000000000001e-78

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

   if -4.2000000000000001e-78 < phi2 < 6.5000000000000002e-66

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 58.1%

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

   10. Applied rewrites 58.1%

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

   if 6.5000000000000002e-66 < phi2

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sin-sum N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sin-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

Alternative 7: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \left(-\lambda_2\right)}\\ \mathbf{if}\;\lambda_2 \leq -0.4:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 9.2 \cdot 10^{-116}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos phi2)))
        (t_2
         (atan2
          (*
           (- (* (cos lambda2) (sin lambda1)) (* (cos lambda1) (sin lambda2)))
           (cos phi2))
          (- t_0 (* t_1 (cos (- lambda2)))))))
   (if (<= lambda2 -0.4)
     t_2
     (if (<= lambda2 9.2e-116)
       (atan2
        (* (sin (- lambda1 lambda2)) (cos phi2))
        (- t_0 (* t_1 (cos lambda1))))
       t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(phi1) * cos(phi2);
	double t_2 = atan2((((cos(lambda2) * sin(lambda1)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (t_1 * cos(-lambda2))));
	double tmp;
	if (lambda2 <= -0.4) {
		tmp = t_2;
	} else if (lambda2 <= 9.2e-116) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * cos(lambda1))));
	} else {
		tmp = t_2;
	}
	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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin(phi1) * cos(phi2)
    t_2 = atan2((((cos(lambda2) * sin(lambda1)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (t_1 * cos(-lambda2))))
    if (lambda2 <= (-0.4d0)) then
        tmp = t_2
    else if (lambda2 <= 9.2d-116) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * cos(lambda1))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin(phi1) * Math.cos(phi2);
	double t_2 = Math.atan2((((Math.cos(lambda2) * Math.sin(lambda1)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos(-lambda2))));
	double tmp;
	if (lambda2 <= -0.4) {
		tmp = t_2;
	} else if (lambda2 <= 9.2e-116) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_0 - (t_1 * Math.cos(lambda1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.sin(phi1) * math.cos(phi2)
	t_2 = math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (t_0 - (t_1 * math.cos(-lambda2))))
	tmp = 0
	if lambda2 <= -0.4:
		tmp = t_2
	elif lambda2 <= 9.2e-116:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_0 - (t_1 * math.cos(lambda1))))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(phi2))
	t_2 = atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(Float64(-lambda2)))))
	tmp = 0.0
	if (lambda2 <= -0.4)
		tmp = t_2;
	elseif (lambda2 <= 9.2e-116)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_0 - Float64(t_1 * cos(lambda1))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = sin(phi1) * cos(phi2);
	t_2 = atan2((((cos(lambda2) * sin(lambda1)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (t_1 * cos(-lambda2))));
	tmp = 0.0;
	if (lambda2 <= -0.4)
		tmp = t_2;
	elseif (lambda2 <= 9.2e-116)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_0 - (t_1 * cos(lambda1))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -0.4], t$95$2, If[LessEqual[lambda2, 9.2e-116], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -0.40000000000000002 or 9.2000000000000001e-116 < lambda2

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

   4. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \]

      2. lower-neg.f64 79.3%

         \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\lambda_2\right)} \]

   6. Applied rewrites 79.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-to-mult-rev N/A

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

      5. lower--.f64 79.4%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 79.4%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.4%

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

   8. Applied rewrites 79.4%

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

   if -0.40000000000000002 < lambda2 < 9.2000000000000001e-116

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 68.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   4. Applied rewrites 68.9%

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

Alternative 8: 89.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (fma (- (sin lambda2)) (cos lambda1) (* (sin lambda1) (cos lambda2)))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((fma(-sin(lambda2), cos(lambda1), (sin(lambda1) * cos(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. +-commutative N/A

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

   5. sin-sum N/A

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

   6. cos-neg-rev N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. sin-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-cos.f64 89.3%

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

3. Applied rewrites 89.3%

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

Alternative 9: 87.5% accurate, 0.7× speedup

Math

\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1)))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}

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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1))) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1))) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. lift-sin.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sin-diff N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 89.3%

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

3. Applied rewrites 89.3%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
4. Add Preprocessing

Alternative 10: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{-23}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(t\_1 \cdot \cos \phi_2\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\phi_1 \leq 12500000000:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{t\_0}\right) \cdot t\_0\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\sin \phi_1 \cdot \cos \phi_2, -t\_1, \sin \phi_2 \cdot \cos \phi_1\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda1) (cos lambda2)))
        (t_1 (cos (- lambda2 lambda1)))
        (t_2 (* (sin (- lambda1 lambda2)) (cos phi2))))
   (if (<= phi1 -4.8e-23)
     (atan2
      t_2
      (- (* (cos phi1) (sin phi2)) (* (* t_1 (cos phi2)) (sin phi1))))
     (if (<= phi1 12500000000.0)
       (atan2
        (* (* (- 1.0 (/ (* (sin lambda2) (cos lambda1)) t_0)) t_0) (cos phi2))
        (sin phi2))
       (atan2
        t_2
        (fma (* (sin phi1) (cos phi2)) (- t_1) (* (sin phi2) (cos phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(lambda2);
	double t_1 = cos((lambda2 - lambda1));
	double t_2 = sin((lambda1 - lambda2)) * cos(phi2);
	double tmp;
	if (phi1 <= -4.8e-23) {
		tmp = atan2(t_2, ((cos(phi1) * sin(phi2)) - ((t_1 * cos(phi2)) * sin(phi1))));
	} else if (phi1 <= 12500000000.0) {
		tmp = atan2((((1.0 - ((sin(lambda2) * cos(lambda1)) / t_0)) * t_0) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_2, fma((sin(phi1) * cos(phi2)), -t_1, (sin(phi2) * cos(phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(lambda2))
	t_1 = cos(Float64(lambda2 - lambda1))
	t_2 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -4.8e-23)
		tmp = atan(t_2, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(t_1 * cos(phi2)) * sin(phi1))));
	elseif (phi1 <= 12500000000.0)
		tmp = atan(Float64(Float64(Float64(1.0 - Float64(Float64(sin(lambda2) * cos(lambda1)) / t_0)) * t_0) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_2, fma(Float64(sin(phi1) * cos(phi2)), Float64(-t_1), Float64(sin(phi2) * cos(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -4.8e-23], N[ArcTan[t$95$2 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 12500000000.0], N[ArcTan[N[(N[(N[(1.0 - N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * (-t$95$1) + N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.7999999999999999e-23

   1. Initial program 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.5%

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 78.5%

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

   3. Applied rewrites 78.5%

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

   if -4.7999999999999999e-23 < phi1 < 1.25e10

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. cos-diff N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
   7. Step-by-step derivation

      1. lower-sin.f64 58.3%

         \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   8. Applied rewrites 58.3%

      \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if 1.25e10 < phi1

   1. Initial program 78.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 78.5%

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

      9. lift-cos.f64 N/A

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

      10. cos-neg-rev N/A

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

      11. lower-cos.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 78.5%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 78.5%

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

   3. Applied rewrites 78.5%

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

Alternative 11: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t\_1 - t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\lambda_2 \leq -6.2 \cdot 10^{-10}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 0.084:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t\_1 - t\_0 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (cos phi2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2
         (atan2
          (* (sin (- lambda2)) (cos phi2))
          (- t_1 (* t_0 (cos (- lambda1 lambda2)))))))
   (if (<= lambda2 -6.2e-10)
     t_2
     (if (<= lambda2 0.084)
       (atan2
        (* (sin (- lambda1 lambda2)) (cos phi2))
        (- t_1 (* t_0 (cos lambda1))))
       t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * cos(phi2);
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = atan2((sin(-lambda2) * cos(phi2)), (t_1 - (t_0 * cos((lambda1 - lambda2)))));
	double tmp;
	if (lambda2 <= -6.2e-10) {
		tmp = t_2;
	} else if (lambda2 <= 0.084) {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - (t_0 * cos(lambda1))));
	} else {
		tmp = t_2;
	}
	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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(phi1) * cos(phi2)
    t_1 = cos(phi1) * sin(phi2)
    t_2 = atan2((sin(-lambda2) * cos(phi2)), (t_1 - (t_0 * cos((lambda1 - lambda2)))))
    if (lambda2 <= (-6.2d-10)) then
        tmp = t_2
    else if (lambda2 <= 0.084d0) then
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - (t_0 * cos(lambda1))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.cos(phi2);
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double t_2 = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), (t_1 - (t_0 * Math.cos((lambda1 - lambda2)))));
	double tmp;
	if (lambda2 <= -6.2e-10) {
		tmp = t_2;
	} else if (lambda2 <= 0.084) {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (t_1 - (t_0 * Math.cos(lambda1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.cos(phi2)
	t_1 = math.cos(phi1) * math.sin(phi2)
	t_2 = math.atan2((math.sin(-lambda2) * math.cos(phi2)), (t_1 - (t_0 * math.cos((lambda1 - lambda2)))))
	tmp = 0
	if lambda2 <= -6.2e-10:
		tmp = t_2
	elif lambda2 <= 0.084:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (t_1 - (t_0 * math.cos(lambda1))))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * cos(phi2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), Float64(t_1 - Float64(t_0 * cos(Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (lambda2 <= -6.2e-10)
		tmp = t_2;
	elseif (lambda2 <= 0.084)
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(t_1 - Float64(t_0 * cos(lambda1))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * cos(phi2);
	t_1 = cos(phi1) * sin(phi2);
	t_2 = atan2((sin(-lambda2) * cos(phi2)), (t_1 - (t_0 * cos((lambda1 - lambda2)))));
	tmp = 0.0;
	if (lambda2 <= -6.2e-10)
		tmp = t_2;
	elseif (lambda2 <= 0.084)
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (t_1 - (t_0 * cos(lambda1))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -6.2e-10], t$95$2, If[LessEqual[lambda2, 0.084], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(t$95$0 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -6.2000000000000003e-10 or 0.084000000000000005 < lambda2

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   3. Step-by-step derivation

      1. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. lower-neg.f64 47.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Applied rewrites 47.6%

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

   if -6.2000000000000003e-10 < lambda2 < 0.084000000000000005

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 68.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   4. Applied rewrites 68.9%

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

Alternative 12: 78.4% accurate, 1.0× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (cos (- lambda2 lambda1)) (cos phi2)) (sin phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
}

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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos((lambda2 - lambda1)) * Math.cos(phi2)) * Math.sin(phi1))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.cos((lambda2 - lambda1)) * math.cos(phi2)) * math.sin(phi1))))

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos((lambda2 - lambda1)) * cos(phi2)) * sin(phi1))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 78.5%

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

   8. lift-cos.f64 N/A

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

   9. cos-neg-rev N/A

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

   10. lower-cos.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lower--.f64 78.5%

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

3. Applied rewrites 78.5%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right) \cdot \sin \phi_1}} \]
4. Add Preprocessing

Alternative 13: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t\_1}{t\_0 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -0.031:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 9.2 \cdot 10^{-116}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin (- lambda1 lambda2)) (cos phi2)))
        (t_2 (atan2 t_1 (- t_0 (* (* (cos lambda2) (sin phi1)) (cos phi2))))))
   (if (<= lambda2 -0.031)
     t_2
     (if (<= lambda2 9.2e-116)
       (atan2 t_1 (- t_0 (* (* (sin phi1) (cos phi2)) (cos lambda1))))
       t_2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	double t_2 = atan2(t_1, (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	double tmp;
	if (lambda2 <= -0.031) {
		tmp = t_2;
	} else if (lambda2 <= 9.2e-116) {
		tmp = atan2(t_1, (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda1))));
	} else {
		tmp = t_2;
	}
	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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin((lambda1 - lambda2)) * cos(phi2)
    t_2 = atan2(t_1, (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))))
    if (lambda2 <= (-0.031d0)) then
        tmp = t_2
    else if (lambda2 <= 9.2d-116) then
        tmp = atan2(t_1, (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda1))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin((lambda1 - lambda2)) * Math.cos(phi2);
	double t_2 = Math.atan2(t_1, (t_0 - ((Math.cos(lambda2) * Math.sin(phi1)) * Math.cos(phi2))));
	double tmp;
	if (lambda2 <= -0.031) {
		tmp = t_2;
	} else if (lambda2 <= 9.2e-116) {
		tmp = Math.atan2(t_1, (t_0 - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos(lambda1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.sin((lambda1 - lambda2)) * math.cos(phi2)
	t_2 = math.atan2(t_1, (t_0 - ((math.cos(lambda2) * math.sin(phi1)) * math.cos(phi2))))
	tmp = 0
	if lambda2 <= -0.031:
		tmp = t_2
	elif lambda2 <= 9.2e-116:
		tmp = math.atan2(t_1, (t_0 - ((math.sin(phi1) * math.cos(phi2)) * math.cos(lambda1))))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2))
	t_2 = atan(t_1, Float64(t_0 - Float64(Float64(cos(lambda2) * sin(phi1)) * cos(phi2))))
	tmp = 0.0
	if (lambda2 <= -0.031)
		tmp = t_2;
	elseif (lambda2 <= 9.2e-116)
		tmp = atan(t_1, Float64(t_0 - Float64(Float64(sin(phi1) * cos(phi2)) * cos(lambda1))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = sin((lambda1 - lambda2)) * cos(phi2);
	t_2 = atan2(t_1, (t_0 - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	tmp = 0.0;
	if (lambda2 <= -0.031)
		tmp = t_2;
	elseif (lambda2 <= 9.2e-116)
		tmp = atan2(t_1, (t_0 - ((sin(phi1) * cos(phi2)) * cos(lambda1))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -0.031], t$95$2, If[LessEqual[lambda2, 9.2e-116], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -0.031 or 9.2000000000000001e-116 < lambda2

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

   4. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \]

      2. lower-neg.f64 79.3%

         \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\lambda_2\right)} \]

   6. Applied rewrites 79.3%

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

   8. Applied rewrites 68.6%

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

   if -0.031 < lambda2 < 9.2000000000000001e-116

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 68.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} \]

   4. Applied rewrites 68.9%

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

Alternative 14: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \cos \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.00066:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 3.7 \cdot 10^{-41}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1
         (atan2
          (* t_0 (cos phi2))
          (-
           (* (cos phi1) (sin phi2))
           (* (* (cos lambda2) (sin phi1)) (cos phi2))))))
   (if (<= phi2 -0.00066)
     t_1
     (if (<= phi2 3.7e-41)
       (atan2
        (* t_0 (fma (* phi2 phi2) -0.5 1.0))
        (- (* phi2 (cos phi1)) (* (cos (- lambda1 lambda2)) (sin phi1))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((t_0 * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(lambda2) * sin(phi1)) * cos(phi2))));
	double tmp;
	if (phi2 <= -0.00066) {
		tmp = t_1;
	} else if (phi2 <= 3.7e-41) {
		tmp = atan2((t_0 * fma((phi2 * phi2), -0.5, 1.0)), ((phi2 * cos(phi1)) - (cos((lambda1 - lambda2)) * sin(phi1))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(t_0 * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(lambda2) * sin(phi1)) * cos(phi2))))
	tmp = 0.0
	if (phi2 <= -0.00066)
		tmp = t_1;
	elseif (phi2 <= 3.7e-41)
		tmp = atan(Float64(t_0 * fma(Float64(phi2 * phi2), -0.5, 1.0)), Float64(Float64(phi2 * cos(phi1)) - Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.00066], t$95$1, If[LessEqual[phi2, 3.7e-41], N[ArcTan[N[(t$95$0 * N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -6.6e-4 or 3.7000000000000002e-41 < phi2

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sin-diff N/A

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

      4. sub-to-mult N/A

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

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

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-cos.f64 89.3%

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

   3. Applied rewrites 89.3%

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

   4. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \]

      2. lower-neg.f64 79.3%

         \[\leadsto \tan^{-1}_* \frac{\left(\left(1 - \frac{\sin \lambda_2 \cdot \cos \lambda_1}{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\lambda_2\right)} \]

   6. Applied rewrites 79.3%

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

   8. Applied rewrites 68.6%

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

   if -6.6e-4 < phi2 < 3.7000000000000002e-41

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{\sin \phi_2} \]

      3. lower-pow.f64 29.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{\sin \phi_2} \]

   7. Applied rewrites 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{\sin \phi_2} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\sin \phi_2} \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right)}{\sin \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      5. lift-pow.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      7. lower-*.f32 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      8. lower-unsound-*.f32 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{-1}{2}}, 1\right)}{\sin \phi_2} \]

      10. lower-unsound-*.f64 29.1%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\sin \phi_2} \]

   9. Applied rewrites 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{-0.5}, 1\right)}{\sin \phi_2} \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

       4. lower-*.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

       7. lower-sin.f64 45.2%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   12. Applied rewrites 45.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 64.7% accurate, 1.1× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (-
   (* 1.0 (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((1.0 * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}

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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((1.0d0 * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((1.0 * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((1.0 * math.sin(phi2)) - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(1.0 * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((1.0 * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi1 around 0

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

4. Applied rewrites 64.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{1} \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
5. Add Preprocessing

Alternative 16: 64.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 420:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (atan2 (* t_0 (cos phi2)) (sin phi2))))
   (if (<= phi2 -0.5)
     t_1
     (if (<= phi2 420.0)
       (atan2
        (* t_0 (fma (* phi2 phi2) -0.5 1.0))
        (- (* phi2 (cos phi1)) (* (cos (- lambda1 lambda2)) (sin phi1))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((t_0 * cos(phi2)), sin(phi2));
	double tmp;
	if (phi2 <= -0.5) {
		tmp = t_1;
	} else if (phi2 <= 420.0) {
		tmp = atan2((t_0 * fma((phi2 * phi2), -0.5, 1.0)), ((phi2 * cos(phi1)) - (cos((lambda1 - lambda2)) * sin(phi1))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(t_0 * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (phi2 <= -0.5)
		tmp = t_1;
	elseif (phi2 <= 420.0)
		tmp = atan(Float64(t_0 * fma(Float64(phi2 * phi2), -0.5, 1.0)), Float64(Float64(phi2 * cos(phi1)) - Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.5], t$95$1, If[LessEqual[phi2, 420.0], N[ArcTan[N[(t$95$0 * N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.5 or 420 < phi2

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -0.5 < phi2 < 420

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{\sin \phi_2} \]

      3. lower-pow.f64 29.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{\sin \phi_2} \]

   7. Applied rewrites 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{\sin \phi_2} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\sin \phi_2} \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right)}{\sin \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      5. lift-pow.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      7. lower-*.f32 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      8. lower-unsound-*.f32 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{-1}{2}}, 1\right)}{\sin \phi_2} \]

      10. lower-unsound-*.f64 29.1%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\sin \phi_2} \]

   9. Applied rewrites 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{-0.5}, 1\right)}{\sin \phi_2} \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

       4. lower-*.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

       7. lower-sin.f64 45.2%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   12. Applied rewrites 45.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 63.7% accurate, 1.2× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (- (* (cos phi1) (sin phi2)) (* (cos (- lambda1 lambda2)) (sin phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos((lambda1 - lambda2)) * sin(phi1))));
}

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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos((lambda1 - lambda2)) * sin(phi1))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - (Math.cos((lambda1 - lambda2)) * Math.sin(phi1))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - (math.cos((lambda1 - lambda2)) * math.sin(phi1))))

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos((lambda1 - lambda2)) * sin(phi1))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

   4. lower-sin.f64 64.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1} \]

4. Applied rewrites 64.7%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
5. Add Preprocessing

Alternative 18: 62.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t\_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -6.4 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 3.4:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (atan2 (* t_0 (cos phi2)) (sin phi2))))
   (if (<= phi2 -6.4e-17)
     t_1
     (if (<= phi2 3.4)
       (atan2
        (* t_0 (fma (* phi2 phi2) -0.5 1.0))
        (* -1.0 (* (cos (- lambda1 lambda2)) (sin phi1))))
       t_1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = atan2((t_0 * cos(phi2)), sin(phi2));
	double tmp;
	if (phi2 <= -6.4e-17) {
		tmp = t_1;
	} else if (phi2 <= 3.4) {
		tmp = atan2((t_0 * fma((phi2 * phi2), -0.5, 1.0)), (-1.0 * (cos((lambda1 - lambda2)) * sin(phi1))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = atan(Float64(t_0 * cos(phi2)), sin(phi2))
	tmp = 0.0
	if (phi2 <= -6.4e-17)
		tmp = t_1;
	elseif (phi2 <= 3.4)
		tmp = atan(Float64(t_0 * fma(Float64(phi2 * phi2), -0.5, 1.0)), Float64(-1.0 * Float64(cos(Float64(lambda1 - lambda2)) * sin(phi1))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -6.4e-17], t$95$1, If[LessEqual[phi2, 3.4], N[ArcTan[N[(t$95$0 * N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -6.4000000000000005e-17 or 3.3999999999999999 < phi2

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   if -6.4000000000000005e-17 < phi2 < 3.3999999999999999

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{\sin \phi_2} \]

      3. lower-pow.f64 29.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{\sin \phi_2} \]

   7. Applied rewrites 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{\sin \phi_2} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\sin \phi_2} \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right)}{\sin \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      5. lift-pow.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      7. lower-*.f32 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      8. lower-unsound-*.f32 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{-1}{2}}, 1\right)}{\sin \phi_2} \]

      10. lower-unsound-*.f64 29.1%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\sin \phi_2} \]

   9. Applied rewrites 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{-0.5}, 1\right)}{\sin \phi_2} \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

       5. lower-sin.f64 43.4%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} \]

   12. Applied rewrites 43.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 47.8% accurate, 2.2× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (sin phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), sin(phi2));
}

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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), sin(phi2))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), Math.sin(phi2));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), math.sin(phi2))

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), sin(phi2));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi1 around 0

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

   1. lower-sin.f64 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

4. Applied rewrites 47.8%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]
5. Add Preprocessing

Alternative 20: 36.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-77}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -9.5e-77)
   (atan2 (* (sin (- lambda2)) (cos phi2)) (sin phi2))
   (atan2
    (* (sin (- lambda1 lambda2)) (cos phi2))
    (* phi2 (+ 1.0 (* -0.16666666666666666 (pow phi2 2.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -9.5e-77) {
		tmp = atan2((sin(-lambda2) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (phi2 * (1.0 + (-0.16666666666666666 * pow(phi2, 2.0)))));
	}
	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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 <= (-9.5d-77)) then
        tmp = atan2((sin(-lambda2) * cos(phi2)), sin(phi2))
    else
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (phi2 * (1.0d0 + ((-0.16666666666666666d0) * (phi2 ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -9.5e-77) {
		tmp = Math.atan2((Math.sin(-lambda2) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (phi2 * (1.0 + (-0.16666666666666666 * Math.pow(phi2, 2.0)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -9.5e-77:
		tmp = math.atan2((math.sin(-lambda2) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (phi2 * (1.0 + (-0.16666666666666666 * math.pow(phi2, 2.0)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -9.5e-77)
		tmp = atan(Float64(sin(Float64(-lambda2)) * cos(phi2)), sin(phi2));
	else
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(phi2 * Float64(1.0 + Float64(-0.16666666666666666 * (phi2 ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -9.5e-77)
		tmp = atan2((sin(-lambda2) * cos(phi2)), sin(phi2));
	else
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (phi2 * (1.0 + (-0.16666666666666666 * (phi2 ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -9.5e-77], N[ArcTan[N[(N[Sin[(-lambda2)], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(phi2 * N[(1.0 + N[(-0.16666666666666666 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -9.5000000000000005e-77

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda1 around 0

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

      1. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. lower-neg.f64 31.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 31.9%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(-\lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   if -9.5000000000000005e-77 < phi2

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)} \]

   7. Applied rewrites 31.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 34.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -11500:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(0.16666666666666666 \cdot {\lambda_2}^{2} - 1\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -11500.0)
   (atan2
    (*
     (* lambda2 (- (* 0.16666666666666666 (pow lambda2 2.0)) 1.0))
     (cos phi2))
    (sin phi2))
   (atan2
    (* (sin (- lambda1 lambda2)) (cos phi2))
    (* phi2 (+ 1.0 (* -0.16666666666666666 (pow phi2 2.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -11500.0) {
		tmp = atan2(((lambda2 * ((0.16666666666666666 * pow(lambda2, 2.0)) - 1.0)) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (phi2 * (1.0 + (-0.16666666666666666 * pow(phi2, 2.0)))));
	}
	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(lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 <= (-11500.0d0)) then
        tmp = atan2(((lambda2 * ((0.16666666666666666d0 * (lambda2 ** 2.0d0)) - 1.0d0)) * cos(phi2)), sin(phi2))
    else
        tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (phi2 * (1.0d0 + ((-0.16666666666666666d0) * (phi2 ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -11500.0) {
		tmp = Math.atan2(((lambda2 * ((0.16666666666666666 * Math.pow(lambda2, 2.0)) - 1.0)) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), (phi2 * (1.0 + (-0.16666666666666666 * Math.pow(phi2, 2.0)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -11500.0:
		tmp = math.atan2(((lambda2 * ((0.16666666666666666 * math.pow(lambda2, 2.0)) - 1.0)) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), (phi2 * (1.0 + (-0.16666666666666666 * math.pow(phi2, 2.0)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -11500.0)
		tmp = atan(Float64(Float64(lambda2 * Float64(Float64(0.16666666666666666 * (lambda2 ^ 2.0)) - 1.0)) * cos(phi2)), sin(phi2));
	else
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(phi2 * Float64(1.0 + Float64(-0.16666666666666666 * (phi2 ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -11500.0)
		tmp = atan2(((lambda2 * ((0.16666666666666666 * (lambda2 ^ 2.0)) - 1.0)) * cos(phi2)), sin(phi2));
	else
		tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), (phi2 * (1.0 + (-0.16666666666666666 * (phi2 ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -11500.0], N[ArcTan[N[(N[(lambda2 * N[(N[(0.16666666666666666 * N[Power[lambda2, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(phi2 * N[(1.0 + N[(-0.16666666666666666 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -11500

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \color{blue}{\cos \lambda_1}, \lambda_2 \cdot \left(\frac{-1}{2} \cdot \sin \lambda_1 + \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \left(\frac{-1}{2} \cdot \sin \lambda_1 + \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \left(\frac{-1}{2} \cdot \sin \lambda_1 + \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \lambda_1, \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      8. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \lambda_1, \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \lambda_1, \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \lambda_1, \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      11. lower-cos.f64 38.6%

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(-0.5, \sin \lambda_1, 0.16666666666666666 \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 38.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(-0.5, \sin \lambda_1, 0.16666666666666666 \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \color{blue}{\left(\frac{1}{6} \cdot {\lambda_2}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(\frac{1}{6} \cdot {\lambda_2}^{2} - \color{blue}{1}\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(\frac{1}{6} \cdot {\lambda_2}^{2} - 1\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(\frac{1}{6} \cdot {\lambda_2}^{2} - 1\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lower-pow.f64 22.8%

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(0.16666666666666666 \cdot {\lambda_2}^{2} - 1\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 22.8%

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \color{blue}{\left(0.16666666666666666 \cdot {\lambda_2}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   if -11500 < phi2

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)} \]

   7. Applied rewrites 31.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_2 \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 31.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -14.5:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(0.16666666666666666 \cdot {\lambda_2}^{2} - 1\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)}\\ \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -14.5)
   (atan2
    (*
     (* lambda2 (- (* 0.16666666666666666 (pow lambda2 2.0)) 1.0))
     (cos phi2))
    (sin phi2))
   (atan2
    (* (sin (- lambda1 lambda2)) (fma (* phi2 phi2) -0.5 1.0))
    (* phi2 (+ 1.0 (* -0.16666666666666666 (pow phi2 2.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -14.5) {
		tmp = atan2(((lambda2 * ((0.16666666666666666 * pow(lambda2, 2.0)) - 1.0)) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * fma((phi2 * phi2), -0.5, 1.0)), (phi2 * (1.0 + (-0.16666666666666666 * pow(phi2, 2.0)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -14.5)
		tmp = atan(Float64(Float64(lambda2 * Float64(Float64(0.16666666666666666 * (lambda2 ^ 2.0)) - 1.0)) * cos(phi2)), sin(phi2));
	else
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * fma(Float64(phi2 * phi2), -0.5, 1.0)), Float64(phi2 * Float64(1.0 + Float64(-0.16666666666666666 * (phi2 ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -14.5], N[ArcTan[N[(N[(lambda2 * N[(N[(0.16666666666666666 * N[Power[lambda2, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(phi2 * N[(1.0 + N[(-0.16666666666666666 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -14.5

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in lambda2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \color{blue}{\cos \lambda_1}, \lambda_2 \cdot \left(\frac{-1}{2} \cdot \sin \lambda_1 + \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      5. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \left(\frac{-1}{2} \cdot \sin \lambda_1 + \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \left(\frac{-1}{2} \cdot \sin \lambda_1 + \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      7. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \lambda_1, \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      8. lower-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \lambda_1, \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \lambda_1, \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \lambda_1, \frac{1}{6} \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      11. lower-cos.f64 38.6%

          \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(-0.5, \sin \lambda_1, 0.16666666666666666 \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   7. Applied rewrites 38.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 + \lambda_2 \cdot \mathsf{fma}\left(-1, \cos \lambda_1, \lambda_2 \cdot \mathsf{fma}\left(-0.5, \sin \lambda_1, 0.16666666666666666 \cdot \left(\lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right)} \cdot \cos \phi_2}{\sin \phi_2} \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \color{blue}{\left(\frac{1}{6} \cdot {\lambda_2}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(\frac{1}{6} \cdot {\lambda_2}^{2} - \color{blue}{1}\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(\frac{1}{6} \cdot {\lambda_2}^{2} - 1\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(\frac{1}{6} \cdot {\lambda_2}^{2} - 1\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

      4. lower-pow.f64 22.8%

         \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \left(0.16666666666666666 \cdot {\lambda_2}^{2} - 1\right)\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   10. Applied rewrites 22.8%

       \[\leadsto \tan^{-1}_* \frac{\left(\lambda_2 \cdot \color{blue}{\left(0.16666666666666666 \cdot {\lambda_2}^{2} - 1\right)}\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   if -14.5 < phi2

   1. Initial program 78.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. Taylor expanded in phi1 around 0

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

      1. lower-sin.f64 47.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

   4. Applied rewrites 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

   5. Taylor expanded in phi2 around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{\sin \phi_2} \]

      3. lower-pow.f64 29.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{\sin \phi_2} \]

   7. Applied rewrites 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{\sin \phi_2} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\sin \phi_2} \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right)}{\sin \phi_2} \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      5. lift-pow.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      6. unpow2 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      7. lower-*.f32 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      8. lower-unsound-*.f32 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{-1}{2}}, 1\right)}{\sin \phi_2} \]

      10. lower-unsound-*.f64 29.1%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\sin \phi_2} \]

   9. Applied rewrites 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{-0.5}, 1\right)}{\sin \phi_2} \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {\phi_2}^{2}}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{\phi_2}^{2}}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + \frac{-1}{6} \cdot {\phi_2}^{\color{blue}{2}}\right)} \]

       4. lower-pow.f64 28.8%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)} \]

   12. Applied rewrites 28.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 29.1% accurate, 2.4× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left(0.008333333333333333 \cdot {\phi_2}^{2} - 0.16666666666666666\right)\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (fma (* phi2 phi2) -0.5 1.0))
  (*
   phi2
   (+
    1.0
    (*
     (pow phi2 2.0)
     (- (* 0.008333333333333333 (pow phi2 2.0)) 0.16666666666666666))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * fma((phi2 * phi2), -0.5, 1.0)), (phi2 * (1.0 + (pow(phi2, 2.0) * ((0.008333333333333333 * pow(phi2, 2.0)) - 0.16666666666666666)))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * fma(Float64(phi2 * phi2), -0.5, 1.0)), Float64(phi2 * Float64(1.0 + Float64((phi2 ^ 2.0) * Float64(Float64(0.008333333333333333 * (phi2 ^ 2.0)) - 0.16666666666666666)))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(phi2 * N[(1.0 + N[(N[Power[phi2, 2.0], $MachinePrecision] * N[(N[(0.008333333333333333 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi1 around 0

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

   1. lower-sin.f64 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

4. Applied rewrites 47.8%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

5. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{\sin \phi_2} \]

   3. lower-pow.f64 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{\sin \phi_2} \]

7. Applied rewrites 29.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{\sin \phi_2} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

   2. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\sin \phi_2} \]

   3. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right)}{\sin \phi_2} \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   5. lift-pow.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   6. unpow2 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   7. lower-*.f32 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   8. lower-unsound-*.f32 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   9. lower-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{-1}{2}}, 1\right)}{\sin \phi_2} \]

   10. lower-unsound-*.f64 29.1%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\sin \phi_2} \]

9. Applied rewrites 29.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{-0.5}, 1\right)}{\sin \phi_2} \]

10. Taylor expanded in phi2 around 0

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)}} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + \color{blue}{{\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)}\right)} \]

    2. lower-+.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)}\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \color{blue}{\frac{1}{6}}\right)\right)} \]

    4. lower-pow.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)} \]

    5. lower--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)} \]

    6. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left(\frac{1}{120} \cdot {\phi_2}^{2} - \frac{1}{6}\right)\right)} \]

    7. lower-pow.f64 28.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \left(1 + {\phi_2}^{2} \cdot \left(0.008333333333333333 \cdot {\phi_2}^{2} - 0.16666666666666666\right)\right)} \]

12. Applied rewrites 28.8%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(0.008333333333333333 \cdot {\phi_2}^{2} - 0.16666666666666666\right)\right)}} \]
13. Add Preprocessing

Alternative 24: 28.8% accurate, 2.8× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\sin \phi_2} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (* (sin (- lambda1 lambda2)) (fma (* phi2 phi2) -0.5 1.0)) (sin phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * fma((phi2 * phi2), -0.5, 1.0)), sin(phi2));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * fma(Float64(phi2 * phi2), -0.5, 1.0)), sin(phi2))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi1 around 0

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

   1. lower-sin.f64 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

4. Applied rewrites 47.8%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

5. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{\sin \phi_2} \]

   3. lower-pow.f64 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{\sin \phi_2} \]

7. Applied rewrites 29.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{\sin \phi_2} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

   2. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\sin \phi_2} \]

   3. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right)}{\sin \phi_2} \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   5. lift-pow.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   6. unpow2 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   7. lower-*.f32 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   8. lower-unsound-*.f32 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   9. lower-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{-1}{2}}, 1\right)}{\sin \phi_2} \]

   10. lower-unsound-*.f64 29.1%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\sin \phi_2} \]

9. Applied rewrites 29.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{-0.5}, 1\right)}{\sin \phi_2} \]
10. Add Preprocessing

Alternative 25: 28.8% accurate, 3.0× speedup

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (fma (* phi2 phi2) -0.5 1.0))
  (* phi2 (+ 1.0 (* -0.16666666666666666 (pow phi2 2.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * fma((phi2 * phi2), -0.5, 1.0)), (phi2 * (1.0 + (-0.16666666666666666 * pow(phi2, 2.0)))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * fma(Float64(phi2 * phi2), -0.5, 1.0)), Float64(phi2 * Float64(1.0 + Float64(-0.16666666666666666 * (phi2 ^ 2.0)))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[(phi2 * phi2), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(phi2 * N[(1.0 + N[(-0.16666666666666666 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Taylor expanded in phi1 around 0

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

   1. lower-sin.f64 47.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2} \]

4. Applied rewrites 47.8%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2}} \]

5. Taylor expanded in phi2 around 0

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

   1. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{\phi_2}^{2}}\right)}{\sin \phi_2} \]

   3. lower-pow.f64 29.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot {\phi_2}^{\color{blue}{2}}\right)}{\sin \phi_2} \]

7. Applied rewrites 29.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {\phi_2}^{2}\right)}}{\sin \phi_2} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {\phi_2}^{2}}\right)}{\sin \phi_2} \]

   2. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + \color{blue}{1}\right)}{\sin \phi_2} \]

   3. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{2} \cdot {\phi_2}^{2} + 1\right)}{\sin \phi_2} \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   5. lift-pow.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left({\phi_2}^{2} \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   6. unpow2 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   7. lower-*.f32 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   8. lower-unsound-*.f32 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot \frac{-1}{2} + 1\right)}{\sin \phi_2} \]

   9. lower-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{\frac{-1}{2}}, 1\right)}{\sin \phi_2} \]

   10. lower-unsound-*.f64 29.1%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\sin \phi_2} \]

9. Applied rewrites 29.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \color{blue}{-0.5}, 1\right)}{\sin \phi_2} \]

10. Taylor expanded in phi2 around 0

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\phi_2}^{2}\right)}} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {\phi_2}^{2}}\right)} \]

    2. lower-+.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{\phi_2}^{2}}\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, \frac{-1}{2}, 1\right)}{\phi_2 \cdot \left(1 + \frac{-1}{6} \cdot {\phi_2}^{\color{blue}{2}}\right)} \]

    4. lower-pow.f64 28.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)} \]

12. Applied rewrites 28.8%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_2 \cdot \phi_2, -0.5, 1\right)}{\phi_2 \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {\phi_2}^{2}\right)}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025205 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))